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\begin{document}

\begin{titlepage}

\title{\textit{\large Re\v{s}er\v{s}n\'i pr\'ace }\large }

\maketitle

\part*{Spekt\'aln\'\i{} vlastnosti oper\'atoru kvazienergie}


\subsection*{Ond\v{r}ej Lev}


\paragraph*{\textmd{\textit{\v{S}kolitel: doc.Ing. Pavel \v{S}\q{t}ov\'\i{}\v{c}ek Dr.Sc. }}}


\paragraph*{\textmd{\textit{Konzultanti: prof. Pierre Duclos, dr. Michel Vittot, Universit\'e
de Toulon et du Var}}}

\thispagestyle{empty}
%\newpage

\end{titlepage}

\section{\'Uvod}

Vy\v{s}et\v{r}ov\'an\'\i{} spektra oper\'atoru a jeho vlastnost\'\i{} je jedna z mnoha matematicky
zaj\'\i{}mav\'ych \'uloh, kter\'e matematick\'e fyzice p\v{r}inesla kvantov\'a mechanika.
Jednou ze t\v{r}\'\i{}d kvantov\v{e} mechanick\'ych syst\'em\r{u} jsou syst\'emy s periodickou
\v{c}asovou z\'avislost\'\i{}. P\v{r}i zkoum\'an\'\i{} vlastnostnost\'\i{} t\v{e}chto syst\'em\r{u},
(jako je nap\v{r}\'\i{}klad jeho dynamick\'a stabilita) je d\r{u}le\v{z}it\'a znalost
charakteru spektra takzvan\'eho Floquetova Hamiltoni\'anu, nebo t\'e\v{z} oper\'atoru
kvazienergie. 

Tato re\v{s}er\v{s}n\'\i{} pr\'ace se zab\'yv\'a vlastnostmi oper\'atoru speci\'aln\'\i{}ho typu
a sice 
\[
K_{F}\left( \omega \right) :=-i\partial _{t}+H+V(\omega t),\]
 kde \( \omega  \) je re\'aln\'e, \( H \) je oper\'ator na Hilbertov\v{e} prostoru
\( \HH  \), \v{c}asov\v{e} nez\'avisl\'y a p\v{r}edpokl\'ad\'ame, \v{z}e m\'a \v{c}ist\v{e}
bodov\'e spektrum \( \sigma (H)=\left\{ H_{n}|n\in \N \right\}  \). P\v{r}i\v{c}em\v{z}
ka\v{z}d\'a vlastn\'\i{} hodnota m\r{u}\v{z}e m\'\i{}t kone\v{c}n\'y stupe\v{n} degenerace.
Z\'akladn\'\i{}m p\v{r}edpokladem je, \v{z}e tyto vlastn\'\i{} hodnoty jsou od sebe dostate\v{c}n\v{e}
vzd\'aleny ve smyslu, kter\'y budeme pozd\v{e}ji precizovat. V(t) bude 2\( \pi  \)
periodick\'y potenci\'al dostate\v{c}n\v{e} mnohokr\'at siln\v{e} diferencovateln\'y.
Budeme jej vy\v{s}et\v{r}ovat jako poruchu.

Po p\v{r}e\v{s}k\'alov\'an\'\i{} \v{c}asu bude m\'\i{}t neporu\v{s}en\'y oper\'ator \( K:=-i\omega \partial _{t}+H \),
p\r{u}sob\'\i{}c\'\i{} na \( L^{2}\left( 0,2\pi \right) \otimes \HH  \), spektrum \( \omega \Z +H(\N ) \),
kter\'e je pro \( H(\N ) \) neomezenou mno\v{z}inu hust\'e v \( \R  \) pro skoro
v\v{s}echna \( \omega  \) pevn\'a a v\v{e}t\v{s}\'\i{} ne\v{z} nula. Tak\v{z}e
nelze pou\v{z}\'\i{}t standartn\'\i{} poruchovou teorii.

Jednou z m\'ala zn\'am\'ych metod pro d\r{u}kaz stability spektra (to je zachov\'an\'\i{}
vlastnosti \v{c}ist\'e bodovosti) v\r{u}\v{c}i \v{c}asov\v{e} z\'avisl\'e poru\v{s}e
je kvantov\'y KAM algoritmus o jeho\v{z} rozvoj se z\'asadn\'\i{}m zp\r{u}sobem zaslou\v{z}ily
francouz\v{s}t\'\i{} matematici J. Bellissard a M.Combescure \cite{2}. P. Duclos
a P. \v{S}\q{t}ov\'\i{}\v{c}ek se touto problematikou zab\'yvali ji\v{z} ve \v{c}l\'anku
\cite{1}.

Snahou t\'eto pr\'ace je zpracovat l\'atku z\'\i{}skanou jednak z cyklu t\v{r}\'\i{} p\v{r}edn\'a\v{s}ek
M. Vittota konan\'ych na kated\v{r}e matematiky FJFI, jednak ze setk\'an\'\i{} s P.
Duclosem p\v{r}i jeho kv\v{e}tnov\'e n\'av\v{s}t\v{e}v\v{e}, ale hlavn\v{e}
z konzultac\'\i{} u doc.Ing. Pavla \v{S}\q{t}ov\'\i{}\v{c}ka, DrSc., vedouc\'\i{}ho t\'eto
pr\'ace. Mezi odli\v{s}nosti proti \cite{1} pat\v{r}\'\i{} absence adiabatick\'e \v{c}\'asti
a p\v{r}ipu\v{s}t\v{e}n\'\i{} p\v{r}\'\i{}padu kone\v{c}n\'e degenerace vlastn\'\i{}ch hodnot
\( H \). Hlavn\'\i{}m rozd\'\i{}lem je vybudov\'an\'\i{} abstraktn\'\i{}ho apar\'atu, kter\'y se mo\v{z}n\'a
zd\'a b\'yt pon\v{e}kud slo\v{z}it\'y a zdlouhav\'y, ale probl\'em t\'\i{}m z\'\i{}sk\'a v\v{e}t\v{s}\'\i{}
matematickou p\v{r}ehlednost a da\v{r}\'\i{} se jej pom\v{e}rn\v{e} dlouho dr\v{z}et
v obecn\'e rovin\v{e}.

Text pr\'ace je roz\v{c}len\v{e}n do \v{s}esti hlavn\'\i{}ch \v{c}ast\'\i{}. V druh\'e
je vybudov\'ana induktivn\'\i{} limita a zvl\'a\v{s}tn\'\i{} prostor pro oper\'ator K, kter\'y
je obecn\v{e} neomezen\'y a je proto nutn\'e jej vy\v{s}et\v{r}ovat zvl\'a\v{s}\q{t}.
T\v{e}\v{z}i\v{s}t\v{e}m algebraick\'e \v{c}\'asti je proveden\'\i{} limity, z kter\'e
potom plyne i konvergence cel\'eho algoritmu v prostoru omezen\'ych oper\'ator\r{u}
na \( \KK . \) 

N\'asleduje t\v{r}et\'\i{} \v{c}\'ast, ve kter\'e se indukc\'\i{} konstruuj\'\i{} vlastn\'\i{} posloupnosti
algoritmu a provede se limitn\'\i{} procedura pro matice. V z\'av\v{e}ru t\'eto \v{c}\'asti
je vyslovena jedna ze st\v{e}\v{z}ejn\'\i{}m v\v{e}t tohoto algoritmu, sice lemma
7. Jeho d\r{u}kaz je nazna\v{c}en v prvn\'\i{}m dodatku.

Ve \v{c}tvrt\'e \v{c}\'asti se ji\v{z} op\v{e}t probl\'em vrac\'\i{} zp\v{e}t do prostoru
\( \KK . \) T\v{e}\v{z}i\v{s}t\v{e}m t\'eto \v{c}\'asti je lemma 8 jeho\v{z}
d\r{u}kazu je v\v{e}nov\'an druh\'y dodatek.

V p\'ate \v{c}\'asti je vyslovena hlavn\'\i{} v\v{e}ta. Jestli\v{z}e spektrum H vyhovuje
podm\'\i{}nk\'am (\ref{1.gap}), (\ref{2.gap}), V(t) je dostate\v{c}n\v{e} hladk\'y
a \( \omega  \) nen\'\i{} rezonantn\'\i{} ferkvence, tak oper\'ator \( K_{F}\left( \omega \right)  \)
m\'a \v{c}ist\v{e} bodov\'e spektrum.

Z\'av\v{e}rem bych r\'ad pod\v{e}koval sv\'emu \v{s}koliteli a vedouc\'\i{}mu pr\'ace,
doc. Ing. Pavlovi \v{S}\q{t}ov\'\i{}\v{c}kovi, DrSc., za trp\v{e}livost p\v{r}i
veden\'\i{} pr\'ace a pomoc p\v{r}i konzultac\'\i{}ch.

Tak\'e bych cht\v{e}l pod\v{e}kovat konzultant\r{u}m prof. Pierre Duclosovi
a dr. Michel Vittotovi z Universit\'e de Toulon et du Var, za cenn\'e rady p\v{r}i
jejich n\'av\v{s}t\v{e}v\'ach na kated\v{r}e matematiky FJFI.

\( \:  \)

\( \quad \quad  \)V Praze dne 1.6. 2000

\( \quad \quad \quad \quad  \) Ond\v{r}ej Lev


\section{abstraktn\'\i{} \v{c}\'ast}


\subsection{Induktivn\'\i{} limita}

Nech\q{t} (\( \gX _{s})^{\infty }_{s=0} \) je usm\v{e}rn\v{e}n\'a posloupnost
line\'arn\'\i{}ch prostor\r{u} 

s normami \( \left\Vert \: \right\Vert _{s} \) a nech\q{t} existuje omezen\'e
line\'arn\'\i{} zobrazen\'\i{}
\[
\iota _{us}:\gX _{s}\rightarrow :\gX _{u}\; \; \textrm{pro}\; s\leq u,\; \; a\; \left| \iota _{us}\right| \leq 1\]
 (\( \iota  \)\( _{ss} \) je identick\'e zobrazen\'\i{}).

P\v{r}edpokl\'adejme, \v{z}e plat\'\i{} n\'asleduj\'\i{}c\'\i{} \v{r}et\v{e}zov\'e pravidlo:
\[
\forall s\leq u\leq v\; \; \; \iota _{vu}\iota _{us}=\iota _{vs}\]
 Ozna\v{c}me
\[
\iota _{s}:=\iota _{s+1,s}\]


Pomoc\'\i{} n\'asleduj\'\i{}c\'\i{}ch pomocn\'ych pojm\r{u} vybudujeme normovanou induktivn\'\i{} limitu
prostor\r{u} \( \gX  \)\( _{s} \) 
\[
\widetilde{\gX }:=\sideset {}{^{\oplus }}\sum _{s=0}^{\infty }\gX _{s}=\left\{ \sum _{j=1}^{n}x_{j}/\; n\in \N ,\; x_{j}\in \gX _{j}\right\} \]
 definujme relaci ekvivalence \( \sim  \) na \( \:  \)\( \widetilde{\gX } \)

\[
\sum _{i\in I}x_{i}\sim \sum _{j\in J}y_{j}\Leftrightarrow \exists v\geq i,j,\: \forall i\in I,j\in J\; \; \; \sum _{i\in I}\iota _{vi}(x_{i})=\sum _{j\in J}\iota _{vj}(y_{j})\]
 \( \widetilde{\widetilde{\gX }}:= \) \( \widetilde{\gX } \)/\( _{\sim } \)
\( \: \:  \)Ozna\v{c}me \( \left\lceil x\right\rceil  \)\( \in \widetilde{\widetilde{\gX }} \)
t\v{r}\'\i{}du podle ekvivalence \( \sim  \) obsahuj\'\i{}c\'\i{} prvek \( x\in \widetilde{\gX } \). 

Lze snadno nahl\'ednout, \v{z}e plat\'\i{} \( \left\lceil x_{s}\right\rceil  \) =
\( \left\{ \iota _{us}(x_{s})/u\geq s\right\} . \) 

Definujme obvykl\'e s\v{c}\'\i{}t\'an\'\i{} t\v{r}\'\i{}d a n\'asoben\'\i{} skal\'arem jako 
\[
\alpha \left\lceil x\right\rceil +\left\lceil y\right\rceil :=\left\lceil \alpha x+y\right\rceil \; \; pro\; \alpha \in \C ;\; \; x\in \gX _{s}\; ,\, y\in \gX _{u};\; u,s\in \N \]
 proto \( \forall n\in \N \; \; \forall x_{j}\in \gX _{j} \)
\[
\left\lceil \sum _{j=1}^{n}x_{j}\right\rceil =\sum _{j=1}^{n}\left\lceil x_{j}\right\rceil =\sum _{j=1}^{n}\left\lceil \iota _{nj}\left( x_{j}\right) \right\rceil =\left\lceil \sum _{j=1}^{n}\iota _{nj}\left( x_{j}\right) \right\rceil \]
 z \v{c}eho\v{z} plyne, \v{z}e
\[
\widetilde{\widetilde{\gX }}=\left\{ \left\lceil x_{s}\right\rceil /s\in \N \, \; x_{s}\in \gX _{s}\right\} .\]
 D\'ale definujme seminormu p na \( \widetilde{\widetilde{\gX }} \) 
\[
p\left( \left\lceil x_{s}\right\rceil \right) :=\limsup _{u\geq s}\left\Vert \iota _{us}(x_{s})\right\Vert _{u}\]
z\v{r}ejm\v{e} p\( \left( \left\lceil x_{s}\right\rceil \right)  \)\( \leq  \)
\( \left\Vert x_{s}\right\Vert _{s} \) 

(pozn\'amka: p je definov\'ano korektn\v{e} a spl\v{n}uje vlastnosti seminormy
.)

Definujme
\[
M:=\left\{ \left\lceil x\right\rceil \in \widetilde{\widetilde{\gX }}/\; p(\left\lceil x\right\rceil )=0\right\} \]
 M je nulov\'y podprostor seminormy p. 

Polo\v{z}me je\v{s}t\v{e} 
\[
\gX :=\widetilde{\widetilde{\gX }}/_{M}\; =\left\{ \left\lceil \left\lceil x_{s}\right\rceil \right\rceil /x_{s}\in \gX _{s}\right\} \]
 
\[
\left\Vert \, \left\lceil \left\lceil x_{s}\right\rceil \right\rceil \, \right\Vert _{\gX }:=p\left( \left\lceil x_{s}\right\rceil \right) =\limsup _{u\geq s}\left\Vert \iota _{us}(x_{s})\right\Vert _{u}\]
 

pozn: \( \left\lceil \left\lceil \right\rceil \right\rceil  \) znamen\'a druhou
t\v{r}\'\i{}du podle ekvivalence. Lze ov\v{e}\v{r}it, \v{z}e na tomto prostoru
je p norma, nez\'avis\'\i{} na v\'yb\v{e}ru reprezentantu z obou t\v{r}\'\i{}d a je t\'e\v{z}
omazen\'a \( \left\Vert \: \right\Vert  \)\( _{s} \) . 

Z\'\i{}skan\'y normovan\'y prostor 
\[
\left( \gX ,\: \left\Vert \: \right\Vert _{\gX }\right) =lim\, norm_{\rightarrow }\left\{ \gX _{s},\: \iota _{us}\right\} \]
 nazveme normovanou algebraickou limitou na\v{s}\'\i{} usm\v{e}rn\v{e}n\'e posloupnosti
Banachov\'ych prostor\r{u}.

Definujme omezen\'e zobrazen\'\i{} spojuj\'\i{}c\'\i{} jednotliv\'e prostory z posloupnosti s induktivn\'\i{}
limitou 
\[
\iota _{\infty s}:\gX _{s}\rightarrow \gX \quad \iota _{\infty s}(x_{s}):=\left\lceil \left\lceil x_{s}\right\rceil \right\rceil \; \; \left| \iota _{\infty s}\right| \leq 1\]
a plat\'\i{}
\[
\forall u\geq s\; \: \: \iota _{\infty u}\iota _{us}=\iota _{\infty s}\]
 \( \,  \) 

Bu\q{d} \( \left\{ A_{s}\in \BB (\gX _{s})\right\} _{s=s_{0}}^{\infty } \)
mno\v{z}ina omezen\'ych oper\'ator\r{u} na \( \gX _{s} \) definovan\'ych pro \( s\geq  \)\( s_{0} \) 

spl\v{n}uj\'\i{}c\'\i{}ch 
\[
A_{s}\iota _{su}=\iota _{su}A_{u}\; \; \textrm{pro}\; s_{0}\leq s\leq u,\; \sup _{s\geq s_{0}}\left| A_{s}\right| <\infty \]
 pak \( \exists _{1} \) A\( _{\infty } \)\( \in \BB (\gX ) \) \( \:  \)
s vlastnost\'\i{} A\( _{\infty } \)\( \iota  \)\( _{\infty s} \) = \( \iota  \)\( _{\infty s} \)A\( _{s} \)
\( \: \forall  \) \( s\geq  \) \( s_{0} \) . 

Explicitn\v{e} lze vyj\'ad\v{r}it 
\[
A_{\infty }(\: \left\lceil \left\lceil x_{s}\right\rceil \right\rceil \: )=\iota _{\infty s}A_{s}\left( x_{s}\right) \]
Plat\'\i{}:
\begin{eqnarray*}
\left\Vert A_{\infty }(\: \left\lceil \left\lceil x_{s}\right\rceil \right\rceil \: )\right\Vert _{\gX } & = & \left\Vert \iota _{\infty s}\left( A_{s}x_{s}\right) \right\Vert _{\gX }=\left\Vert \, \left\lceil \left\lceil A_{s}x_{s}\right\rceil \right\rceil \, \right\Vert _{\gX }=\limsup _{u\geq s}\left\Vert \iota _{us}(A_{s}x_{s})\right\Vert _{u}\\
 & = & \limsup _{u\geq s}\left\Vert A_{u}\left( \iota _{us}x_{s}\right) \right\Vert _{u}\leq \sup _{v\geq s_{0}}\left| A_{v}\right| \, \limsup _{u\geq s}\left\Vert \iota _{us}x_{s}\right\Vert _{u}\\
 & = & \sup _{v\geq s_{0}}\left| A_{v}\right| \, \left\Vert \, \left\lceil \left\lceil x_{s}\right\rceil \right\rceil \, \right\Vert _{\gX }.
\end{eqnarray*}
Z\'\i{}skali jsme tedy odhad normy
\[
\left| A_{\infty }\right| \leq \sup _{u\geq s_{0}}\left| A_{u}\right| \]


Proto\v{z}e obecn\v{e} nen\'\i{} prostor \( \gX , \) Banach\r{u}v z\'upln\'\i{}me jej.
Existuje tedy Banach\r{u}v prostor \( \left( \gX _{\infty },\left\Vert \: \right\Vert \right)  \)
s vlastnostmi: \( \gX \subset \gX _{\infty },\; \forall x\in \gX \; \left\Vert x\right\Vert =\left\Vert x\right\Vert _{\gX },\; \gX \;  \)je
hust\'y v \( \left( \gX _{\infty },\left\Vert \: \right\Vert \right)  \) 

V dal\v{s}\'\i{}m budeme zobrazen\'\i{} \( \iota _{\infty s} \) ch\'apat jako zobrazen\'\i{}
z \( \gX _{s}\; do\; \gX _{\infty } \).

Prostor 
\[
\left( \gX _{\infty },\left\Vert \; \right\Vert \right) \]


nazveme z\'upln\v{e}nou induktivn\'\i{} limitou.

M\'ame-li d\'ano zobrazen\'\i{} A\( _{\infty } \)\( \in \BB (\gX ) \) , m\r{u}\v{z}eme
jej spojit\v{e} roz\v{s}\'\i{}\v{r}it na cel\'y prostor \( \gX _{\infty } \).

Tak\v{z}e shrneme-li: pro v\'y\v{s}e uvedenou posloupnost oper\'ator\r{u} \( A_{s} \)
existuje pr\'av\v{e} jeden oper\'ator s vlastnostmi:
\begin{equation}
\label{Vlastnosti_indukt.lim}
\widetilde{A_{\infty }}\in \BB \left( \gX _{\infty }\right) :\: \widetilde{A_{\infty }}\iota _{\infty s}=\iota _{\infty s}A_{s}\; \forall s\geq s_{0},\; \left| \widetilde{A_{\infty }}\right| \leq \sup _{u\geq s_{0}}\left| A_{u}\right| 
\end{equation}


Oper\'ator s t\v{e}mito vlastnostmi nazveme induktivn\'\i{} limitou posloupnosti oper\'ator\r{u}
\( A_{s} \) .


\subsection{Roz\v{s}\'\i{}\v{r}en\'\i{}}

Nech\q{t} pro ka\v{z}d\'e \( s\geq 0 \) je d\'ana mno\v{z}ina \( \Omega  \)\( _{s} \),
podmno\v{z}ina re\'aln\'e p\v{r}\'\i{}mky.\( \; \Omega _{s}\subset \R  \) . 

Tyto mno\v{z}iny a\q{t} jsou v n\'asleduj\'\i{}c\'\i{}m vztahu
\[
\Omega _{s+1}\subset \Omega _{s}\; \; \textrm{pro}\; s\geq 0\]
 Ozna\v{c}me
\[
\Omega _{\infty }:=\bigcap _{s=0}^{\infty }\: \Omega _{s}\]
a p\v{r}edpokl\'adejme, \v{z}e \( \Omega _{\infty } \) je nepr\'azdn\'a. 

Nech\q{t} K\( _{0} \) je n\v{e}jak\'a vektorov\'a funkce definovan\'a na \( \Omega  \)\( _{0} \).
To znamen\'a, \v{z}e je definov\'ano zobrazen\'\i{} do n\v{e}jak\'eho (zat\'\i{}m libovoln\'eho)
line\'arn\'\i{}ho prostoru \( \ZZ  \)
\[
K_{0}:\Omega _{0}\to \ZZ \; \; \; \omega \mapsto K_{0}\left( \omega \right) \in \ZZ \]
 Definujme
\[
K_{s}:=K_{0}/_{\Omega _{s}}\; tedy\; K_{s}:\Omega _{s}\to \ZZ \; a\; K_{s}\left( \omega \right) =K_{0}\left( \omega \right) \; \forall \omega \in \Omega _{s}.\]
 A stejn\v{e} i \( K_{\infty } \) pro \( s:=\infty  \).

Tyto vektorov\'e funkce se tedy li\v{s}\'\i{} pouze sv\'ym defini\v{c}n\'\i{}m oborem.

Definujme n\'asoben\'\i{} vektorov\'e funkce skal\'arem \( \alpha K_{s} \) pro \( \alpha  \)\( \in \C  \)

\[
\alpha K_{s}:\Omega _{s}\to \ZZ \; \; \; (\alpha K_{s})\, \left( \omega \right) :=\alpha \, K_{s}\left( \omega \right) \in \ZZ \]
 Ozna\v{c}me
\[
\C K_{s}:=\left\{ \alpha K_{s}/\alpha \in \C \right\} \]
 jednorozm\v{e}rn\'y komplexn\'\i{} line\'arn\'\i{} prostor s b\'az\'\i{} K\( _{s} \). 

Roz\v{s}i\v{r}me nyn\'\i{} prostory a dodefinujme zobrazen\'\i{} n\'asledovn\v{e} : 
\begin{eqnarray*}
\widetilde{\iota _{us}}:\C K_{s}\oplus \gX _{s}\to \C K_{u}\oplus \gX _{u} &  & \\
\widetilde{\iota _{us}}\left( \alpha K_{s}+x_{s}\right) :=\alpha K_{u}+\iota _{us}\left( x_{s}\right) \: \; \textrm{pro} & \alpha \in \C ,\: u\geq s\geq 0,\: x_{s}\in \gX _{s} & 
\end{eqnarray*}
 Z\v{r}ejm\v{e} bude znovu platit
\[
\widetilde{\iota _{vu}}\widetilde{\iota _{us}}=\widetilde{\iota _{vs}}\; \: \textrm{pro}\; s\leq u\leq v.\]
 Podobn\v{e} roz\v{s}\'\i{}\v{r}\'\i{}me i definice \( \iota  \)\( _{\infty s} \)
na \( \C K_{s}\oplus \gX _{s}\to \C K_{\infty }\oplus \gX _{\infty } \)
\[
\widetilde{\iota _{\infty u}}\left( K_{u}\right) :=K_{\infty }\]
 a v dal\v{s}\'\i{}m si vlnku odpust\'\i{}me.


\subsection{Konvergence v algebraick\'e \v{c}\'asti}

Nech\q{t} \( \DD  \)\( \in \BB (\gX _{\infty }) \) je induktivn\'\i{} limita oper\'ator\r{u} 

\( \DD  \)\( _{s} \) \( \in  \)\textbf{\( \BB \left( \gX _{s}\right)  \)}
s vlastnostmi: 
\begin{equation}
\label{normDsleq1}
|\DD _{s}|\leq 1,\textrm{ }|1-\DD _{s}|\leq 1,\quad \DD \iota _{\infty s}=\iota _{\infty s}\DD _{s}\quad \forall s.
\end{equation}


\begin{lem}
P\v{r}edpokl\'adejme, \v{z}e jsou d\'any posloupnosti \( \left\{ V_{s}\right\} _{s=o}^{\infty }, \)
\( \left\{ W_{s}\right\} _{s=0}^{\infty } \) \( V_{s},W_{s}\in \gX _{s} \)
a posloupnost zobrazen\'\i{} \( \left\{ \Theta _{u}^{s}\right\}  \)\( _{u=s}^{\infty } \)
definovan\'ych na \( \C K_{u+1}\oplus \gX _{u+1} \) s hodnotami v \( \gX _{u+1} \) 

(tedy i \( \Theta  \)\( _{u}^{s} \)\( \left( K_{u+1}\right)  \)\( \in \gX _{u+1} \)
pro \( s\leq u \) ).

Nav\'\i{}c \( \Theta _{u}^{s}/_{\gX _{u+1}} \) nech\q{t} je omezen\'e a 
\[
\Theta ^{s}_{v\; }\iota _{v+1,u+1}=\iota _{v+1,u+1\; }\Theta _{u}^{s}\quad \textrm{pro }s\leq u\leq v.\]


Polo\v{z}me \( V_{-1}=0 \) , \( W_{-1}=0 \), \( T_{-1}=1:\C K_{0}\oplus \gX _{0}\rightarrow \C K_{0}\oplus \gX _{0} \)
a
\begin{equation}
\label{Ts_as_prod}
T_{s}:=e^{\Theta ^{s}_{s}}e^{\Theta ^{s-1}_{s}}\dots e^{\Theta ^{0}_{s}}:\C K_{s+1}\oplus \gX _{s+1}\rightarrow \C K_{s+1}\oplus \gX _{s+1}\quad \textrm{pro }s\geq 0\, .
\end{equation}
 Kde e\( ^{\Theta _{u}^{s}} \) je na \( K_{u+1} \) definov\'ano
\[
e^{\Theta _{u}^{s}}\left( K_{u+1}\right) :=K_{u+1}+\sum ^{\infty }_{j=0}\frac{\left( \Theta _{u}^{s}\right) ^{j}}{\left( j+1\right) !}\left( \Theta _{u}^{s}\left( K_{u+1}\right) \right) \]


\( \,  \)

Nech\q{t} plat\'\i{} n\'asleduj\'\i{}c\'\i{} rekurz\'\i{}vn\'\i{} vztahy:


\begin{eqnarray}
W_{0} & = & V_{0},\nonumber \\
W_{s+1} & = & \iota _{s}(W_{s})+T_{s}(V_{s+1}-\iota _{s}(V_{s}))\label{recruleWs} \\
 &  & +\left( e^{\Theta ^{s}_{s}}-\frac{e^{\Theta ^{s}_{s}}-1}{\Theta ^{s}_{s}}\right) \; \iota _{s}\; (1-\DD _{s})(W_{s}-\iota _{s-1}(W_{s-1})),\nonumber \\
\Theta ^{s}_{s}(\: \iota _{s}\: (K_{s}+\DD _{s}(W_{s})\, )) & = & -\iota _{s}\, (1-\DD _{s})(W_{s}-\iota _{s-1}(W_{s-1})),\quad s=0,1,2,\dots .\label{defcondTheta} 
\end{eqnarray}
Potom plat\'\i{}
\begin{equation}
\label{T(K+V)}
T_{s-1}(K_{s}+V_{s})=K_{s}+\DD _{s}(W_{s})+(1-\DD _{s})(W_{s}-\iota _{s-1}(W_{s-1})),\quad s=0,1,2,\dots .
\end{equation}

\end{lem}
\textit{D\r{u}kaz}: Indukc\'\i{} podle \( s \). Pro \( s=0 \) je to z\v{r}ejm\'e.
Induk\v{c}n\'\i{} krok \( s\to s+1 \):
\begin{eqnarray*}
T_{s}(K_{s+1}+V_{s+1}) & = & T_{s}\iota _{s}(K_{s}+V_{s})+T_{s}(V_{s+1}-\iota _{s}(V_{s}))\\
 & = & e^{\Theta ^{s}_{s}}\, \iota _{s}\, T_{s-1}(K_{s}+V_{s})+T_{s}(V_{s+1}-\iota _{s}(V_{s}))\\
 & = & e^{\Theta ^{s}_{s}}\, \iota _{s}\, (K_{s}+\DD _{s}(W_{s})+(1-\DD _{s})(W_{s}-\iota _{s-1}(W_{s-1})))\\
 &  & +T_{s}(V_{s+1}-\iota _{s}(V_{s}))\\
 & = & \iota _{s}(K_{s}+\DD _{s}(W_{s}))+\frac{e^{\Theta ^{s}_{s}}-1}{\Theta ^{s}_{s}}\Theta ^{s}_{s}\: \iota _{s}\: (K_{s}+\DD _{s}(W_{s}))\\
 &  & +e^{\Theta ^{s}_{s}}\: \iota _{s}\: (1-\DD _{s})(W_{s}-\iota _{s-1}(W_{s-1}))+T_{s}(V_{s+1}-\iota _{s}(V_{s}))\\
 & = & K_{s+1}-(1-\DD _{s+1})\iota _{s}(W_{s})+\iota _{s}(W_{s})+T_{s}(V_{s+1}-\iota _{s}(V_{s}))\\
 &  & +\left( e^{\Theta ^{s}_{s}}-\frac{e^{\Theta ^{s}_{s}}-1}{\Theta ^{s}_{s}}\right) \: \iota _{s}\: (1-\DD _{s})(W_{s}-\iota _{s-1}(W_{s-1}))\\
 & = & K_{s+1}-(1-\DD _{s+1})\iota _{s}(W_{s})+W_{s+1}\\
 & = & K_{s+1}+\DD _{s+1}(W_{s+1})+(1-\DD _{s+1})(W_{s+1}-\iota _{s}(W_{s}))\, .
\end{eqnarray*}


\( \,  \)

\textit{Pozn\'amka} \textit{: (Podle \ref{defcondTheta})
\[
\Theta _{u}^{s}\left( K_{u+1}\right) =\Theta _{u}^{s}\left( \iota _{u+1,s+1}\, \iota _{s}K_{s}\right) =\iota _{u+1,s+1}\, \Theta _{s}^{s}\left( \iota _{s}K_{s}\right) \in \gX _{u+1}\]
} 

Definujme
\[
\phi (x)=\frac{1}{x}\left( e^{x}-\frac{e^{x}-1}{x}\right) =\sum ^{\infty }_{k=0}\frac{k+1}{(k+2)!}\, x^{k}\, .\]


\begin{lem}
P\v{r}edpokl\'adejme, \v{z}e posloupnosti \( \{V_{s}\}_{s=0}^{\infty } \),
\( \{W_{s}\}_{s=0}^{\infty } \) and \( \{\Theta ^{s}_{u}\}_{u=s}^{\infty } \)
jsou stejn\'e jako v lemmatu 1 . 

Ozna\v{c}me
\[
v_{s}=|V_{s}-\iota _{s-1}(V_{s-1})|,\textrm{ }w_{s}=|W_{s}-\iota _{s-1}(W_{s-1})|\]
(kde \( v_{0}=|V_{0}| \) , \( w_{0}=|W_{0}| \)). D\'ale nech\q{t} existuje
posloupnost kladn\'ych \v{c}\'\i{}sel, \( \{F_{s}\}_{s=0}^{\infty } \),takov\'a \v{z}e

\begin{equation}
\label{Theta_leq_Fw}
|\Theta ^{s}_{u}/_{\gX _{u+1}}|\leq F_{s}w_{s},\quad \forall s,u\textrm{ }\; u\geq s,
\end{equation}
konstanta \( A\geq 0 \), \v{z}e
\begin{equation}
\label{Thmconv_assumA}
F_{s}v_{s}^{\, 2}\leq A\, v_{s+1},\textrm{ }\forall s,
\end{equation}
a plat\'\i{}
\begin{eqnarray}
B & =\sum _{s=0}^{\infty }F_{s}v_{s}<\infty .\label{Thmconv_assumB} 
\end{eqnarray}
Ozna\v{c}me
\[
C=\sup _{s}F_{s}v_{s}\]
(\( C<\infty  \) podle (\ref{Thmconv_assumB})). Jestli\v{z}e \( d>0 \) vyhovuje
nerovnici
\begin{equation}
\label{ABd_ineq}
e^{dB}+A\phi (dC)\, d^{2}\leq d
\end{equation}
potom
\begin{equation}
\label{ws_leq_dvs}
w_{s}\leq d\, v_{s},\textrm{ }\forall s.
\end{equation}

\end{lem}
\begin{proof}
Provedeme indukc\'\i{} podle \( s \) . Pro \( s=0 \) je \( v_{0}=w_{0}=|V_{0}| \)
a (\ref{ws_leq_dvs}) plat\'\i{} proto\v{z}e (\ref{ABd_ineq}) implikuje, \v{z}e
\( d\geq 1 \). Induk\v{c}n\'\i{} krok \( s\to s+1 \): podle (\ref{recruleWs}),
(\ref{Ts_as_prod}), (\ref{normDsleq1}) a (\ref{ABd_ineq}), a s vyu\v{z}it\'\i{}m
monotonie \( \phi (x) \) dostaneme
\begin{eqnarray*}
w_{s+1} & \leq  & |T_{s}|\, v_{s+1}+\phi (\Theta _{s}^{s}/_{\gX _{s+1}}|)|\Theta _{s}^{s}/_{\gX _{s+1}}|\, w_{s}\\
 & \leq  & \exp \left( \sum ^{s}_{j=0}F_{j}w_{j}\right) v_{s+1}+\phi (F_{s}w_{s})F_{s}w_{s}^{\, 2}\\
 & \leq  & \exp \left( d\sum ^{s}_{j=0}F_{j}v_{j}\right) v_{s+1}+\phi (d\, F_{s}v_{s})F_{s}d^{2}v_{s}^{\, 2}\\
 & \leq  & e^{dB}v_{s+1}+\phi (dC)d^{2}A\, v_{s+1}\\
 & \leq  & d\, v_{s+1}.
\end{eqnarray*}

\end{proof}
\textit{Pozn\'amka} [P. Duclos ] Pro
\[
B\leq \frac{1}{3}\ln 2\quad \textrm{a}\quad A\, \phi (3C)\leq \frac{1}{9}\]
nerovnost (\ref{ABd_ineq}) plat\'\i{} pro \( d=3 \).

P\v{r}ipome\v{n}me, \v{z}e \( \Theta ^{s}_{\infty }/_{\gX _{\infty }}\in \BB (\gX _{\infty }) \)
je induktivn\'\i{} limita oper\'ator\r{u} \( \Theta  \)\( _{u}^{s} \) s vlastnost\'\i{}

\[
(\Theta _{\infty }^{s}/_{\gX _{\infty }})(\iota _{\infty ,u+1}/_{\gX _{u+1}})=(\iota _{\infty ,u+1}/_{\gX _{u+1}})(\Theta _{u}^{s}/_{\gX _{u+1}}),\textrm{ }\forall u\geq s.\]


D\'\i{}ky (\ref{Theta_leq_Fw}) a (\ref{Vlastnosti_indukt.lim}) m\r{u}\v{z}e b\'yt
jeho norma odhadnuta v\'yrazem
\begin{equation}
\label{Tinfty_leq_Fw}
|\Theta ^{s}_{\infty }/_{\gX _{\infty }}|\leq F_{s}w_{s}.
\end{equation}


\begin{lem}
Jestli\v{z}e existuje \( d>0 \) takov\'e, \v{z}e podm\'\i{}nka (\ref{ABd_ineq})
je spln\v{e}na a nav\'\i{}c
\[
F_{\mathrm{inf}}=\inf _{s}F_{s}>0\]
potom existuj\'\i{} limity v \'upln\'em prostoru \( \gX _{\infty } \)


\[
V':=\lim _{s\to \infty }\iota _{\infty s}(V_{s})\quad W:=\lim _{s\to \infty }\iota _{\infty s}(W_{s})\]
 a limita
\[
T=\lim _{s\to \infty }e^{\Theta _{\infty }^{s}/_{\gX _{\infty }}}\dots e^{\Theta ^{0}_{\infty }/_{\gX _{\infty }}}\]
jako zobrazen\'\i{} z \( \BB \left( \gX _{\infty }\right)  \).
\end{lem}
\begin{proof}
Pro \( u\geq s \)
\[
|\iota _{\infty u}(V_{u})-\iota _{\infty s}(V_{s})|=\left| \sum _{j=s+1}^{u}\iota _{\infty j}(V_{j}-\iota _{j-1}(V_{j-1}))\right| \leq \sum _{j=s+1}^{u}v_{j}\]
 Proto\v{z}e
\[
\sum _{s=0}^{\infty }v_{s}\leq \frac{1}{F_{\mathrm{inf}}}\sum _{s=0}^{\infty }F_{s}v_{s}<\infty \]
je posloupnost \( \{\iota _{\infty s}(V_{s})\} \) Cauchyovsk\'a. D\'\i{}ky (\ref{ws_leq_dvs})
je posloupnost \( \{\iota _{\infty s}(W_{s})\} \) tak\'e Cauchyovsk\'a. Proto v
\'upln\'em \( \gX _{\infty } \) \( \, \,  \)limity existuj\'\i{}. 

Polo\v{z}me
\[
\tilde{T}_{s}=e^{\Theta _{\infty }^{s-1}/_{\gX _{\infty }}}\dots e^{\Theta ^{0}_{\infty }/_{\gX _{\infty }}}\quad \textrm{pro }s\geq 1\]
Jestli\v{z}e \( u\geq s \) potom podle (\ref{Tinfty_leq_Fw}) a (\ref{ws_leq_dvs}),
dost\'av\'ame 
\begin{eqnarray*}
|\tilde{T}_{u}-\tilde{T}_{s}| & \leq  & \left( \exp \left( \sum ^{u-1}_{j=s}|\Theta ^{j}_{\infty /_{\gX _{\infty }}}|\right) -1\right) \exp \left( \sum ^{s-1}_{j=0}|\Theta ^{j}_{\infty /_{\gX _{\infty }}}|\right) \\
 & \leq  & \exp \left( d\sum ^{u-1}_{j=0}F_{j}v_{j}\right) -\exp \left( d\sum ^{s-1}_{j=0}F_{j}v_{j}\right) .
\end{eqnarray*}
Z p\v{r}edpokladu (\ref{Thmconv_assumB}) vypl\'yv\'a, \v{z}e \( \{\tilde{T}_{s}\} \)
je Cauchyovsk\'a posloupnost v \( \BB (\gX _{\infty }) \) 

a proto \( T\in \BB (\gX _{\infty }) \) existuje. 

\( \,  \)
\end{proof}
\textbf{D\r{u}sledek:} Zachov\'ame-li p\v{r}edpoklady p\v{r}edch\'azej\'\i{}c\'\i{}ch tvrzen\'\i{},

tak existuje \v{c}\'\i{}slo \( \alpha >0 \) , takov\'e \v{z}e
\[
\forall s\geq 0\, \; \; \left| W_{s}\right| _{s}\leq \alpha \]


\begin{proof}
\( \forall j\geq 0 \) plat\'\i{}
\begin{eqnarray*}
\left| W_{j}\right| _{j}-\left| W_{j-1}\right| _{j-1}\leq \left| W_{j}\right| _{j}-\left| \iota _{j-1}\left( W_{j-1}\right) \right| _{j}\leq \left| W_{j}-\iota _{j-1}\left( W_{j-1}\right) \right| _{j-1}=w_{j}\leq dv_{j} & 
\end{eqnarray*}


Z toho plyne odhad 
\begin{eqnarray*}
\left| W_{s}\right| _{s} & = & \sum _{j=1}^{s}\left( \left| W_{j}\right| _{j}-\left| W_{j-1}\right| _{j-1}\right) +\left| W_{0}\right| \\
 & \leq  & d\, \sum _{j=1}^{s}v_{j}+\left| V_{0}\right| _{0}\leq d\frac{1}{F_{inf}}\sum _{j=0}^{\infty }F_{j}v_{j}+v_{0}\\
 & \leq  & \frac{dB}{F_{inf}}+v_{0}=:\: \alpha 
\end{eqnarray*}
 
\end{proof}
\begin{lem}
P\v{r}i zachov\'an\'\i{} dosavadn\'\i{}ho ozna\v{c}en\'\i{} a p\v{r}edpoklad\r{u} plat\'\i{},
\v{z}e posloupnost \( \left\{ \widetilde{T_{s}}\left( K_{\infty }\right) -K_{\infty }\right\}  \)
kde
\[
\widetilde{T_{s}}\left( K_{\infty }\right) :=e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta ^{0}_{\infty }}\left( K_{\infty }\right) \]
 je Cauchyovsk\'a posloupnost v \( \gX _{\infty } \)
\end{lem}
\begin{proof}
D\'\i{}ky zn\'am\'ym vlastnostem \( \Theta ^{s}_{\infty } \) a \( \,  \)\( \iota  \)\( _{\infty s} \)
m\'ame \( \forall s \) 
\begin{equation}
\label{(Theta_naK)}
\Theta ^{s}_{\infty }\left( K_{\infty }\right) =\Theta ^{s}_{\infty }\left( \iota _{\infty ,s+1}K_{s+1}\right) =\iota _{\infty ,s+1}\Theta ^{s}_{s}\left( \iota _{s}K_{s}\right) 
\end{equation}
 a d\'\i{}ky \textit{(\ref{defcondTheta}) zase} 
\begin{eqnarray*}
\Theta ^{s}_{s}\left( \iota _{s}K_{s}\right)  & = & \Theta ^{s}_{s}\left( \iota _{s}\: \left( K_{s}+\DD _{s}\left( W_{s}\right) \right) \right) -\Theta _{s}^{s}\left( \iota _{s\, }(\DD _{s}\left( Ws\right) )\right) \\
 & = & -\iota _{s\, }\left( 1-\DD _{s}\right) \left( W_{s}-\iota _{s-1}\left( W_{s-1}\right) \right) -\Theta _{s}^{s}\left( \iota _{s}\left( \DD _{s}\left( W_{s}\right) \, \right) \, \right) 
\end{eqnarray*}
 tak\v{z}e podle (\ref{Theta_leq_Fw}), (\ref{(Theta_naK)}) a (\ref{normDsleq1})
z\'\i{}sk\'ame odhad normy 
\begin{equation}
\label{Norm_theta_naK}
\left\Vert \Theta ^{s}_{\infty }\left( K_{\infty }\right) \right\Vert =\left\Vert \iota _{\infty ,s+1}\Theta ^{s}_{s}\left( K_{s+1}\right) \right\Vert \leq \, 1\, w_{s}+|\Theta ^{s}_{s}/_{\gX _{s+1}}|\, \left\Vert W_{s}\right\Vert _{s}\leq w_{s}+\left\Vert W_{s}\right\Vert _{s}\, F_{s}w_{s}
\end{equation}
 Proto\v{z}e
\[
\forall x\geq 0\; \, \; \frac{e^{x}-1}{x}:=\sum _{n=0}^{\infty }\frac{1}{\left( n+1\right) !}x^{n}\leq e^{x}\]
 a \( \,  \)\( \Theta _{\infty }^{s}\left( K_{\infty }\right) \in \gX _{\infty } \)

\begin{eqnarray*}
\left\Vert e^{\Theta _{\infty }^{s-1}}\left( K_{\infty }\right) -K_{\infty }\right\Vert  & \leq  & \sum _{n=0}^{\infty }\frac{1}{\left( n+1\right) !}\left\Vert (\Theta _{\infty }^{s-1}\, )^{n}\, \left( \Theta _{\infty }^{s-1}\left( K_{\infty }\right) \right) \right\Vert \\
 & \leq  & \sum _{n=0}^{\infty }\frac{1}{\left( n+1\right) !}\, |\Theta ^{s-1}_{\infty }/_{\gX _{\infty }}|^{n}\, \left\Vert \Theta _{\infty }^{s-1}\left( K_{\infty }\right) \, \right\Vert \\
 & \leq  & e^{|\Theta ^{s-1}_{\infty }/_{\gX _{\infty }}|}\, \left\Vert \Theta _{\infty }^{s-1}\left( K_{\infty }\right) \right\Vert 
\end{eqnarray*}
Proto\v{z}e pro ka\v{z}dou kone\v{c}nou posloupnost line\'arn\'\i{}ch zobrazen\'\i{}
na libovoln\'em line\'arn\'\i{}m prostoru plat\'\i{}
\begin{eqnarray*}
a_{j}\dots a_{0}-1 & = & a_{j}\dots a_{1}\, \left( a_{0}-1\right) +\\
 & + & a_{j}\dots a_{2}\, \left( a_{1}-1\right) +\\
 & + & \dots +(a_{j}-1)
\end{eqnarray*}
 je podle p\v{r}edchoz\'\i{}ch odhad\r{u} 
\begin{eqnarray*}
\left\Vert e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta _{\infty }^{0}}\left( K_{\infty }\right) -K_{\infty }\right\Vert  & = & \left\Vert \right. e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta _{\infty }^{1}}\left( e^{\Theta _{\infty }^{0}}(K_{\infty })-K_{\infty }\right) +\\
 &  & +e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta _{\infty }^{2}}\left( e^{\Theta _{\infty }^{1}}(K_{\infty })-K_{\infty }\right) +\\
 &  & +\dots +\left( e^{\Theta _{\infty }^{s-1}}(K_{\infty })-K_{\infty }\right) \left. \right\Vert \\
 & \leq  & e^{|\Theta _{\infty }^{s-1}/_{\gX _{\infty }}|}\dots e^{|\Theta ^{1}_{\infty }/_{\gX _{\infty }}|}\; e^{|\Theta _{\infty }^{0}/_{\gX _{\infty }}|}\left\Vert \Theta _{\infty }^{o}\left( K_{\infty }\right) \right\Vert \\
 &  & +\dots +e^{|\Theta _{\infty }^{s-1}/_{\gX _{\infty }}|}\; \left\Vert \Theta _{\infty }^{s-1}\left( K_{\infty }\right) \right\Vert 
\end{eqnarray*}
 Tak\v{z}e pro \( t>s \) dostaneme 
\begin{eqnarray*}
\left| \widetilde{T_{t}}\left( K_{\infty }\right) -\widetilde{T_{s}}\left( K_{\infty }\right) \right|  & = & \left\Vert e^{\Theta _{\infty }^{t-1}}\dots e^{\Theta _{\infty }^{0}}\left( K_{\infty }\right) -e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta _{\infty }^{0}}\left( K_{\infty }\right) \right\Vert \\
 & \leq  & \left\Vert (e^{\Theta _{\infty }^{t-1}}\dots e^{\Theta _{\infty }^{s}}-1)(e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta _{\infty }^{0}}\left( K_{\infty }\right) -K_{\infty }+K_{\infty })\right\Vert \\
 & \leq  & \left( e^{|\Theta _{\infty }^{t-1}/_{\gX _{\infty }}|\; \dots +|\Theta _{\infty }^{s}/_{\gX _{\infty }}|}\; -1\right) \, \left\Vert e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta _{\infty }^{0}}\left( K_{\infty }\right) -K_{\infty }\right\Vert \\
 &  & +\left\Vert e^{\Theta _{\infty }^{t-1}}\dots e^{\Theta _{\infty }^{s}}\left( K_{\infty }\right) -K_{\infty }\right\Vert \\
 & \leq  & \left( \exp \left( \sum _{j=s}^{t-1}|\Theta _{\infty }^{j}/_{\gX _{\infty }}|\right) -1\right) \times \\
 &  & \times \left\{ \exp \left( \sum _{j=0}^{s-1}|\Theta _{\infty }^{j}/_{\gX _{\infty }}|\right) \, \left\Vert \Theta _{\infty }^{0}\left( K_{\infty }\right) \right\Vert +\dots +e^{|\Theta _{\infty }^{s-1}/_{\gX _{\infty }}|}\; \left\Vert \Theta _{\infty }^{s-1}\left( K_{\infty }\right) \right\Vert \right\} \\
 &  & +\left( \exp \left( \sum _{j=s}^{t-1}|\Theta _{\infty }^{j}/_{\gX _{\infty }}|\right) \; \left\Vert \Theta _{\infty }^{s}\left( K_{\infty }\right) \right\Vert +\dots +e^{|\Theta _{\infty }^{t-1}/_{\gX _{\infty }}|}\; \left\Vert \Theta _{\infty }^{t-1}\left( K_{\infty }\right) \right\Vert \right) \\
 & \leq  & \left( \exp \left( \sum _{j=s}^{t-1}|\Theta _{\infty }^{j}/_{\gX _{\infty }}|\right) -1\right) \, \exp \left( \sum _{j=0}^{s-1}|\Theta _{\infty }^{j}/_{\gX _{\infty }}|\right) \, \left( \sum _{j=0}^{s-1}\left\Vert \Theta _{\infty }^{j}\left( K_{\infty }\right) \right\Vert \right) \\
 &  & +\exp \left( \sum _{j=s}^{t-1}|\Theta _{\infty }^{j}/_{\gX _{\infty }}|\right) \, \left( \sum _{j=s}^{t-1}\left\Vert \Theta _{\infty }^{j}\left( K_{\infty }\right) \right\Vert \right) 
\end{eqnarray*}
 Ale proto\v{z}e podle (\ref{Tinfty_leq_Fw}), (\ref{ws_leq_dvs}) a (\ref{Thmconv_assumB})
\[
\sum _{j=0}^{\infty }|\Theta _{\infty }^{j}/_{\gX _{\infty }}|\leq d\sum _{j=0}^{\infty }F_{j}\, v_{j}=d\, B<\infty \]
 a podle (\ref{Norm_theta_naK}), (\ref{ws_leq_dvs}) a (\ref{Thmconv_assumB})
a p\v{r}edpokladu lemmatu 4 m\'ame
\begin{eqnarray*}
\sum _{j=0}^{\infty }\left\Vert \Theta _{\infty }^{j}\left( K_{\infty }\right) \right\Vert  & \leq  & \sum _{j=0}^{\infty }\left( w_{j}+\left\Vert W_{j}\right\Vert _{j}\, F_{j\, }w_{j}\right) \leq d\, \sum _{j=0}^{\infty }v_{j}+\alpha \, d\, \sum _{j=0}^{\infty }F_{j\, }v_{j}\\
 & \leq  & \frac{d}{F_{inf}}B+\alpha \, d\, B<\infty 
\end{eqnarray*}
 tak\v{z}e posloupnost je Cauchyovsk\'a
\begin{eqnarray*}
\left| \widetilde{T_{t}}\left( K_{\infty }\right) -\widetilde{T_{s}}\left( K_{\infty }\right) \right|  & \leq  & \left( \exp \left( \sum _{j=s}^{t-1}\left\Vert \Theta _{\infty }^{j}\left( K_{\infty }\right) \right\Vert \right) -1\right) \, e^{d\, B}\, \left( \frac{1}{F_{inf}}+\alpha \right) \, d\, B\\
 &  & +\exp \left( \sum _{j=s}^{t-1}\left\Vert \Theta _{\infty }^{j}\left( K_{\infty }\right) \right\Vert \right) \, \sum _{j=s}^{t-1}\left\Vert \Theta _{\infty }^{j}\left( K_{\infty }\right) \right\Vert 
\end{eqnarray*}
 \( \,  \)
\end{proof}
\textbf{Z\'av\v{e}r:} P\v{r}edpokl\'adejme, \v{z}e z\r{u}stanou v platnosti
p\v{r}edpoklady lemmatu 1, 2 a 3. 

Podle tvrzen\'\i{} lemmatu 3 existuj\'\i{} limity v \'upln\'em prostoru \( \gX _{\infty } \)
\[
V'=\lim _{s\to \infty }\iota _{\infty s}(V_{s})\quad W=\lim _{s\to \infty }\iota _{\infty s}(W_{s})\]
 a limita \( \widetilde{T_{s}} \) jako zobrazen\'\i{} z \( \BB \left( \gX _{\infty }\right)  \)
do L\( ^{\infty } \)
\[
T=\lim _{s\to \infty }e^{\Theta _{\infty }^{s-1}/_{\gX _{\infty }}}\dots e^{\Theta ^{0}_{\infty }/_{\gX _{\infty }}}\]


Dodefinujme nyn\'\i{} T na prostoru \( \C K_{\infty } \) line\'arn\v{e} a polo\v{z}me

\[
T\left( K_{\infty }\right) :=\lim _{s\to \infty }\left( e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta ^{0}_{\infty }}\, \left( K_{\infty }\right) -K_{\infty }\right) \, +K_{\infty }\]
 co\v{z} je prvek prostoru \( \C K_{\infty }\otimes \gX _{\infty }, \) proto\v{z}e
posloupnost \( \left\{ \left( e^{\Theta _{\infty }^{s-1}}\dots e^{\Theta ^{0}_{\infty }}\, \left( K_{\infty }\right) -K_{\infty }\right) \right\}  \)\( \,  \)
je Cauchyovsk\'a v \'upln\'em prostoru \( \gX _{\infty } \) . Tud\'\i{}\v{z} m\'a smysl
definovat limitu
\[
T=\lim _{s\to \infty }\widetilde{T_{s}}\; \; :\C K_{\infty }\oplus \gX _{\infty }\to \C K_{\infty }\oplus \gX _{\infty }\]
 Proto\v{z}e v\v{s}ak podle (\ref{T(K+V)}) plat\'\i{}, \v{z}e 
\begin{eqnarray}
\widetilde{T_{s}}\left( K_{\infty }+\iota _{\infty s}V_{s}\right)  & = & e^{\Theta ^{s-1}_{\infty }}e^{\Theta _{\infty }^{s-2}}\dots e^{\Theta _{\infty }^{0}}\, \left( \iota _{\infty s}\left( K_{s}+V_{s}\right) \right) \nonumber \\
 & = & \iota _{\infty s\, }e^{\Theta ^{s-1}_{s-1}}e^{\Theta _{s-1}^{s-2}}\dots e^{\Theta _{s}^{0}}\: \left( K_{s}+V_{s}\right) \nonumber \\
 & = & \iota _{\infty s}T_{s-1}\left( K_{s}+V_{s}\right) \nonumber \\
 & = & \iota _{\infty s}\left\{ K_{s}+\DD _{s}\left( W_{s}\right) +\left( 1-\DD _{s}\right) \left( W_{s}-\iota _{s-1}\left( W_{s-1}\right) \right) \right\} \nonumber \\
 & = & K_{\infty }+\DD \left( \iota _{\infty s}W_{s}\right) +\left( 1-\DD \right) \left( \iota _{\infty s}\left( W_{s}\right) -\iota _{\infty ,s-1}\left( W_{s-1}\right) \right) \label{lim_T(K+V)} 
\end{eqnarray}


a p\v{r}ejdeme-li k limit\v{e} \( \,  \) \( s\rightarrow \infty  \) dost\'av\'ame
kone\v{c}n\'y v\'yraz.
\[
T\left( K_{\infty }+V'\right) =K_{\infty }+\DD \left( W\right) \]
 Kde rovnost plat\'\i{} v \( \C K_{\infty }\otimes \gX _{\infty } \)

\( \,  \) 

\( \,  \)


\subsection{O zobrazen\'\i{} Gama}


\subsubsection{\'Uvod}

P\v{r}edpokl\'adejme, \v{z}e \( \dim \gX <\infty  \), \( \dim \gY <\infty  \)
(Toto \( \gX  \) neznamen\'a pomocn\'y prostor p\v{r}i definici induktivn\'\i{} limity.),
\( A\in \LL (\gY ) \) , \( B\in \LL (\gX ) \), \( A \) i \( B \) jsou hermitovsk\'e
a \( V\in \LL (\gX ,\gY ) \). Jestli\v{z}e \( \gamma  \) je jednoduch\'a uzav\v{r}en\'a
a kladn\v{e} orientovan\'a k\v{r}ivka v komplexn\'\i{} rovin\v{e}, takov\'a \v{z}e
\( \Spec (A) \) le\v{z}\'\i{} ve vnit\v{r}ku \( \gamma  \) zat\'\i{}mco \( \Spec (B) \)
le\v{z}\'\i{} v jeho dopl\v{n}ku. Tak potom rovnice
\[
AW-WB=V\]
m\'a \v{r}e\v{s}en\'\i{} \( W\in \LL (\gX ,\gY ) \) definovan\'e vztahem
\begin{equation}
\label{inverse_comm_as_int}
W=\frac{1}{2\pi \i }\oint _{\gamma }(A-z)^{-1}V(B-z)^{-1}dz\, .
\end{equation}


\nid \textit{Ov\v{e}\v{r}en\'\i{}.} 
\begin{eqnarray*}
AW-WB & = & \frac{1}{2\pi \i }\oint _{\gamma }(A-z)^{-1}(AV-VB)(B-z)^{-1}dz\\
 & = & \frac{1}{2\pi \i }\oint _{\gamma }\left( V(B-z)^{-1}-(A-z)^{-1}V\right) \, dz\\
 & = & 0+\frac{\i }{2\pi }\left( \oint _{\gamma }(A-z)^{-1}dz\right) V\\
 & = & V\, .
\end{eqnarray*}


Ozna\v{c}me \( M_{1}=\dim \gX  \), \( M_{2}=\dim \gY  \). V dal\v{s}\'\i{}m vyu\v{z}ijeme
dva n\'asleduj\'\i{}c\'\i{} odhady 

normy \( X\in \LL (\gX ,\gY ) \):
\begin{eqnarray}
|X|^{2} & \leq  & \sum ^{M_{2}}_{i=1}\sum ^{M_{1}}_{j=1}|X_{ij}|^{2}=\Tr X^{\ast }X\quad (\textrm{Hilbert}-\textrm{Schmidtova norma}),\label{estm_Xnorm_ab} \\
|X|^{2} & \geq  & \max \left( \max _{1\leq i\leq M_{2}}\sum ^{M_{1}}_{j=1}|X_{ij}|^{2},\max _{1\leq j\leq M_{1}}\sum ^{M_{2}}_{i=1}|X_{ij}|^{2}\right) ,\label{estm_Xnorm_be} 
\end{eqnarray}
kde \( (X_{ij}) \) jsou slo\v{z}ky matice \( X \) v libovoln\'ych ortonorm\'aln\'\i{}ch
baz\'\i{}ch \( \gX  \) a \( \gY  \).

\textbf{Tvrzen\'\i{}.}

Jestli\v{z}e \( \sup \Spec (A)<\inf \Spec (B) \) pak
\[
|W|\leq \frac{|V|}{\dist (\Spec (A),\Spec (B))}\, ,\]
v opa\v{c}n\'em p\v{r}\'\i{}pad\v{e}, kdy \( \Spec (A) \) a \( \Spec (B) \) jsou
prom\'\i{}ch\'ana,
\[
|W|\leq (\dim \gX \wedge \dim \gY )^{1/2}\frac{|V|}{\dist (\Spec (A),\Spec (B))}\, .\]


\begin{proof}
(1) Pro \( d=\inf \Spec (B)-\sup \Spec (A)>0 \) po obvykl\'e limitn\'\i{} procedu\v{r}e
za k\v{r}ivku \( \gamma  \) v (\ref{inverse_comm_as_int}) zvol\'\i{}me p\v{r}imku
rovnob\v{e}\v{z}nou s imagin\'arn\'\i{} osou, kter\'a re\'alnou osu protne 

v bodu \( x_{0}=(\sup \Spec (A)+\inf \Spec (B))/2 \). 

Tak\v{z}e 
\begin{eqnarray*}
|W|^{2} & \leq  & \frac{1}{2\pi }\int ^{\infty }_{-\infty }|(A-x_{0}-\i s)^{-1}||V||(B-x_{0}-\i s)^{-1}|\, ds\\
 & = & \frac{|V|}{2\pi }\int ^{\infty }_{-\infty }\frac{ds}{\left( \frac{d}{2}\right) ^{2}+s^{2}}\\
 & = & \frac{|V|}{d}\, .
\end{eqnarray*}
Kde jsme pou\v{z}ili \( |(A-z)^{-1}|=\dist (z,\Spec (A))^{-1} \).

(2) V druh\'em p\v{r}\'\i{}pad\v{e} zvol\'\i{}me b\'aze v \( \gX  \) a \( \gY  \) tak,
aby \( A \) a \( B \) v nich vyj\'ad\v{r}en\'e byly diagon\'aln\'\i{}, \( A=\diag (a_{1},\dots ,a_{M_{2}}) \)
a \( B=(b_{1},\dots ,b_{M_{1}}) \). Ozna\v{c}me \( d:=\dist (\Spec (A),\Spec (B)) \).
Potom \( W_{ij}=V_{ij}/(a_{i}-a_{j}) \), a pou\v{z}ijeme-li (\ref{estm_Xnorm_ab}),
(\ref{estm_Xnorm_be}) dost\'av\'ame odhad: 
\begin{eqnarray*}
|W|^{2} & \leq  & \sum ^{M_{2}}_{i=1}\sum ^{M_{1}}_{j=1}\left| \frac{V_{ij}}{a_{i}-b_{j}}\right| ^{2}\leq \sum ^{M_{2}}_{i=1}\sum ^{M_{1}}_{j=1}\frac{|V_{ij}|^{2}}{d^{2}}\\
 & \leq  & \sum ^{M_{2}}_{i=1}\frac{|V|^{2}}{d^{2}}=M_{2}\frac{|V|^{2}}{d^{2}}\, .
\end{eqnarray*}
A symetricky, \( |W|\leq M_{1}^{1/2}|V|/d \), z \v{c}eho\v{z} plyne po\v{z}adovan\'e.
\end{proof}

\subsubsection{Volba normy }

Nech\q{t} 
\[
\HH =\sideset {}{^{\oplus }}\sum _{n\in \N }\HH _{n}\]
je rozklad Hilbertova prostoru na direktn\'\i{} sou\v{c}et kone\v{c}n\v{e}dimenzion\'aln\'\i{}ch
podprostor\r{u} a \( J\subset \R  \). Pro libovolnou dvojici kladn\'ych re\'aln\'ych
\v{c}\'\i{}sel \( \phi  \) a \( E \) definujeme prostor 
\[
\gA \subset L^{\infty }\left( J\times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\HH _{m},\HH _{n})\right) \]
jeho\v{z} prvky tvo\v{r}\'\i{} matice se slo\v{z}kami 
\[
\VV _{knm}(\omega )\in \LL (\HH _{m},\HH _{n})\]
kter\'e maj\'\i{} kone\v{c}nou normu 
\[
\n \VV \n =\sup _{\substack {\omega ,\omega '\in J\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |\VV _{knm}(\omega )|+\phi \, |\partial \VV _{knm}(\omega ,\omega ')|\right) e^{|k|/E}\]
kde \( \partial  \) ozna\v{c}uje oper\'ator diference 
\begin{equation}
\label{diference}
\partial \VV (\omega ,\omega ')=\frac{\VV (\omega )-\VV (\omega ')}{\omega -\omega '}\, .
\end{equation}
 Poznamenejme, \v{z}e pro takov\'yto oper\'ator plat\'\i{} pravidlo 
\begin{equation}
\label{partialLeibniz}
\partial (\UU \VV )(\omega ,\omega ')=\partial \UU (\omega ,\omega ')\, \VV (\omega ')+\UU (\omega )\partial \VV (\omega ,\omega ')\, .
\end{equation}
 a \v{z}e podle na\v{s}\'\i{} notace pro ka\v{z}dou dvojici \( (\omega ,k)\in J\times \Z  \)
a ka\v{z}d\'y dvojindex \( (n,m)\in \N \times \N  \), je \( \VV _{knm}(\omega ) \)
prvek prostoru \( \LL (\HH _{m},\HH _{n}) \).

\textbf{Tvrzen\'\i{}. }

Takto definovan\'y normovan\'y prostor \( \gA  \) tvo\v{r}\'\i{} normovanou algebru
s n\'asoben\'\i{}m 
\begin{equation}
\label{prodUV}
(\UU \VV )_{knm}(\omega )=\sum _{\ell \in \Z }\sum _{p\in \N }\UU _{k-\ell ,n,p}(\omega )\, \VV _{\ell pm}(\omega )\, .
\end{equation}
 

\begin{proof}
Je t\v{r}eba dok\'azat, \v{z}e 
\begin{equation}
\label{algnorm}
\n \UU \VV \n \leq \n \UU \n \, \n \VV \n \, .
\end{equation}
 Pro jednoduchost ozna\v{c}me (v tomto d\r{u}kazu) 
\begin{eqnarray*}
\XX _{p}(\omega ) & = & \sum _{\ell \in \Z }\sum _{m\in \N }|\VV _{\ell pm}(\omega )|\, e^{|\ell |/E},\\
\partial \XX _{p}(\omega ,\omega ') & = & \sum _{\ell \in \Z }\sum _{m\in \N }|\partial \VV _{\ell pm}(\omega ,\omega ')|\, e^{|\ell |/E}.
\end{eqnarray*}
Zde \( \partial \XX  \) je pouze ozna\v{c}en\'\i{}, kter\'e nem\'a v\'yznam \( \partial  \)
z \( \XX  \). Plat\'\i{} 
\begin{eqnarray*}
 &  & \sum _{k}\sum _{m}|(\UU \VV )_{knm}(\omega )|\, e^{|k|/E}\\
 &  & \qquad \leq \sum _{k}\sum _{m}\sum _{\ell }\sum _{p}|\UU _{k-\ell ,n,p}(\omega )|\, e^{|k-\ell |/E}|\VV _{\ell pm}(\omega )|\, e^{|\ell |/E}\\
 &  & \qquad =\sum _{k}\sum _{m}\sum _{\ell }\sum _{p}|\UU _{knp}(\omega )|\, e^{|k|/E}|\VV _{\ell pm}(\omega )|\, e^{|\ell |/E}\\
 &  & \qquad =\sum _{k}\sum _{p}|\UU _{knp}(\omega )|\, e^{|k|/E}\XX _{p}(\omega )\, .
\end{eqnarray*}
Podobn\v{e} podle (\ref{partialLeibniz}),
\begin{eqnarray*}
\sum _{k}\sum _{m}|\partial (\UU \VV )_{knm}(\omega )|\, e^{|k|/E} & \leq  & \sum _{k}\sum _{m}\sum _{\ell }\sum _{p}\left( |\UU _{knp}(\omega )|\, e^{|k|/E}|\partial \VV _{\ell pm}(\omega ,\omega ')|\, e^{|\ell |/E}\right. \\
 &  & \\
 &  & \qquad \left. +|\partial \UU _{knp}(\omega ,\omega ')|\, e^{|k|/E}|\VV _{\ell pm}(\omega ')|\, e^{|\ell |/E}\right) \\
 & = & \sum _{k}\sum _{p}\left( |\UU _{knp}(\omega )|\, \partial \XX _{p}(\omega ,\omega ')\right. \\
 &  & \qquad \left. +|\partial \UU _{knp}(\omega ,\omega ')|\, \XX _{p}(\omega ')\right) e^{|k|/E}\, .
\end{eqnarray*}
A kombinace t\v{e}chto dvou nerovnost\'\i{} d\'a 
\begin{eqnarray*}
 &  & \sum _{k}\sum _{m}\left( |(\UU \VV )_{knm}(\omega )|+\phi \, |\partial (\UU \VV )_{knm}(\omega ,\omega ')|\right) e^{|k|/E}\\
 &  & \qquad \leq \sum _{k}\sum _{p}\left( |\UU _{knp}(\omega )|(\XX _{p}(\omega )+\phi \, \partial \XX _{p}(\omega ,\omega '))+\phi \, |\partial \UU _{knp}(\omega ,\omega ')|\, \XX _{p}(\omega ')\right) e^{|k|/E}\\
 &  & \qquad \leq \sup _{\omega ,\omega '}\sup _{p}(\XX _{p}(\omega )+\phi \, \partial \XX _{p}(\omega ,\omega '))\sum _{k}\sum _{p}(|\UU _{knp}(\omega )|+\phi \, |\partial \UU _{knp}(\omega ,\omega ')|)e^{|k|/E}\\
 &  & \qquad =\n \VV \n \, \sum _{k}\sum _{p}(|\UU _{knp}(\omega )|+\phi \, |\partial \UU _{knp}(\omega ,\omega ')|)e^{|k|/E}\, .
\end{eqnarray*}
K d\r{u}kazu (\ref{algnorm}) sta\v{c}\'\i{} aplikovat \( \sup _{\omega ,\omega '}\sup _{n} \)
na tuto nerovnost. 
\end{proof}
M\v{e}jme nyn\'\i{} dva p\'ary kladn\'ych re\'aln\'ych \v{c}\'\i{}sel \( (\phi _{1},E_{1}) \)
a \( (\phi _{2},E_{2}) \), \v{z}e plat\'\i{} 
\begin{equation}
\label{condsonFw12}
\rho =\frac{1}{E_{1}}-\frac{1}{E_{2}}\geq 0\quad \textrm{a}\quad \phi _{2}\leq \phi _{1}\, .
\end{equation}
K nim definujeme dva normovan\'e prostory, \( \gA _{1} \) and \( \gA _{2} \).
D\'ale p\v{r}edpokl\'adejme, \v{z}e je d\'an prvek 
\begin{equation}
\label{Gammaintro}
\Gamma \in \Map \left( J\times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\LL (\HH _{m},\HH _{n}))\right) \, ,
\end{equation}
takov\'y, \v{z}e pro ka\v{z}d\'y p\'ar \( (\omega ,k)\in J\times \Z  \) a ka\v{z}dou
dvojici index\r{u} \( (n,m)\in \N \times \N  \), \( \Gamma _{knm}(\omega ) \)
n\'ale\v{z}\'\i{} \( \LL (\LL (\HH _{m},\HH _{n})) \). \( \Gamma  \) definuje p\v{r}irozen\v{e}
line\'arn\'\i{} zobrazen\'\i{}, kter\'e pro jednoduchost ozna\v{c}\'\i{}me tak\'e \( \Gamma  \),
z \( \gA _{1} \) do \( \gA _{2} \), dan\'e p\v{r}edpisem 
\begin{equation}
\label{Gammamap}
\Gamma (\VV )_{knm}(\omega )=\Gamma _{knm}(\omega )(\VV _{knm}(\omega ))\, .
\end{equation}
Pro oper\'ator diference m\r{u}\v{z}eme v tomto p\v{r}\'\i{}pad\v{e} pou\v{z}\'\i{}t
pravidla 
\[
\partial \left( \Gamma (\VV )\right) (\omega ,\omega ')=\partial \Gamma (\omega ,\omega ')\, (\VV (\omega '))+\Gamma (\omega )(\partial \VV (\omega ,\omega '))\, .\]


\textbf{Tvrzen\'\i{}.}

Pro normu \( \Gamma :\gA _{1}\to \gA _{2} \) plat\'\i{} odhad 
\begin{equation}
\label{nGammaI}
\n \Gamma \n \leq \sup _{\substack {\omega ,\omega '\in J\cr \omega \neq \omega '}}\, \sup _{k\in \Z }\, \sup _{(n,m)\in \N \times \N }e^{-\rho |k|}\left( |\Gamma _{knm}(\omega )|+\phi _{2}\, |\partial \Gamma _{knm}(\omega ,\omega ')|\right) \, .
\end{equation}


\begin{proof}
Poznamenejme, \v{z}e je-li pot\v{r}eba, lze zam\v{e}nit \( \omega  \) a
\( \omega ' \) v \( |\partial \UU (\omega ,\omega ')| \). Potom 
\begin{eqnarray*}
 &  & \sum _{k}\sum _{m}\left( |\Gamma _{knm}(\omega )(\VV _{knm}(\omega ))|+\phi _{2}\, |\partial \left( \Gamma _{knm}(\VV _{knm})\right) (\omega ,\omega ')|\right) e^{|k|/E_{2}}\\
 &  & \qquad \leq \sum _{k}\sum _{m}\left( |\VV _{knm}(\omega )|(|\Gamma _{knm}(\omega )|+\phi _{2}\, |\partial \Gamma _{knm}(\omega ,\omega ')|)e^{-\rho |k|}\right. \\
 &  & \qquad \qquad \left. +\phi _{2}\, |\partial \VV _{knm}(\omega ,\omega ')|\, |\Gamma _{knm}(\omega ')|e^{-\rho |k|}\right) e^{|k|/E_{1}}\\
 &  & \qquad \leq \sup _{\omega ,\omega '}\, \sup _{k}\, \sup _{(n,m)}e^{-\rho |k|}\left( |\Gamma _{knm}(\omega )|+\phi _{2}\, |\partial \Gamma _{knm}(\omega ,\omega ')|\right) \\
 &  & \qquad \qquad \times \sum _{k}\sum _{m}\left( |\VV _{knm}(\omega )|+\phi _{1}\, |\partial \VV _{knm}(\omega ,\omega ')|\right) e^{|k|/E_{1}}\, .
\end{eqnarray*}
K dokon\v{c}en\'\i{} d\r{u}kazu sta\v{s}\'\i{} prov\'est v t\'eto nerovnosti \( \sup _{\omega ,\omega '}\, \sup _{n} \)
.
\end{proof}

\subsubsection{Odhad normy Gama. }

Nech\q{t} \( \{H_{m}\}_{m\in \N } \) je ost\v{r}e rostouc\'\i{} posloupnost kladn\'ych
re\'aln\'ych \v{c}\'\i{}sel. Polo\v{z}me 
\[
\Delta _{mn}=H_{m}-H_{n},\]
a
\[
\delta =\left( \sup _{m}\sum _{n\neq m}\frac{1}{|\Delta _{nm}|}\right) ^{-1}.\]
P\v{r}edpokl\'adejme, \v{z}e \( \delta >0 \). Plat\'\i{} \( \delta \leq |\Delta _{nm}| \)
pro ka\v{z}d\'e \( m\neq n \). D\'ale nech\q{t} \( \Omega >0 \) je pevn\'e \v{c}\'\i{}slo
a plat\'\i{} 
\[
J\subset \big [\, \frac{8}{9}\Omega ,\frac{9}{8}\Omega \, \big ]\]
\v{R}ekneme, \v{z}e multiindex \( (k,n,m)\in \Z \times \N \times \N  \) je
kritick\'y pr\'av\v{e} kdy\v{z} \( m\neq n \) a
\[
\frac{k\Omega }{\Delta _{mn}}\in \, \big ]\frac{1}{2},2\big [\, .\]
 V opa\v{c}n\'em p\v{r}\'\i{}pad\v{e} \v{r}ekneme, \v{z}e kritick\'y nen\'\i{}. Zde p\v{r}edpokl\'ad\'ame,
\v{z}e \( k\neq 0 \) nebo \( m\neq n \); p\v{r}\'\i{}pad \( k=0 \) a \( m=n \)
budeme v\v{z}dy vy\v{s}et\v{r}ovat zvl\'a\v{s}\q{t}.

P\v{r}\'\i{}m\'ym d\r{u}sledkem t\v{e}chto definic je 

\begin{lem}
Jestli\v{z}e \( \omega \in J \), \( k\neq 0 \) nebo \( m\neq n \), a \( (k,n,m) \)
nen\'\i{} kritick\'y tak 
\[
|k\omega +H_{n}-H_{m}|\geq \min \left\{ \frac{7}{16}\, \delta ,\, \frac{8}{9}\, \Omega \right\} .\]

\end{lem}
\begin{proof}
Rozli\v{s}\'\i{}me dva p\v{r}\'\i{}pady. Pro \( m\neq n \) 
\[
|k\omega +H_{n}-H_{m}|\geq \delta \, \left| \frac{k\omega }{\Delta _{mn}}-1\right| .\]
 A jestli\v{z}e \( (k,n,m) \) nen\'\i{} kritick\'y, plat\'\i{}: 
\[
\frac{k\omega }{\Delta _{mn}}=\frac{\omega }{\Omega }\frac{k\Omega }{\Delta _{mn}}\in \, ]-\infty ,\frac{9}{16}\, ]\cup [\, \frac{16}{9},\infty [\, ,\]
a tedy 
\[
\left| \frac{k\omega }{\Delta _{mn}}-1\right| \geq \frac{7}{16}.\]


V druh\'em p\v{r}\'\i{}pad\v{e}, kdy \( m=n \) a \( k\neq 0 \), jednodu\v{s}e
dost\'av\'ame 
\[
|k\omega +H_{n}-H_{m}|\geq \omega \geq \frac{8}{9}\, \Omega .\]

\end{proof}
D\'ale p\v{r}edpokl\'adejme, \v{z}e pro zobrazen\'\i{} \( \Gamma  \) jako prvek (\ref{Gammaintro})
plat\'\i{} n\'asleduj\'\i{}c\'\i{} 
\begin{eqnarray*}
\Gamma _{knm}(\omega ) & = & 0\qquad \textrm{pro }k=0\textrm{ a }n=m,\\
|\Gamma _{knm}(\omega )| & \leq  & \frac{\delta }{\phi _{2}|\Delta _{mn}||k|^{1/2}}\, e^{\rho |k|/2}\qquad \textrm{pro }\omega \in J\textrm{ a }(k,n,m)\textrm{ kriticky}',\\
|\Gamma _{knm}(\omega )| & \leq  & \frac{4}{|k|\Omega +|\Delta _{mn}|}\qquad \textrm{pro }\omega \in J\textrm{ a }(k,n,m)\textrm{ nekriticky}',
\end{eqnarray*}
 a d\'ale \( \Gamma _{knm}(\omega ) \) je invertibiln\'\i{} pro v\v{s}echna \( (\omega ,k,n,m)\in J\times \Z \times \N \times \N  \),
takov\'a \v{z}e \( k\neq 0 \) nebo \( n\neq m \), a v tomto p\v{r}\'\i{}pad\v{e}
plat\'\i{}
\[
|\partial \Gamma _{knm}^{-1}(\omega ,\omega ')|\leq |k|+\frac{1}{2}.\]


\begin{lem}
Jestli\v{z}e 
\[
\phi _{2}\leq \min \left\{ \frac{1}{4}\, \Omega ,\frac{1}{4}\, \delta \right\} \]
potom plat\'\i{} odhad 
\[
\n \Gamma \n \leq \frac{5}{2\, \phi _{2}}.\]
 
\end{lem}
\begin{proof}
Pro odhad \( \n \Gamma \n  \) vyu\v{z}ijeme vztah (\ref{nGammaI}). Plat\'\i{}

\begin{eqnarray}
|\partial \Gamma _{knm}(\omega ,\omega ')| & = & |\Gamma _{knm}(\omega )\, \partial \Gamma _{knm}^{-1}(\omega ,\omega ')\, \Gamma _{knm}(\omega ')|\nonumber \label{partGamma} \\
 & \leq  & |\Gamma _{knm}(\omega )||\Gamma _{knm}(\omega ')|\left( |k|+\frac{1}{2}\right) \, .\label{partGamma} 
\end{eqnarray}
A pro \( (k,n,m) \) kritick\'y, proto\v{z}e \( \delta \leq |\Delta _{nm}| \),
plat\'\i{}
\begin{eqnarray*}
e^{-\rho |k|}\left( |\Gamma _{knm}(\omega )|+\phi _{2}\, |\partial \Gamma _{knm}(\omega ,\omega ')|\right)  & \leq  & e^{-\rho |k|}\left( \frac{1}{\phi _{2}|k|^{1/2}}\, e^{\rho |k|/2}+\frac{|k|+\frac{1}{2}}{\phi _{2}|k|}\, e^{\rho |k|}\right) \\
 & \leq  & \frac{1}{\phi _{2}}\, \left( 1+1+\frac{1}{2|k|}\right) \leq \frac{5}{2\, \phi _{2}}.
\end{eqnarray*}


Jestli\v{z}e \( (k,n,m) \) kritick\'y nen\'\i{}, tak
\begin{eqnarray*}
e^{-\rho |k|}\left( |\Gamma _{knm}(\omega )|+\phi _{2}\, |\partial \Gamma _{knm}(\omega ,\omega ')|\right)  & \leq  & e^{-\rho |k|}\frac{4}{|k|\Omega +|\Delta _{mn}|}\left( 1+\phi _{2}\, \frac{4|k|+2}{|k|\Omega +|\Delta _{mn}|}\right) \\
 & \leq  & 4\, \max \left\{ \frac{1}{\Omega }e^{-\rho },\frac{1}{\delta }\right\} \, \left( 1+\phi _{2}\, \max \left\{ \frac{4}{\Omega },\frac{2}{\delta }\right\} \right) \\
 & \leq  & \frac{2}{\phi _{2}}\, .
\end{eqnarray*}

\end{proof}
\( \,  \)


\section{Konkr\'etn\'\i{} \v{c}\'ast}


\subsection{\'Uvod}

Nech\q{t} \( \,  \)\( \HH  \) je separabiln\'\i{} Hilbert\r{u}v prostor.Definujeme 

\( \KK \: =L^{2}\left( 0,2\pi \right)  \)\( \otimes \HH  \)

Bu\q{d} \( H \) samosdru\v{z}en\'y oper\'ator na \( \HH  \) , jeho\v{z} spektum
je diskr\'etn\'\i{} 
\[
\sigma \left( H\right) =\left\{ H_{n}|\, n\in \N \right\} \subset \R \]
 H m\'a spektr\'aln\'\i{} rozklad
\[
H=\sum _{n\in \N }H_{n}\, Q_{n}\; \; ,\, \; \sup _{n\in \N }\left| H_{n}\right| =\infty \]
 kde Q\( _{n} \) jsou ortogon\'aln\'\i{} projektory na podprostor odpov\'\i{}daj\'\i{}c\'\i{} vlastn\'\i{}mu
\v{c}\'\i{}slu H\( _{n} \) :
\[
Q_{n}Q_{m}=\delta _{mn}Q_{m}\quad \sum _{n\in \N }Q_{n}=id_{\HH }\]
 Ozna\v{c}me vzd\'alenost dvou vlastn\'\i{}ch \v{c}\'\i{}sel H
\[
\Delta _{mn}:=H_{m}-H_{n}\]
a p\v{r}edpokl\'adejme, \v{z}e
\begin{equation}
\label{1.gap}
\delta :=\left( \sup _{m\in \N }\, \, \sum _{n\neq m}\frac{1}{\left| \Delta _{nm}\right| }\right) ^{-1}\; >0
\end{equation}


a existuje \( r>0 \) , \v{z}e je spln\v{e}na i podm\'\i{}nka 


\begin{equation}
\label{2.gap}
\Delta _{r}\left( \Omega \right) :=\left( \sum _{m,n\in \N ,\Delta _{mn}>\frac{\Omega }{2}}\left| \Delta _{mn}\right| ^{-r}M_{m}M_{n}\left( \min \left( M_{m},M_{n}\right) \right) ^{\frac{1}{2}}\right) ^{r}<\infty 
\end{equation}


Polo\v{z}me 
\[
\Delta _{0}:=\inf _{\substack {m,n\in \N \cr m\neq n}}\left| \Delta _{mn}\right| .\]
 Proto\v{z}e \( \delta >0 \) tak i pro \( \Delta _{0} \) plat\'\i{}, \v{z}e \( \Delta _{0}>0 \).

Ozna\v{c}me stopu\( \;  \) M\( _{n}:=\Tr \left( Q_{n}\right)  \) a p\v{r}edpokl\'adejme,
\v{z}e \( \forall n\in \N \; \: M_{n}<\infty  \) . 

Polo\v{z}me
\[
D:=-i\, \partial _{t}\; na\; L^{2}\left( 0,2\pi \right) \]
s periodickou podm\'\i{}nkou. Potom
\[
D=\sum _{k\in \Z }\, k\, P_{k}\]
 kde P\( _{k} \) je ortogon\'aln\'\i{} projektor na vlastn\'\i{} \v{c}\'\i{}slo k \( \in \Z  \).
Tedy \( \sigma  \)\( \left( D\right) =\Z  \)

Definujme pro \( \omega  \)\( \in \R  \) oper\'ator na \( \KK  \)
\[
\mathbf{K}:=\omega D\otimes Id_{\HH }+Id_{L^{2}}\otimes H\]
 \textbf{K} je samozdru\v{z}en\'y proto\v{z}e H i D jsou samozdru\v{z}en\'e.

\textbf{K} m\'a \textbf{}spektr\'aln\'\i{} rozklad
\[
\mathbf{K}=\sum _{k\in \Z }\; \sum _{n\in \N }\, \left( \omega \, k+H_{n}\right) P_{k}\otimes Q_{n}\]
Snadno se p\v{r}esv\v{e}d\v{c}\'\i{}me, \v{z}e pro spektrum K plat\'\i{} 
\[
\sigma \left( K\right) =\omega \Z +\sigma \left( H\right) \]


\textbf{Tvrzen\'\i{}:} Nech\q{t} \( \;  \)H\( _{n} \) je posloupnost re\'aln\'ych
\v{c}\'\i{}sel, takov\'a \v{z}e
\[
\sup _{n\in \N }\left| H_{n}\right| =\infty \]
 potom mno\v{z}ina pevn\v{e} zvolen\'ych \v{c}\'\i{}sel \( \omega >0 \) , 

takov\'ych \v{z}e mno\v{z}ina \( \left\{ \omega \, k+H_{n}|\, k\in \Z \; n\in \N \right\}  \)
nen\'\i{} hust\'a v \( \R  \) , 

je nulov\'e Lebesgueovy m\'\i{}ry.

D\r{u}kaz: nap\v{r} \cite{3}, Apendix A.

\( \,  \)

Nech\q{t} funkce \( t\longmapsto V\left( t\right)  \) je omezen\'a m\v{e}\v{r}iteln\'a
2\( \pi  \) periodick\'a s hodnotami v prostoru hermitovsk\'ych oper\'ator\r{u}
na \( \HH  \).

Definujme oper\'ator \textbf{V} na \( \KK  \)
\[
\left( \mathbf{V}\psi \right) \left( t\right) :=V\left( t\right) \psi \left( t\right) \]
 Na\v{s}\'\i{} \'ulohou bude vy\v{s}et\v{r}ov\'an\'\i{} vlastnost\'\i{} spektra poru\v{s}en\'eho
\v{c}asov\v{e} z\'avisl\'eho Floquetova 

Hamiltoni\'anu
\[
-i\omega \, \partial _{t}\otimes Id_{\HH }+Id_{L^{2}}\otimes H+\mathbf{V}\]
 Tento oper\'ator je tak\'e samozdru\v{z}en\'y, proto\v{z}e \textbf{K} je samozdru\v{z}en\'y
a V\( \left( t\right)  \) je hermitovsk\'y.

Pro X oper\'ator na \( \KK  \) definujeme ``slo\v{z}ky''
\begin{eqnarray}
X_{k\ell mn}:=tr\left( P_{k}\, Q_{m}\, X\, P_{\ell }\, Q_{n}\, \right) =\int _{0}^{2\pi }e^{ikt}Q_{m}Xe^{-i\ell t}Q_{n}\, dt & \label{def_slozek} 
\end{eqnarray}
 
\[
X_{k\ell mn}\in \LL \left( \HH _{M_{m}},\HH _{M_{n}}\right) \]
 tr zna\v{c}\'\i{} stopu v L\( ^{2}\left( 0,2\pi \right)  \). Kv\r{u}li p\v{r}ehlednosti
jsme vynechali tenzorov\'e sou\v{c}iny s odpov\'\i{}daj\'\i{}c\'\i{}mi identick\'ymi oper\'atory. 

Polo\v{z}me pro \( \Omega >0 \)
\begin{eqnarray*}
\Omega _{0}:=[\frac{8}{9}\Omega ,\frac{9}{8}\Omega ] & 
\end{eqnarray*}


Definujeme 
\[
\ZZ \: =\textrm{Map}\left( \Z \times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\HH _{m},\HH _{n})\right) \]


Jeho\v{z} prvky tvo\v{r}\'\i{} matice (obecn\v{e} neomezen\'e) se slo\v{z}kami

\[
Y_{kjnm}\in \LL (\HH _{M_{m}},\HH _{M_{n}})\]


Definujme vektorovou funkci na tomto prostoru \( K_{0}:\Omega _{0}\to \ZZ  \)
\[
\left( K_{0}\right) _{k\ell mn}\left( \omega \right) :=\mathbf{K}_{k\ell mn}\left( \omega \right) =k\omega \delta _{k\ell }+H_{m}\delta _{mn}\; \, pro\; \omega \in \Omega _{0},\: k,\ell \in \Z \, \: m,n\in \N \]
 podobn\v{e} pro potenci\'al V, kter\'y je konstantn\'\i{} funkc\'\i{} v \( \omega  \)
\[
\left( V\right) _{k\ell mn}:=\mathbf{V}_{k\ell mn}\]
 Proto\v{z}e v\v{s}ak \textbf{V} p\r{u}sob\'\i{} jako n\'asoben\'\i{} funkc\'\i{}, plat\'\i{}
\begin{eqnarray*}
V_{k\ell mn} & = & tr\left( P_{k}\, Q_{m}\, \mathbf{V}\, P_{\ell }\, Q_{n}\, \right) =\int _{0}^{2\pi }e^{ikt}Q_{m}V\left( t\right) e^{-i\ell t}Q_{n}\, dt\\
 & = & \int _{0}^{2\pi }e^{i\left( k-\ell \right) t}Q_{m}V\left( t\right) Q_{n}\, dt=V_{k-\ell ,0,m,n}=:V_{k-\ell ,m,n}
\end{eqnarray*}
Jeliko\v{z} matice V z\'avis\'\i{} pouze na rozd\'\i{}lu k a \( \ell  \) budeme ji d\'ale
br\'at jako matici se t\v{r}emi indexy.

\textbf{P\v{r}edpoklad :} Nech\q{t} \( \left| V_{kmn}\right| <\infty  \)
jako\v{z}to oper\'atorov\'a norma na \( \LL \left( \HH _{M_{m}},\HH _{M_{n}}\right)  \) 

Nech\q{t} \v{c}\'\i{}slo
\[
\sigma :=\left[ r+\frac{3}{2}\right] \]
 je takov\'e \v{z}e
\begin{equation}
\label{epsilon}
\epsilon :=\sup _{n\in \N }\, \left[ \sum _{m\in \Z }\left( \sum _{k\in \Z }\left| V_{kmn}\right| \, \left( max\left\{ \left| k\right| ,1\right\} \right) ^{\sigma }\right) \right] <\infty 
\end{equation}
 co\v{z} znamen\'a, \v{z}e V jako\v{z}to oper\'atorov\'a funkce je \( \sigma  \)
kr\'at spojit\v{e} diferencovateln\'a na \( \left( 0,2\pi \right)  \). (viz \cite{1},
adiabatick\'a \v{c}\'ast)


\subsection{Konstrukce}

Nech\q{t} jsou d\'any posloupnosti E\( _{s} \) a \( \phi  \)\( _{s} \) s vlastnostmi

\[
E_{-1}=0,\; \; \phi _{0}\leq \frac{1}{4}min\left\{ \Omega ,\; \delta \right\} \]
 E\( _{s} \) roste do nekone\v{c}na a \( \phi  \)\( _{s} \) kles\'a do nuly,
tedy
\[
E_{s}\nearrow \infty ,\; \; \phi _{s}\searrow 0\]


Ozna\v{c}me
\[
\rho _{s}:=\frac{1}{E_{s}}-\frac{1}{E_{s+1}}>0\]
 

Nyn\'\i{} indukc\'\i{} vybudujeme posloupnosti \( \gX _{s},V_{s},W_{s},\Theta _{u}^{s},\iota _{us} \)
atd, jejich\v{z} existenci jsme p\v{r}edpokl\'adali v algebraick\'e \v{c}\'asti.


\subsubsection{Induk\v{c}n\'\i{} krok, s=0}

Definujme prostor 
\[
\gX _{0}\subset l^{\infty }\left( \Omega _{0}\times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\HH _{m},\HH _{n})\right) \]
jeho\v{z} prvky tvo\v{r}\'\i{} matice se slo\v{z}kami
\[
X_{knm}(\omega )\in \LL (\HH _{M_{m}},\HH _{M_{n}})\, ,\, \; \omega \in \Omega _{0}\]
 kter\'e maj\'\i{} kone\v{c}nou normu 0
\[
\n X\n _{0}=\sup _{\substack {\omega ,\omega '\in \Omega _{0}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |X_{knm}(\omega )|+\phi _{0}\, |\partial X_{knm}(\omega ,\omega ')|\right) e^{|k|/E_{0}}\]
 kde derivace je zavedena stejn\v{e} vztahem (\ref{diference})

Definujme n\'asoben\'\i{} t\v{e}chto matic jako 
\begin{equation}
\label{XkratY}
(X\, Y)_{knm}(\omega )=\sum _{\ell \in \Z }\sum _{p\in \N }X_{k-\ell ,n,p}(\omega )\, Y_{\ell pm}(\omega )\, .
\end{equation}
 D\'\i{}ky tvrzen\'\i{} z algebraick\'e \v{c}\'asti tvo\v{r}\'\i{} normovan\'y prostor \( \gX _{0} \)
s t\'\i{}mto n\'asoben\'\i{}m normovanou algebru.

\textit{Pozn\'amka:} \textit{\( K_{0}\notin \gX _{0} \)}

Polo\v{z}me
\[
(O_{o})_{kmn}:=V_{kmn}\, \chi \left( 0\leq \left| k\right| <E_{0}\right) \]
 obecn\v{e} pro \( s\geq 1 \)
\begin{equation}
\label{defO}
(O_{s})_{kmn}:=V_{kmn}\, \chi \left( E_{s-1}\leq \left| k\right| <E_{s}\right) 
\end{equation}
 (O nen\'\i{} funkce \( \omega  \)). Plat\'\i{} odhad 
\begin{eqnarray*}
\n O_{o}\n _{0} & = & \sup _{\substack {\omega ,\omega '\in \Omega _{0}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |\left( O_{0}\right) _{knm}(\omega )|+\phi _{0}\, |\partial \left( O_{0}\right) _{knm}(\omega ,\omega ')|\right) e^{|k|/E_{0}}\\
 & = & \sup _{n\in \N }\sum _{0\leq |k|<E_{0}}\sum _{m\in \N }\left| \left( O_{0}\right) _{kmn}\right| e^{\left| k\right| /E_{0}}\leq \epsilon e<\infty 
\end{eqnarray*}
 tak\v{z}e
\[
O_{0}\in \gX _{0}\]
 Definujme
\[
\DD _{0}\in \BB \left( \gX _{0}\right) \; \; (\DD _{0}\left( X\right) )_{knm}:=X_{0nm}\delta _{nm}\delta _{0k}\]
 Polo\v{z}me
\[
V_{0}:=O_{o},\; W_{0}:=V_{o}\in \gX _{0}\]
 
\[
d_{1}:=\dist \left( \Spec \left( k\omega -\Delta _{mn}+(W_{0})_{0mm}\right) ,\Spec \left( W_{0}\right) _{0nn}\right) \]
 a definujme
\[
\Omega _{1}:=\left\{ \omega \in \Omega _{0}|\; \forall m,n\in \N \, \: ,\forall k\in \Z ,\; \; \; d_{1}\geq \psi _{1}\left( k,m,n\right) \right\} \]
kde
\begin{eqnarray*}
\psi _{1}\left( k,m,n\right)  & := & \frac{\left| k\right| \Omega +\left| \Delta _{mn}\right| }{4}\; \; \; \; \; pro\, \: \frac{k\Omega }{\Delta _{mn}}\notin [\frac{1}{2},\, 2]\: a\: m\neq n\\
 & := & \phi _{0}\, e^{\frac{-\left| k\right| \rho _{1}}{2}\, }\sqrt{\left| k\right| }\, \left| \frac{\Delta _{mn}}{\delta }\right| \, \left( M_{m}\wedge M_{n}\right) ^{\frac{1}{2}}\; \; jinak
\end{eqnarray*}
 

Polo\v{z}me \( \rho _{1}=\frac{1}{E_{0}}-\frac{1}{E_{1}} \)

Z defini\v{c}n\'\i{}ho vztahu je vid\v{e}t, \v{z}e
\[
\Omega _{1}\subset \Omega _{0}\]
Definujme
\[
K_{1}:=K_{0}/_{\Omega _{1}}\]
 a prostor
\[
\gX _{1}\subset l^{\infty }\left( \Omega _{1}\times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\HH _{m},\HH _{n})\right) \]
 jeho\v{z} prvky tvo\v{r}\'\i{} matice se slo\v{z}kami 
\[
X_{knm}(\omega )\in \LL (\HH _{M_{m}},\HH _{M_{n}})\, ,\, \; \omega \in \Omega _{1}\]
kter\'e maj\'\i{} kone\v{c}nou normu 
\[
\n X\n _{1}=\sup _{\substack {\omega ,\omega '\in \Omega _{1}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |X_{knm}(\omega )|+\phi _{1}\, |\partial X_{knm}(\omega ,\omega ')|\right) e^{|k|/E_{1}}\]
 

Zobrazen\'\i{} \( \iota  \)\( _{0}:\C K_{0}\oplus \gX _{0}\to \C K_{1}\oplus \gX _{1} \)
definujeme
\[
\iota _{0}\left( \lambda K_{0}\right) :=\lambda K_{1},\; \; \iota _{0}\left( X\right) :=X/_{\Omega _{1}}\; \; \textrm{pro}\: \lambda \in \C ,\: X\in \gX _{0}\]
kde
\[
\left( X/_{\Omega _{1}}\right) _{knm}\left( \omega \right) =X_{knm}\left( \omega \right) \; \; \forall \omega \in \Omega _{1}\]
 d\'\i{}ky \( \Omega _{1}\subset \Omega _{0} \) , \( \rho  \)\( _{1}>0 \), \( \phi  \)\( _{1}>\phi _{0} \)
plat\'\i{}, \v{z}e
\[
\left| \iota _{0}/_{\gX _{0}}\right| \leq 1\]
 Jestli\v{z}e je O\( _{1} \) definov\'ano ve shod\v{e} s (\ref{defO}) je 
\begin{eqnarray*}
\n O_{1}\n _{1} & = & \sup _{\substack {\omega ,\omega '\in \Omega _{1}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |\left( O_{1}\right) _{knm}(\omega )|+\phi _{1}\, |\partial \left( O_{1}\right) _{knm}(\omega ,\omega ')|\right) e^{|k|/E_{1}}\\
 & = & \sup _{n\in \N }\sum _{E_{o}\leq |k|<E_{1}}\sum _{m\in \N }\left| V_{kmn}\right| e^{\left| k\right| /E_{1}}\\
 & \leq  & \sup _{n\in \N }\sum _{E_{o}\leq |k|<E_{1}}\sum _{m\in \N }\left| V_{knm}\right| \frac{\left( \max \left( \left| k\right| ,1\right) \right) ^{\sigma }}{\left( E_{0}\right) ^{\sigma }}\, e\leq \frac{\epsilon e}{E_{0}^{\sigma }}<\infty 
\end{eqnarray*}
 

Tak\v{z}e O\( _{1}\in \gX _{1} \) a i
\[
V_{1}:=\iota _{0}V_{0}+O_{1}\in \gX _{1}\]
 \( \,  \)

Pokusme se nyn\'\i{} \v{r}e\v{s}it pro dan\'e \( \omega  \) a X rovnici pro Y 
\begin{equation}
\label{[K+D(W),Y]}
\left( 1-\widetilde{\DD }\right) X\left( \omega \right) =\left[ K_{0}+\widetilde{\DD }\left( W_{0}\right) ,Y\right] \left( \omega \right) 
\end{equation}
 \( X,Y\in l^{\infty }\left( \Omega _{0}\times \Z \times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\HH _{m},\HH _{n})\right)  \) 

tedy
\[
X_{kjnm}(\omega )\in \LL (\HH _{M_{m}},\HH _{M_{n}})\, ,\, \; \omega \in \Omega _{0}\]
 a kde \( \widetilde{\DD } \) je diagonalizuj\'\i{}c\'\i{} oper\'ator
\[
\left( \widetilde{\DD }\left( X\right) \right) _{kjnm}\left( \omega \right) :=\left( X\right) _{kknn}\left( \omega \right) \delta _{kj}\delta _{nm}\]


tedy pro \noun{\( n\neq m,k\neq j \)
\begin{eqnarray}
X_{kjnm}\left( \omega \right)  & = & \left\{ \sum _{\ell \in \Z }\sum _{p\in \N }\left( K_{0}+\widetilde{\DD }\left( W_{0}\right) \right) _{k\ell np}Y_{\ell jpm}-Y_{k\ell np}\left( K_{0}+\widetilde{\DD }\left( W_{0}\right) \right) _{\ell jpm}\right\} \left( \omega \right) \nonumber \\
 & = & \left( k\omega +H_{n}+\left( W_{0}\right) _{kknn}\left( \omega \right) \right) Y_{kjnm}\left( \omega \right) -Y_{kjnm}\left( \omega \right) \left( j\omega +H_{m}+\left( W_{0}\right) _{jjmm}\left( \omega \right) \right) \nonumber \\
 & = & \left( \left( k-j\right) \omega -\Delta _{mn}+\left( W_{0}\right) _{kknn}\left( \omega \right) \right) Y_{kjnm}\left( \omega \right) -Y_{kjnm}\left( \omega \right) \left( W_{0}\right) _{jjmm}\left( \omega \right) \label{[X,Y]} 
\end{eqnarray}
}

Ale to je \'uloha, kterou jsme vy\v{s}et\v{r}ovali v odstavci o zobrazen\'\i{} Gama
pro
\[
A=\left( (k-j)\omega -\Delta _{mn}+\left( W_{0}\right) _{kknn}\left( \omega \right) \right) \in \LL \left( \HH _{M_{n}}\right) \]
 
\[
B=\left( W_{0}\right) _{jjmm}\left( \omega \right) \in \LL \left( \HH _{M_{m}}\right) \]
 \v{R}e\v{s}en\'\i{} existuje a jeho normu lze odhadnout 
\begin{equation}
\label{norm_slozekgama1}
|Y_{kjnm}\left( \omega \right) |\leq \frac{|X_{kjnm}\left( \omega \right) |}{\dist (\Spec (A),\Spec (B))}\, ,
\end{equation}


jestli\v{z}e \( \sup \Spec (A)<\inf \Spec (B) \) 

v opa\v{c}n\'em p\v{r}\'\i{}pad\v{e}, kdy \( \Spec (A) \) jsou \( \Spec (B) \)
prom\'\i{}ch\'ana
\begin{equation}
\label{norm_slozekgama2}
|Y_{kjnm}\left( \omega \right) |\leq (M_{n}\wedge M_{m})^{1/2}\frac{|X_{kjnm}\left( \omega \right) |}{\dist (\Spec (A),\Spec (B))}\, .
\end{equation}


Ale \( \,  \)
\[
\dist (\Spec (A),\Spec (B))=\dist (\Spec \left( \left( k-j\right) \omega -\Delta _{mn}+\left( W_{0}\right) _{kknn}\left( \omega \right) \right) ,\Spec \left( W_{0}\right) _{jjmm}\left( \omega \right) )\]


\textit{Pozn\'amka:} Jestli\v{z}e ve v\'yrazu (\ref{[X,Y]}) z\'avis\'\i{} hodnota matice
X pouze na rozd\'\i{}lu prvn\'\i{}ch dvou index\r{u} \v{c}ili
\begin{equation}
\label{X(k,j)=X(k-j,0)}
X_{k,j,n,m}=X_{k-j,0,n,m}
\end{equation}


Tak stejnou vlastnost mus\'\i{} m\'\i{}t i Y. D\'\i{}ky tomu \v{z}e K je line\'arn\'\i{} v prvn\'\i{}m
indexu a je diagon\'aln\'\i{}. Tedy
\[
Y_{k,j,n,m}=Y_{k-j,0,n,m}\]
 A tot\'e\v{z} plat\'\i{} i obr\'acen\v{e}. M\'ali tuto vlastnost Y, zd\v{e}d\'\i{} ji i
X definovan\'e vztahem (\ref{[K+D(W),Y]}).

\( \:  \)

\textit{Pozn\'amka :} Z dal\v{s}\'\i{}ho postupu bude vid\v{e}t, \v{z}e v\v{s}echny
\( V_{s} \) budou z\'aviset pouze na rozd\'\i{}lu prvn\'\i{}ch dvou index\r{u}. Tuto vlastnost,
podle p\v{r}edchoz\'\i{} pozn\'amky budou m\'\i{}t i v\v{s}echny \( W_{s} \). \( K_{s} \)
sice tuto vlastnost nem\'a, ale to n\'am nebude vadit, proto\v{z}e bude vystupovat
pouze prost\v{r}ednictv\'\i{}m komut\'atoru. To je tak\'e d\r{u}vod k zaveden\'\i{} na\v{s}\'\i{}
``divn\'e'' definici n\'asoben\'\i{} ``divn\'ych'' matic s lich\'ym po\v{c}tem index\r{u}. 

Komut\'ator tedy budeme d\'ale ch\'apat ve smyslu n\'asoben\'\i{} (\ref{XkratY}) a 
\[
\left[ K_{0},Y\right] _{knm}\left( \omega \right) :=\left( k\omega -\Delta _{mn}\right) Y_{knm}\left( \omega \right) .\]
 Proto\v{z}e plat\'\i{}
\[
\left( W_{0}\right) _{k,j,n,m}\left( \omega \right) =\left( V_{0}\right) _{k,j,n,m}\left( \omega \right) =\left( V_{0}\right) _{k-j,0,n,m}=:\, \left( W_{0}\right) _{k-j,n,m}\]
 dost\'av\'ame 
\begin{eqnarray*}
\dist (\Spec (A),\Spec (B)) & = & \dist (\Spec \left( \left( k-j\right) \omega -\Delta _{mn}+\left( W_{0}\right) _{kknn}\left( \omega \right) \right) ,\Spec \left( W_{0}\right) _{jjmm}\left( \omega \right) )\\
 & = & \dist (\Spec \left( k\omega -\Delta _{mn}+\left( W_{0}\right) _{0nn}\left( \omega \right) \right) ,\Spec \left( W_{0}\right) _{0mm}\left( \omega \right) )\\
 & = & d_{1}
\end{eqnarray*}
Definujeme oper\'ator
\[
\Gamma _{0}:=\iota _{0}\: ad_{K_{0}+\DD _{0}\left( W_{0}\right) }^{-1}\left( 1-\DD _{0}\right) \; :\gX _{0}\to \gX _{1}\]

\begin{equation}
\label{defcondGama0}
X\longmapsto Y\Leftrightarrow \forall \omega \in \Omega _{1}\; \; \left( 1-\DD _{0}\right) X\left( \omega \right) =\left[ K_{0}+\DD _{0}\left( W_{0}\right) ,Y\right] \left( \omega \right) 
\end{equation}


kde \( \left[ \; ,\; \right]  \) znamen\'a komut\'ator matic ve smyslu p\v{r}edchoz\'\i{}
pozn\'amky.

Pomoc\'\i{} slo\v{z}ek zaps\'ano tak\'e
\[
(\Gamma _{0})_{knm}\left( \omega \right) \left( X_{knm}\left( \omega \right) \right) =Y_{knm}\left( \omega \right) \]
 pro \'uplnost polo\v{z}me \( \left( \Gamma _{0}\right) _{0nn}:=0 \)

Normu \( (\Gamma _{0})_{knm} \) lze d\'\i{}ky (\ref{norm_slozekgama1}) respektive
(\ref{norm_slozekgama2}) pro ka\v{z}d\'e \( \omega \in \Omega _{1} \) odhadnout 


\begin{eqnarray*}
\left| (\Gamma _{0})_{knm}\left( \omega \right) \right|  & = & 0\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; pro\; k=0\; a\; n=m\\
 & \leq  & \frac{4}{\left| k\right| \Omega +\left| \Delta _{mn}\right| }\; \; \; \; \; \; \; \; pro\, \frac{k\Omega }{\Delta _{mn}}\notin [\frac{1}{2},\, 2]a\, m\neq n\\
 & \leq  & \frac{\delta }{\left| \Delta _{mn}\right| \phi _{0}\sqrt{\left| k\right| }}\: e^{\frac{1}{2}\rho _{1}\left| k\right| }\; \; jinak
\end{eqnarray*}
 

Zb\'yv\'a ov\v{e}\v{r}it, \v{z}e \( \left( \Gamma _{0}\right) _{knm}\left( \omega \right)  \)
je pro ka\v{z}d\'e \( \left( \omega ,k,n,m\right) \in \Omega _{1}\times \Z \times \N \times \N  \),
kde \( k\neq 0 \) nebo \( n\neq m \) invertibiln\'\i{} a v tomto p\v{r}\'\i{}pad\v{e}
plat\'\i{} 
\[
\left| \partial \left( \Gamma _{0}\right) ^{-1}_{knm}\left( \omega ,\omega '\right) \right| \leq \left| k\right| +\frac{1}{2}\]


Definujeme
\[
\left( \Gamma ^{-1}_{0}Y\right) _{knm}\left( \omega \right) :=\left( k\omega -\Delta _{mn}+\left( W_{0}\right) _{0nn}\left( \omega \right) \right) Y_{knm}\left( \omega \right) -Y_{knm}\left( \omega \right) \left( W_{0}\right) _{0mm}\left( \omega \right) ,\quad \omega \in \Omega _{1}\]


Prost\'ym dosazen\'\i{}m zjist\'\i{}me, \v{z}e jde o inverzn\'\i{} matici.

Na tuto definici aplikujeme oper\'ator diference, p\v{r}i\v{c}em\v{z} na lev\'e
stran\v{e} vyu\v{z}ijeme formule (\ref{partGamma}), zat\'\i{}mco na prav\'e (\ref{partialLeibniz}). 

Porovn\'an\'\i{}m dostaneme p\v{r}edpis 
\[
\partial \left( \Gamma _{0}^{-1}\right) _{knm}\left( \omega ,\omega '\right) \left( Y_{knm}\left( \omega '\right) \right) =\left( k+\partial \left( W_{0}\right) _{0nn}\left( \omega ,\omega '\right) \right) Y_{knm}\left( \omega '\right) -Y_{knm}\left( \omega '\right) \left( \partial \left( W_{0}\right) _{knm}\left( \omega ,\omega '\right) \right) \]


Proto\v{z}e \( W_{0}=V_{0} \) a \( V_{0} \) jak v\'\i{}me nez\'avis\'\i{} na \( \omega  \),
dost\'av\'ame odhad
\[
\left| \partial \left( \Gamma _{0}\right) ^{-1}_{knm}\left( \omega ,\omega '\right) \right| \leq \left| k\right| \]


Proto\v{z}e \( \phi _{0}\leq \frac{1}{4}min\left\{ \Omega \, ,\delta \, \right\}  \)
a jsou spln\v{e}ny p\v{r}edpoklady lemmatu 6, tak plat\'\i{}
\[
\left| \Gamma _{0}\right| \leq \frac{5}{2\phi _{0}}\]
 Definujme zobrazen\'\i{} \( \Theta _{0}^{0}:\C K_{1}\oplus \gX _{1}\to \C K_{1}\oplus \gX _{1} \)
stejn\v{e} pro K\( _{1} \) i pro \( X\in \gX _{1} \)
\[
\Theta _{0}^{0}\left( X\right) :=\left[ \Gamma _{0}\left( W_{0}\right) ,\: X\right] =\left[ \iota _{0}\: ad_{K_{0}+\DD _{0}\left( W_{0}\right) }^{-1}\left( 1-\DD _{0}\right) \left( W_{0}\right) ,\: X\right] \]
a snadn\'ymi \'upravami zjist\'\i{}me, \v{z}e
\[
\left| \Theta _{0}^{0}/_{\gX _{1}}\right| \leq 2\, \left| \Gamma _{0}\right| \, \n W_{0}\n _{0}\leq \frac{5}{\phi _{0}}v_{0}\]
 Nyn\'\i{} ov\v{e}\v{r}\'\i{}me platnost podm\'\i{}nky (\ref{defcondTheta}) pro zobrazen\'\i{}
\( \Theta _{0}^{0} \) 
\[
\Theta _{0}^{0}\left( \iota _{0}\left( K_{0}+\DD _{0}\left( W_{0}\right) \right) \right) =\left[ \Gamma _{0}\left( W_{0}\right) ,\: \iota _{0}\left( K_{0}+\DD _{0}\left( W_{0}\right) \right) \right] \]
ale podle definice zobrazen\'\i{} \( \Gamma _{0} \)
\[
\Gamma _{0}\left( W_{0}\right) =Y\Leftrightarrow \forall \omega \in \Omega _{1}\; \; \left( 1-\DD _{0}\right) W_{0}\left( \omega \right) =\left[ K_{0}+\DD _{0}\left( W_{0}\right) ,Y\right] \left( \omega \right) \]
 z toho plyne
\[
\Theta _{0}^{0}\left( \iota _{0}\left( K_{0}+\DD _{0}\left( W_{0}\right) \right) \right) \left( \omega \right) =-\left( 1-\DD _{0}\right) \left( W_{0}\right) \left( \omega \right) \quad \forall \omega \in \Omega _{1}\]
 a tedy i
\begin{equation}
\label{defcondTheta0}
\Theta _{0}^{0}\left( \iota _{0}\left( K_{0}+\DD _{0}\left( W_{0}\right) \right) \right) =-\iota _{0}\left( 1-\DD _{0}\right) \left( W_{0}\right) 
\end{equation}
 Definujme oper\'ator \( T_{0}:\C K_{1}\oplus \gX _{1}\to \C K_{1}\oplus \gX _{1} \)
\[
T_{0}:=e^{\Theta _{0}^{0}}\]
 ve smyslu
\[
T_{0}/_{\gX _{1}}:=\sum _{j=0}^{\infty }\frac{\left( \Theta _{0}^{0}/_{\gX _{1}}\right) ^{j}}{j!}\]
kde \v{r}ada na prav\'e stran\v{e} konverguje v oper\'atorov\'e norm\v{e}, proto\v{z}e
\( \Theta ^{0}_{0}/_{\gX _{1}} \) je omezen\'y 

a d\'ale
\[
T_{0}\left( K_{1}\right) :=K_{1}+\sum _{j=0}^{\infty }\frac{\left( \Theta _{0}^{0}\right) ^{j}}{\left( j+1\right) !}\left( \Theta _{0}^{0}\left( K_{1}\right) \right) \]
 (suma m\'a smysl nebo\q{t} podle (\ref{defcondTheta0}) \( \Theta _{0}^{0}\left( K_{1}\right) \in \gX _{1} \)). 

Kone\v{c}n\v{e} polo\v{z}me
\[
W_{1}:=\iota _{0}\left( W_{0}\right) +T_{0}\left( V_{1}-\iota _{0}\left( V_{0}\right) \right) +\left( e^{\Theta _{0}^{0}}-\frac{e^{\Theta _{0}^{0}}-1}{\Theta _{0}^{0}}\right) \: \iota _{o}\left( 1-\DD _{0}\right) W_{0}\]



\subsubsection{Induk\v{c}n\'\i{} krok \protect\( s\to s+1\protect \)}

P\v{r}edpokl\'adejme, \v{z}e pro s p\v{r}irozen\'e zn\'ame klesaj\'\i{}c\'\i{} posloupnost
mno\v{z}in \( \left\{ \Omega _{j}\right\} _{j=0}^{s} \)
\[
\Omega _{s}\subset \Omega _{s-1}\subset \dots \subset \Omega _{0}\]
 

vektory
\[
K_{j}=K_{j-1}/_{\Omega _{j}}\; \; \; 0\leq j\leq s\]
a posloupnost prostor\r{u} \( \left\{ \gX _{j}\right\} _{j=0}^{s} \) 
\[
\gX _{j}\subset l^{\infty }\left( \Omega _{j}\times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\HH _{m},\HH _{n})\right) \]
 jeho\v{z} prvky tvo\v{r}\'\i{} matice se slo\v{z}kami
\[
X_{knm}(\omega )\in \LL (\HH _{M_{m}},\HH _{M_{n}})\, ,\, \; \omega \in \Omega _{j}\]
 kter\'e maj\'\i{} kone\v{c}nou normu 


\[
\n X\n _{j}=\sup _{\substack {\omega ,\omega '\in \Omega _{j}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |X_{knm}(\omega )|+\phi _{j}\, |\partial X_{knm}(\omega ,\omega ')|\right) e^{|k|/E_{j}}\]


N\'asoben\'\i{} matic zad\'ame jako (\ref{XkratY}), zase \( \gX _{j} \) bude normovan\'a
algebra.

D\'ale a\q{t} jsou zn\'ama zobrazen\'\i{} \( \iota  \)\( _{j-1}:\C K_{j-1}\oplus \gX _{j-1}\to \C K_{j}\oplus \gX _{j} \)
pro \( 1\leq j\leq s \) , kter\'a p\r{u}sob\'\i{} 
\[
\iota _{j-1}\left( \lambda K_{j-1}\right) =\lambda K_{j}\: ,\; \; \iota _{j-1}\left( X\right) =X/_{\Omega _{j}}\; \; pro\; \lambda \in \C ,\; X\in \gX _{j-1}\]


d\'\i{}ky \( \Omega _{j}\subset \Omega _{j-1} \) , \( \frac{1}{E_{j-1}}-\frac{1}{E_{j}}>0 \),
\( \phi  \)\( _{j}>\phi _{j-1} \) plat\'\i{}, \v{z}e
\[
\left| \iota _{j-1}/_{\gX _{j-1}}\right| \leq 1\]
Polo\v{z}me pro \( 0\leq t\leq j\leq s \)
\[
\iota _{jt}:=\iota _{j-1}\dots \iota _{t}\: :\C K_{t}\oplus \gX _{t}\to \C K_{j}\oplus \gX _{j}\]
 Z definice \( \iota  \)\( _{t} \) je z\v{r}ejm\'e, \v{z}e jsou spln\v{e}ny
podm\'\i{}nky
\begin{equation}
\label{vlastnostii}
\iota _{jt}\iota _{tu}=\iota _{ju}\; \; \left| \iota _{ju}/_{\gX _{u}}\right| \leq 1\; \; pro\; 0\leq u\leq t\leq j\leq s
\end{equation}


Definujme \( \DD _{j}\in \BB \left( \gX _{j}\right)  \) 
\begin{equation}
\label{def_Diags}
\left( \DD _{j}\left( X\right) \right) _{kmm}:=X_{0mm}\delta _{0k}\delta _{mn}\; X\in \gX _{j}
\end{equation}


z\v{r}ejm\v{e} 
\[
\DD _{j}\, \iota _{jt}/_{\gX _{t}}=\iota _{jt}/_{\gX _{t}}\, \DD _{t}\; \; pro\; 0\leq t\leq j\leq s\]
 Nech\q{t} zn\'ame je\v{s}t\v{e} posloupnosti vektor\r{u} \( \left\{ V_{j}\right\} _{j=0}^{s} \),
\( \left\{ W_{j}\right\} _{j=0}^{s} \), \( V_{j},\: W_{j}\in \gX _{j} \) 

a zobrazen\'\i{} \( \left\{ \Theta _{j}^{t}\right\} _{j=t}^{s-1} \) definovan\'ych
na \( \C K_{j+1}\oplus \gX _{j+1} \) s hodnotami v \( \gX _{j+1} \)

(tedy i \( \Theta  \)\( _{j}^{t} \)\( \left( K_{j+1}\right)  \)\( \in \gX _{j+1} \)
pro \( 0\leq t\leq j\leq s-1 \) )

Nav\'\i{}c definujeme-li
\[
\textrm{ }w_{j}=|W_{j}-\iota _{j-1}(W_{j-1})|\]
 po\v{z}adujeme i
\[
\left| \Theta _{j}^{k}/_{\gX _{j+1}}\right| \leq \frac{5}{\phi _{k}}w_{k}\]
a
\[
\Theta ^{j}_{v\; }\iota _{v+1,u+1}=\iota _{v+1,u+1\; }\Theta _{u}^{s}\quad \textrm{pro }j\leq u\leq v\leq s-1.\]
 Polo\v{z}\'\i{}me
\[
T_{j}=e^{\Theta ^{j}_{j}}e^{\Theta _{j}^{j-1}}\dots e^{\Theta ^{0}_{j}}:\C K_{j+1}\oplus \gX _{j+1}\rightarrow \C K_{j+1}\oplus \gX _{j+1}\quad \textrm{pro }s\geq j\geq 0\, .\]


posledn\'\i{}m p\v{r}edpokladem je aby \( \forall j,\: 0\leq j\leq s \) platili
rekurz\'\i{}vn\'\i{} relace
\[
W_{j+1}=\iota _{j}(W_{j})+T_{j}(V_{j+1}-\iota _{j}(V_{j}))+\left( e^{\Theta _{j}^{j}}-\frac{e^{\Theta ^{j}_{j}}-1}{\Theta _{j}^{j}}\right) \; \iota _{j}\; (1-\DD _{j})(W_{j}-\iota _{j-1}(W_{j-1}))\]

\[
\Theta _{j}^{j}(\: \iota _{j}\: (K_{j}+\DD _{j}(W_{j})\, ))=-\iota _{j}\, (1-\DD _{j})(W_{j}-\iota _{j-1}(W_{j-1}))\]
 

\( \;  \)

Nyn\'\i{} p\v{r}istoup\'\i{}me k vlastn\'\i{}mu induk\v{c}n\'\i{}mu kroku.

Jestli\v{z}e je O\( _{s+1} \) definov\'ano ve shod\v{e} s (\ref{defO}) pak
podobn\v{e} jako p\v{r}i odhadu O\( _{1} \)

se uk\'a\v{z}e
\[
\n O_{s+1}\n _{s+1}\leq \frac{\epsilon e}{\left( E_{s}\right) ^{\sigma }}<\infty \]
Tak\v{z}e O\( _{s+1}\in \gX _{s+1} \) a i
\[
V_{s+1}:=\iota _{s}V_{s}+O_{s+1}\in \gX _{s+1}\]
polo\v{z}me
\[
\dist (\Spec \left( k\omega -\Delta _{mn}+\left( W_{s}\right) _{0nn}\left( \omega \right) \right) ,\Spec \left( W_{s}\right) _{0mm}\left( \omega \right) )=:d_{s+1}\]


Definujme 
\[
\Omega _{s+1}:=\left\{ \omega \in \Omega _{s}|\; \forall m,n\in \N ,\, \: \forall k\in \Z ,\; \; d_{s+1}\geq \psi _{s+1}\left( k,m,n\right) \right\} \]


kde
\begin{eqnarray*}
\psi _{s+1}\left( k,m,n\right)  & := & \frac{\left| k\right| \Omega +\left| \Delta _{mn}\right| }{4}\; \; \; \; \; pro\, \: \frac{k\Omega }{\Delta _{mn}}\notin [\frac{1}{2},\, 2]\: a\: m\neq n\\
 & := & \phi _{s}\, e^{\frac{-\left| k\right| \rho _{s+1}}{2}\, }\sqrt{\left| k\right| }\, \left| \frac{\Delta _{mn}}{\delta }\right| \, \left( M_{m}\wedge M_{n}\right) ^{\frac{1}{2}}\; \; jinak
\end{eqnarray*}


\( \rho _{s+1}=\frac{1}{E_{s}}-\frac{1}{E_{s+1}} \)

Definujeme prostor 
\[
\gX _{s+1}\subset l^{\infty }\left( \Omega _{s+1}\times \Z \times \N \times \N ,\sum _{n\in \N }\sideset {}{^{\oplus }}\sum _{m\in \N }\LL (\HH _{m},\HH _{n})\right) \]
 jeho\v{z} prvky tvo\v{r}\'\i{} matice se slo\v{z}kami
\[
X_{knm}(\omega )\in \LL (\HH _{M_{m}},\HH _{M_{n}})\, ,\, \; \omega \in \Omega _{s+1}\]
 kter\'e maj\'\i{} kone\v{c}nou normu 


\[
\n X\n _{s+1}=\sup _{\substack {\omega ,\omega '\in \Omega _{s+1}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |X_{knm}(\omega )|+\phi _{s+1}\, |\partial X_{knm}(\omega ,\omega ')|\right) e^{|k|/E_{s+1}}\]


Zavedeme \( \Gamma _{s} \)
\[
\Gamma _{s}:=\iota _{s}\: ad_{K_{s}+\DD _{s}\left( W_{s}\right) }^{-1}\left( 1-\DD _{s}\right) \; :\gX _{s}\to \gX _{s+1}\]


kde v\'yraz na prav\'e stran\v{e} ch\'apeme ve smyslu (\ref{defcondGama0}), pouze
index 0 nahrad\'\i{}me s.

Ov\v{e}\v{r}\'\i{}me, \v{z}e \( \left( \Gamma _{s}\right) _{knm}\left( \omega \right)  \)
je pro ka\v{z}d\'e \( \left( \omega ,k,n,m\right) \in \Omega _{s}\times \Z \times \N \times \N  \)
, kde \( k\neq 0 \) nebo \( n\neq m \) invertibiln\'\i{} a v tomto p\v{r}\'\i{}pad\v{e}
plat\'\i{} 
\[
\left| \partial \left( \Gamma _{s}\right) ^{-1}_{knm}\left( \omega ,\omega '\right) \right| \leq \left| k\right| +\frac{1}{2}\]


Definujeme
\[
\left( \Gamma ^{-1}_{s}Y\right) _{knm}\left( \omega \right) :=\left( k\omega -\Delta _{mn}+\left( W_{s}\right) _{0nn}\left( \omega \right) \right) Y_{knm}\left( \omega \right) -Y_{knm}\left( \omega \right) \left( W_{s}\right) _{0mm}\left( \omega \right) ,\quad \omega \in \Omega _{1}\]


Prost\'ym dosazen\'\i{}m op\v{e}t zjist\'\i{}me, \v{z}e jde o inverzn\'\i{} matici.

Podobn\v{e} jako v p\v{r}\'\i{}pad\v{e} s=0 dostaneme p\v{r}edpis 
\[
\partial \left( \Gamma _{s}^{-1}\right) _{knm}\left( \omega ,\omega '\right) \left( Y_{knm}\left( \omega '\right) \right) =\left( k+\partial \left( W_{s}\right) _{0nn}\left( \omega ,\omega '\right) \right) Y_{knm}\left( \omega '\right) -Y_{knm}\left( \omega '\right) \left( \partial \left( W_{s}\right) _{knm}\left( \omega ,\omega '\right) \right) \]


Z toho plyne odhad 
\[
\left| \partial \left( \Gamma _{s}\right) ^{-1}_{knm}\left( \omega ,\omega '\right) \right| \leq \left| k\right| +\left| \partial \left( W_{s}\right) _{0nn}\left( \omega ,\omega '\right) \right| +\left| \partial \left( W_{s}\right) _{0mm}\left( \omega ,\omega '\right) \right| \]


Proto budeme p\v{r}epokl\'adat pro ka\v{z}d\'e \( n\in \N ;\: \omega ,\omega '\in \Omega _{s} \)

\begin{equation}
\label{derivace_W}
\left| \partial \left( W_{s}\right) _{0nn}\left( \omega ,\omega '\right) \right| \leq \frac{1}{4}.
\end{equation}
 Platnost t\'eto podm\'\i{}nky, posta\v{c}uj\'\i{}c\'\i{} pro konvergenci ov\v{e}\v{r}\'\i{}me
pro konkr\'etn\'\i{} \( \phi _{s} \) a \( E_{s} \) v z\'av\v{e}ru.

Potom se stejn\v{e} jako u s=0 dok\'a\v{z}e, \v{z}e
\[
\left| \Gamma _{s}\right| \leq \frac{5}{2\phi _{s}}\]
Definujme zobrazen\'\i{} \( \Theta _{s}^{s}:\C K_{s+1}\oplus \gX _{s+1}\to \C K_{s+1}\oplus \gX _{s+1} \)
op\v{e}t stejn\'ym p\v{r}edpisem pro K\( _{s+1} \) 

i pro \( X\in \gX _{s+1} \) a sice:
\begin{eqnarray*}
\Theta _{s}^{s}\left( X\right)  & := & \left[ \Gamma _{s}\left( W_{s}-\iota _{s-1}\left( W_{s-1}\right) \right) ,\: X\right] \\
 & = & \left[ \iota _{s}\: ad_{K_{s}+\DD _{s}\left( W_{s}\right) }^{-1}\left( 1-\DD _{s}\right) \left( W_{s}-\iota _{s-1}\left( W_{s-1}\right) \right) ,\: X\right] 
\end{eqnarray*}
 a snadn\'ymi \'upravami zjist\'\i{}me, \v{z}e
\[
\left| \Theta _{s}^{s}/_{\gX _{s+1}}\right| \leq 2\, \left| \Gamma _{s}\right| \, \n W_{s}-\iota _{s-1}\left( W_{s-1}\right) \n _{s+1}\leq \frac{5}{\phi _{s}}w_{s}\]


Ov\v{e}\v{r}en\'\i{} (\ref{defcondTheta}) bude zase velmi podobn\'e p\v{r}\'\i{}padu
s=0.

Pro \( 0\leq j\leq s \) definujeme \( \Theta _{s}^{j}:\C K_{s+1}\oplus \gX _{s+1}\to \C K_{s+1}\oplus \gX _{s+1} \)
pro \( X\in \gX _{s+1} \) i \( X\in \C K_{s+1} \) vztahem
\[
\Theta _{s}^{j}\left( X\right) :=\left[ \iota _{s+1,j}\: ad_{K_{j}+\DD _{j}\left( W_{j}\right) }^{-1}\left( 1-\DD _{j}\right) \left( W_{j}-\iota _{j-1}\left( W_{j-1}\right) \right) ,\: X\right] \]


D\'\i{}ky (\ref{vlastnostii}) dostaneme stejn\'y odhad pro normu
\[
\left| \Theta _{s}^{j}/_{\gX _{s+1}}\right| \leq 2\, \left| \Gamma _{j}\right| \, \n W_{j}-\iota _{j-1}\left( W_{j-1}\right) \n _{j}\leq \frac{5}{\phi _{j}}w_{j}\]


a z t\'eho\v{z} plyne i
\begin{equation}
\label{[theta,i]}
\Theta _{s}^{j}\, \iota _{s+1,v+1}=\iota _{s+1,v+1}\Theta _{v}^{j}
\end{equation}


s vyu\v{z}it\'\i{}m (kde M je libovoln\'a mno\v{z}ina na kterou z\'u\v{z}ujeme)
\[
\left( X\, Y\right) /_{M}=X/_{M}\, Y/_{M}\]
 Polo\v{z}me
\[
T_{s}=e^{\Theta ^{s}_{s}}e^{\Theta ^{s-1}_{s}}\dots e^{\Theta ^{0}_{s}}:\C K_{s+1}\oplus \gX _{s+1}\rightarrow \C K_{s+1}\oplus \gX _{s+1}\]
a kone\v{c}n\v{e}
\[
W_{s+1}=\iota _{s}(W_{s})+T_{s}(V_{s+1}-\iota _{s}(V_{s}))+\left( e^{\Theta ^{s}_{s}}-\frac{e^{\Theta ^{s}_{s}}-1}{\Theta ^{s}_{s}}\right) \; \iota _{s}\; (1-\DD _{s})(W_{s}-\iota _{s-1}(W_{s-1}))\]


T\'\i{}mto jsme ukon\v{c}ili induk\v{c}n\'\i{} krok. 

Nyn\'\i{} zn\'ame posloupnosti prostor\r{u} \( \gX _{s} \) matice \( V_{s},W_{s} \)
zobrazen\'\i{} \( \Theta ^{s}_{u} \) i \( \iota _{us} \) a \( \DD _{s} \) .

Posledn\'\i{}m krokem t\'eto \v{c}\'asti bude konstukce induktivn\'\i{} limity \( \gX _{\infty } \)
a vyu\v{z}it\'\i{} abstraktn\'\i{} \v{c}\'asti pro n\'a\v{s} probl\'em.


\subsection{\protect\( \gX _{\infty }\protect \)}

Ke konstrukci \( \gX _{\infty } \)u\v{z}ijeme postupu ze za\v{c}\'atku algebraick\'e
\v{c}\'asti. 

Proto\v{z}e \( \iota _{us}\left( X\right) =X/_{\Omega _{u}}\; \; \textrm{pro}\; X\in \gX _{s} \)
tak na\v{s}e relace ekvivalence \( \sim  \) na \( \widetilde{\gX } \) p\v{r}edstaven\'em
na za\v{c}\'atku m\'a tvar
\[
X_{s}\sim Y_{u}\Leftrightarrow \exists v\geq s,u\quad X_{s}/_{\Omega _{v}}=Y_{u}/_{\Omega _{v}}\quad \textrm{pro}\; X_{s}\in \gX _{s},Y_{u}\in \gX _{u}\]
T\v{r}\'\i{}da podle t\'eto ekvivalence obsahuj\'\i{}c\'\i{} prvek \( X_{s}\in \gX _{s} \)
se d\'a vyj\'ad\v{r}it 
\[
\left\lceil X_{s}\right\rceil =\left\{ X_{s}/_{\Omega _{u}}\, |u\geq s\right\} .\]


Tak\v{z}e 
\[
\widetilde{\widetilde{\gX }}=\left\{ \left\lceil X_{s}\right\rceil |\, X_{s}\in \gX _{s}\right\} \]


Hodnotu seminormy p spo\v{c}teme pro \( X\in \gX _{s} \)
\begin{eqnarray*}
p\left( \left\lceil X\right\rceil \right)  & = & \limsup _{u\to \infty }\left\Vert X/_{\Omega _{u}}\right\Vert _{u}\\
 & = & \limsup _{u\to \infty }\sup _{\substack {\omega ,\omega '\in \Omega _{u}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |X_{knm}(\omega )|+\phi _{u}\, |\partial X_{knm}(\omega ,\omega ')|\right) e^{|k|/E_{u}}\\
 & \leq  & \sup _{\substack {\omega ,\omega '\in \Omega _{s}\cr \omega \neq \omega '}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }\left( |X_{knm}(\omega )|+\phi _{s}\, |\partial X_{knm}(\omega ,\omega ')|\right) e^{|k|/E_{s}}\\
 & = & \left\Vert X\right\Vert _{s}
\end{eqnarray*}
 \textbf{Tvrzen\'\i{}:} Jestli\v{z}e jsou spln\v{e}ny vlastnosti posloupnost\'\i{} \( E_{s},\phi _{s} \)
uveden\'e na za\v{c}\'atku konkretn\'\i{} \v{c}\'asti, tak plat\'\i{}: 
\[
p\left( \left\lceil X\right\rceil \right) =\limsup _{u\to \infty }\sup _{\substack {\omega \in \Omega _{u}}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }|X_{knm}(\omega )|\]


D\r{u}kaz je snadn\'y. Prav\'a strana rovnice z tvrzen\'\i{} je men\v{s}\'\i{} nebo rovna
lev\'e a zbytek je d\r{u}sledek p\v{r}edpoklad\r{u}.

Ozna\v{c}\'\i{}me 
\[
M:=\left\{ \left\lceil X\right\rceil \in \widetilde{\widetilde{\gX }}|\, p\left( \left\lceil X\right\rceil \right) =0\right\} \]
 

Potom 
\[
\gX =\widetilde{\widetilde{\gX }}/_{M}\; s\: normou\: \left\Vert \left\lceil \left\lceil X\right\rceil \right\rceil \right\Vert _{\gX }=p\left( \left\lceil X\right\rceil \right) =\limsup _{u\to \infty }\sup _{\substack {\omega \in \Omega _{u}}}\, \sup _{n\in \N }\, \sum _{k\in \Z }\sum _{m\in \N }|X_{knm}(\omega )|\]
 a zobazen\'\i{} \( \iota _{\infty s}/_{\gX _{s}}\left( X\right) :=\left\lceil \left\lceil X\right\rceil \right\rceil  \)
spln\'\i{} \v{r}et\v{e}zov\'e pravidlo i odhad normy:
\[
\left| \iota _{\infty s}/_{\gX _{s}}\right| \leq 1\; \forall s\in \N \]


\( \left( \gX _{\infty },\left\Vert \; \right\Vert \right)  \) z\'\i{}sk\'ame pomoc\'\i{}
standartn\'\i{} z\'upl\v{n}ovac\'\i{} procedury.

Jestli\v{z}e \( \DD _{s} \) definujeme jako v induk\v{c}n\'\i{}m kroku vztahem
(\ref{def_Diags}), potom \( \left\{ \DD _{s}\right\} _{s=0}^{\infty },\left\{ 1-\DD _{s}\right\} _{s=0}^{\infty } \)
jsou posloupnosti omezen\'ych oper\'ator\r{u} s vlastnostmi 
\[
\DD _{s}\, \iota _{st}/_{\gX _{t}}=\iota _{st}/_{\gX _{t}}\, \DD _{t}\; \; \textrm{pro}\; 0\leq t\leq s\]


a tot\'e\v{z} plat\'\i{} i o \( 1-\DD _{s} \), tak\v{z}e existuje induktivn\'\i{} limita
t\v{e}chto oper\'ator\r{u}, pro kterou plat\'\i{}
\[
\DD \, \iota _{\infty t}/_{\gX _{t}}=\iota _{\infty t}/_{\gX _{t}}\, \DD _{t}\; \; \textrm{pro}\; 0\leq t\]


A d\'\i{}ky (\ref{Vlastnosti_indukt.lim}) z\r{u}stane v platnosti 
\[
\left| \DD \right| \leq 1,\; \left| 1-\DD \right| \leq 1\]


T\'\i{}m je konstrukce \( \gX _{\infty } \) dokon\v{c}ena. 


\subsection{Z\'av\v{e}r:}

\begin{lem}
Nech\q{t} je spln\v{e}na podm\'\i{}nka (\ref{2.gap}) pro n\v{e}jak\'e \( r>0 \).
A pro 
\[
\sigma :=\left[ r+\frac{3}{2}\right] \]

\end{lem}
kde \( \left[ \: \right]  \) zna\v{c}\'\i{} celou \v{c}\'ast, je spln\v{e}na podm\'\i{}nka
(\ref{epsilon}) na \( \epsilon  \).

Jestli\v{z}e polo\v{z}\'\i{}me 
\[
\phi _{s}:=\frac{\epsilon R_{2}}{a^{sr}}s^{\alpha },\; E_{s}:=a^{s}R_{1}\]


kde \( R_{1},R_{2},a,\alpha  \) jsou zat\'\i{}m libovoln\'a re\'aln\'a \v{c}\'\i{}sla a \( \epsilon  \)
je definovan\'e vztahem (\ref{epsilon}).

Posloupnost \( W_{s} \) je definovan\'a pomoc\'\i{} induk\v{c}n\'\i{}ho kroku pro tyto
posloupnosti \( \phi _{s},E_{s} \). 

Tak potom existuje \( \epsilon _{0}>0 \) , \v{z}e plat\'\i{} li pro \( \epsilon  \)
definovan\'e (\ref{epsilon}), \v{z}e \( \epsilon >\epsilon _{0} \), lze konstanty
\( R_{1},R_{2},a,\alpha  \) volit tak, \v{z}e mno\v{z}ina 
\[
\Omega _{\infty }:=\bigcap _{s=0}^{\infty }\Omega _{s}\]


je kladn\'e Lebesgueovy m\'\i{}ry. 

N\'aznak d\r{u}kazu je obsahem prvn\'\i{}ho dodatku.

\( \,  \)

P\v{r}edpokl\'adejme, \v{z}e \( \Omega _{\infty } \) je nepr\'azdn\'a mno\v{z}ina.

D\'ale p\v{r}edpokl\'adejme, \v{z}e \( K_{\infty }:\Omega _{\infty }\to \ZZ  \)
je definovan\'e vztahem
\[
\left( K_{\infty }\right) _{kjnm}\left( \omega \right) =\left( \omega k+H_{n}\right) \delta _{kj}\delta _{nm}\]


V m\'a kone\v{c}n\'e \( \epsilon  \) definovan\'e vztahem (\ref{epsilon}), 

existuj\'\i{} posloupnosti \( F_{s},v_{s} \) spl\v{n}uj\'\i{}c\'\i{} po\v{z}adavky (\ref{Thmconv_assumA}),
(\ref{Thmconv_assumB}) a 
\[
B\leq \frac{1}{3}\ln 2\quad \textrm{a}\quad A\, \phi (3C)\leq \frac{1}{9}\]
tedy nerovnost (\ref{ABd_ineq}) plat\'\i{} pro \( d=3 \). 

A nav\'\i{}c plat\'\i{} 
\[
\frac{5}{F_{0}}\leq \min \left\{ \frac{1}{4}\Omega ,\frac{1}{4}\delta \right\} \]
 Potom tvrd\'\i{}me, \v{z}e existuje \( W\in L^{\infty } \), posloupnost vektor\r{u}
\( A_{s}\in \gX _{s} \) spl\v{n}uj\'\i{}c\'\i{}ch
\[
\left( A_{s}\right) _{knm}\left( \omega \right) =-\left( \left( \overline{A_{s}}\right) _{-kmn}\left( \omega \right) \right) ^{t}\quad \forall \omega \in \Omega _{s}\]
 

kde pruh ozna\v{c}uje kompexn\'\i{} sdru\v{z}en\'\i{} a t znamen\'a transponov\'an\'\i{} matice
\( \left( A_{s}\right) _{knm}\left( \omega \right) \in \LL \left( \HH _{M_{n}},\HH _{M_{m}}\right)  \) 

a plat\'\i{}
\[
\sum _{s=o}^{\infty }\left\Vert A_{s}\right\Vert _{s}<\infty .\]


Definujeme-li posloupnost zobrazen\'\i{} \( \,  \)\( \Theta _{u}^{s}:\C K_{u+1}\oplus \gX _{u+1}\to \gX _{u+1} \)
vztahem 
\[
\Theta _{u}^{s}\left( X\right) :=\left[ \iota _{u+1,s+1}A_{s+1},X\right] \quad \textrm{pro}\; u\geq s\]


Tak potom existuje zobrazen\'\i{} \( T:\C K_{\infty }\oplus \gX _{\infty }\to \C K_{\infty }\oplus \gX _{\infty } \)

\[
T=\lim _{s\to \infty }e^{\Theta _{s}^{s}}\dots e^{\Theta _{s}^{0}}\]


ve smyslu stejn\'em jako v z\'av\v{e}ru algebraick\'e \v{c}\'asti, 

a plat\'\i{} 
\[
T\left( K_{\infty }+V'\right) =K_{\infty }+\DD \left( W\right) \]


\begin{proof}
Definujeme-li posloupnosti \( \phi _{s} \) a E\( _{s} \) vztahem
\[
\phi _{s}:=\frac{5}{F_{s}},\; E_{s}:=\frac{\epsilon ^{\frac{1}{\sigma }}}{e\, \left( v_{s}\right) ^{\frac{1}{\sigma }}}\]


provedeme induk\v{c}kn\'\i{} krok, tak algebraick\'a \v{c}\'ast n\'am zaru\v{c}uje konvergenci
pro
\begin{equation}
\label{As}
A_{s}:=\Gamma _{s}\left( W_{s}-\iota _{s-1}\left( W_{s-1}\right) \right) =\iota _{s}\: ad_{K_{s}+\DD _{s}\left( W_{s}\right) }^{-1}\left( 1-\DD _{s}\right) \left( W_{s}-\iota _{s-1}\left( W_{s-1}\right) \right) \in \gX _{s+1}
\end{equation}


Mus\'\i{}me v\v{s}ak je\v{s}t\v{e} ov\v{e}\v{r}it podm\'\i{}nku, \v{z}e pro ka\v{z}d\'e
\( n\in \N ;\: \omega ,\omega '\in \Omega _{s} \) 
\[
\left| \partial \left( W_{s}\right) _{0nn}\left( \omega ,\omega '\right) \right| \leq \frac{1}{4}.\]


Proto\v{z}e (\( \partial W_{0}=\partial V_{0}=0 \))
\begin{eqnarray*}
\left| \partial \left( W_{s}\right) _{0nn}\left( \omega ,\omega '\right) \right|  & \leq  & \left| \sum _{k=1}^{s}\partial \left( W_{k}\right) _{0nn}\left( \omega ,\omega '\right) -\partial \left( W_{k-1}\right) _{0nn}\left( \omega ,\omega '\right) +0\right| \\
 & \leq  & \sum _{k=1}^{s}\frac{1}{\phi _{k}}\phi _{k}\left| \partial \left( W_{k}\right) _{0nn}\left( \omega ,\omega '\right) -\partial \left( W_{k-1}\right) _{0nn}\left( \omega ,\omega '\right) \right| \\
 & \leq  & \frac{1}{5}\sum _{k=1}^{s}F_{k}\left| W_{k}-\iota _{k-1}W_{k-1}\right| \leq \frac{1}{5}\sum _{k=0}^{\infty }F_{k}w_{k}\leq \frac{dB}{5}
\end{eqnarray*}
 

Ale podle p\v{r}edpoklad\r{u} \( d=3,B\leq \frac{1}{3}\ln 2 \), tak\v{z}e
\( \frac{dB}{5}\leq \frac{\ln 2}{5}<\frac{1}{4}. \) T\'\i{}m je podm\'\i{}nka ov\v{e}\v{r}ena.

\v{Z}e \( A_{s} \) maj\'\i{} vlastnost po\v{z}adovanou tvrzen\'\i{}m v\v{e}ty se dok\'a\v{z}e
indukc\'\i{} pomoc\'\i{} n\'asleduj\'\i{}c\'\i{} \'uvahy;

tato vlastnost vlastn\v{e} p\v{r}edstavuje antihermitovost matice \( A_{s} \),
kterou jsme definovali jako inverzi komut\'atoru. Proto\v{z}e v\v{s}echny \( V_{s} \)
jsou hermitovsk\'e (jsou pouhou \v{c}\'ast\'\i{} potenci\'alu V) tak \( A_{0} \), kter\'e
m\'a vyhovovat vztahu
\[
\left( 1-\DD _{0}\right) V_{0}\left( \omega \right) =\left[ K_{0}+\DD _{0}\left( V_{0}\right) ,A_{0}\right] \left( \omega \right) \quad pro\; \omega \in \Omega _{1}\]


mus\'\i{} b\'yt antihermitovsk\'e (\( K_{s} \) je samozdru\v{z}en\'a pro v\v{s}echna
s ).

\( W_{1} \) mus\'\i{} b\'yt z rekurzivn\'\i{} definice hermitovsk\'a, proto\v{z}e \( \Theta ^{0}_{0} \)
p\r{u}sob\'\i{} jako komut\'ator s antihermitovsk\'ym \( A_{0} \) . Indukc\'\i{} se pokra\v{c}uje
stejn\v{e} pro \( W_{s} \) a \( A_{s} \).
\end{proof}

\section{Limita v \protect\( \BB \left( \KK \right) \protect \)}

\textbf{Tvrzen\'\i{}:} Nech\q{t} \( \mathbf{X} \) je omezen\'y oper\'ator na \( \KK  \).
Definujeme-li slo\v{z}ky \textbf{X} p\v{r}edpisem (\ref{def_slozek})

tak pro oper\'atorovou normu \textbf{X} plat\'\i{} odhad 
\[
\n \mathbf{X}\n \leq \max \left\{ \sup _{k\in \Z ,m\in \N }\sum _{l\in \Z }\sum _{n\in \N }\left| X_{klmn}\right| ,\sup _{l\in \Z ,n\in \N }\sum _{k\in \Z }\sum _{m\in \N }\left| X_{klmn}\right| \right\} .\]


D\r{u}kaz je velice podobn\'y odhadu pro Schur-Holmgrenovu normu.

\( \:  \)

Definujme Banach\r{u}v prostor tvo\v{r}en\'y oper\'atorov\'ymi funkcemi
\[
L^{\infty }:=\left\{ \mathbf{X}:\Omega _{\infty }\to \BB \left( \KK \right) |\sup _{\omega \in \Omega _{\infty }}\n \mathbf{X}\left( \omega \right) \n =:\n \mathbf{X}\n _{\infty }<\infty \right\} \]


Pro \( X\in \gX _{s} \) hermitovskou matice ve smyslu
\[
\left( X\right) _{kmn}\left( \omega \right) =\left( \overline{\left( X\right) _{-knm}}\right) ^{t}\left( \omega \right) \quad \forall k\in \Z ,\forall m,n\in \N ,\forall \omega \in \Omega _{s}\]
 

definujeme zobrazen\'\i{} \( \kappa _{s} \) do \( L^{\infty } \) , kter\'e matici
X p\v{r}i\v{r}ad\'\i{} oper\'ator \textbf{X} pomoc\'\i{} slo\v{z}ek.

Toto zobrazen\'\i{} je dob\v{r}e definov\'ano, proto\v{z}e
\begin{eqnarray*}
\n \mathbf{X}\left( \omega \right) \n  & \leq  & \max \left\{ \sup _{m\in \N }\sum _{k\in \Z }\sum _{n\in \N }\left| X_{kmn}\left( \omega \right) \right| ,\sup _{n\in \N }\sum _{k\in \Z }\sum _{m\in \N }\left| X_{kmn}\left( \omega \right) \right| \right\} \\
 & = & \sup _{n\in \N }\sum _{k\in \Z }\sum _{m\in \N }\left| X_{kmn}\left( \omega \right) \right| 
\end{eqnarray*}
 Z toho plynou odhady (normy \( \left\Vert \: \right\Vert ,\left\Vert \: \right\Vert _{\infty } \)
jsou definov\'any jako v induk\v{c}n\'\i{}m kroku) 
\[
\n \mathbf{X}\n _{\infty }\leq \n \iota _{\infty s}\left( X\right) \n \leq \n X\n _{s}\]
 \v{C}ili toto zobrazen\'\i{} definuje omezen\'y oper\'ator, kter\'y je nav\'\i{}c hermitovsk\'y.

Z p\v{r}edchoz\'\i{}ho lze tak\'e vid\v{e}t, \v{z}e lze toto p\v{r}i\v{r}azen\'\i{}
prov\'est i pro \( \left\lceil \left\lceil X\right\rceil \right\rceil \in \gX  \),
jeho\v{z} reprezentant je hermitovsk\'y a je nez\'avisl\'e na v\'yb\v{e}ru reprezentanu
z obou t\v{r}\'\i{}d. Ozna\v{c}\'\i{}me ho \( \kappa  \)\( _{\infty } \). 

Tato zobrazen\'\i{} dodefinujeme op\v{e}t i na podprostor \( \C K_{s} \) 
\begin{equation}
\label{operator_K}
\left( \kappa _{s}K_{s}\right) \left( \omega \right) :=\mathbf{K}\left( \omega \right) =\sum _{k\in \Z }\sum _{n\in \N }\left( \omega k+H_{n}\right) P_{k}\otimes Q_{n}\quad pro\; \omega \in \Omega _{\infty },s\in \N \cup \left\{ \infty \right\} 
\end{equation}


Z p\v{r}edpisu tak\'e plyne, \v{z}e pro ka\v{z}dou hermitovskou \( X\in \gX _{s} \)

\begin{equation}
\label{kapa}
\kappa _{s}\left( X\right) =\kappa _{\infty }\iota _{\infty s}\left( X\right) \quad a\quad \kappa _{s}\left( X\right) =\kappa _{u}\left( \iota _{us}X\right) ,\forall s\leq u
\end{equation}


\begin{lem}
Nech\q{t} \( K_{s},V_{s},\gX _{s} \) jsou definov\'any jako v induk\v{c}n\'\i{}m
kroku 

\textbf{a K} je definov\'ano vztahem (\ref{operator_K}).

Jestli\v{z}e \( T_{s-1} \) je zobrazen\'\i{} \( \C K_{s}\oplus \gX _{s} \) do
sebe definovan\'e p\v{r}edpisem 
\[
T_{s-1}=e^{\Theta _{s-1}^{s-1}}\dots e^{\Theta _{s-1}^{0}}\]
kde
\[
\Theta _{u}^{s}\left( X\right) =\left[ \iota _{u+1,s+1}A_{s+1},X\right] \quad \textrm{pro}\; u\geq s\]


a \( A_{s+1} \) je antihermitovsk\'a z \( \gX _{s+1} \).

Tak potom pro unit\'arn\'\i{} zobrazen\'\i{} \( U_{s}\in L^{\infty } \) definovan\'e na \( \KK  \)
p\v{r}edpisem 
\begin{equation}
\label{Us}
U_{s}\left( \omega \right) =e^{-i\kappa _{s+1}\left( iA_{s+1}\right) \left( \omega \right) }\dots e^{-i\kappa _{1}\left( iA_{1}\right) \left( \omega \right) }
\end{equation}


plat\'\i{}, \v{z}e \( U_{s} \) zachov\'av\'a defini\v{c}n\'\i{} obor \textbf{\( \mathbf{K}\left( \omega \right)  \),}
tedy
\[
U_{s}\left( \omega \right) \Dom \left( \mathbf{K}\left( \omega \right) \right) =\Dom \left( \mathbf{K}\left( \omega \right) \right) \quad \forall \omega \in \Omega _{\infty }\]


a pro ka\v{z}d\'e \( v \) z defini\v{c}n\'\i{}ho oboru \textbf{K\( \left( \omega \right)  \)}
a \( \omega \in \Omega _{\infty } \) plat\'\i{}
\[
\left\{ \kappa _{s}T_{s-1}\left( K_{s}+V_{s}\right) \right\} \left( \omega \right) \: \left( v\right) =U_{s}\left( \mathbf{K}\left( \omega \right) +\kappa _{s}V_{s}\right) U_{s}^{-1}\: \left( v\right) \]


D\r{u}kaz je v dodatku.
\end{lem}
\textbf{D\r{u}sledek :} Jestli\v{z}e, posloupnosti \( V_{s},W_{s},\Theta _{u}^{s} \)
jsou zvoleny jako v induk\v{c}n\'\i{}m kroku, \( A_{s} \) je definov\'ano vztahem
(\ref{As}) a aplikujeme-li zobrazen\'\i{} \( \kappa _{\infty } \) na rovnost (\ref{lim_T(K+V)})
dost\'av\'ame \( \forall v\in \Dom \left( \mathbf{K}\left( \omega \right) \right)  \)
d\'\i{}ky (\ref{kapa}) na lev\'e stran\v{e} 


\begin{eqnarray*}
\kappa _{s}\left\{ \widetilde{T_{s}}\left( K_{\infty }+\iota _{\infty s}\left( V_{s}\right) \right) \right\} \left( \omega \right) \, \left( v\right)  & = & \kappa _{\infty }\iota _{\infty s}T_{s-1}\left( K_{s}+V_{s}\right) \left( \omega \right) \, \left( v\right) \\
 & = & U_{s}\left( \omega \right) \left( \mathbf{K}\left( \omega \right) +\kappa _{s}V_{s}\right) U_{s}^{-1}\left( \omega \right) \: \left( v\right) 
\end{eqnarray*}


a na prav\'e stran\v{e}
\begin{eqnarray*}
\kappa _{\infty }\left\{ K_{\infty }+\DD \left( \iota _{\infty s}W_{s}\right) +\left( 1-\DD \right) \left( \iota _{\infty s}W_{s}-\iota _{\infty s-1}W_{s-1}\right) \right\} \left( \omega \right) \, \left( v\right) =\; \qquad \qquad  &  & \\
\mathbf{K}\left( \omega \right) \, v+\kappa _{\infty }\DD \left( \iota _{\infty s}W_{s}\right) \left( \omega \right) \, \left( v\right) +\kappa _{\infty }\left( 1-\DD \right) \left( \iota _{\infty s}W_{s}-\iota _{\infty s-1}W_{s-1}\right) \left( \omega \right) \, \left( v\right)  &  & 
\end{eqnarray*}


Proto\v{z}e 
\begin{eqnarray*}
\n \kappa _{s}V_{s}-\mathbf{V}\n _{\infty } & \leq  & \sup _{n\in \N }\sum _{k\in \Z }\sum _{m\in \N }\left| \left( V_{s}\right) _{kmn}-\left( V\right) _{kmn}\right| \\
 & = & \sup _{n\in \N }\sum _{\left| k\right| \geq E_{s}}\sum _{m\in \N }\left| \left( V\right) _{kmn}\right| \to 0
\end{eqnarray*}
 

posloupnost \( \kappa _{s}V_{s} \) konverguje k potenci\'alu \textbf{V} v oper\'atorov\'e
norm\v{e}.

D\'ale pro posloupnost zobrazen\'\i{} \( U_{s} \) definovan\'e vztahem (\ref{Us}),
plat\'\i{} odhad 
\begin{eqnarray*}
\n U_{s}-U_{t}\n _{\infty } & \leq  & \left\{ \exp \left( \sum _{j=t-1}^{s}\n \kappa _{j}A_{j}\n _{\infty }\right) -1\right\} \exp \left( \sum _{j=0}^{t}\n \kappa _{j}A_{j}\n _{\infty }\right) \\
 & \leq  & \exp \left( \sum _{j=0}^{s}\n A_{j}\n _{j}\right) -\exp \left( \sum _{j=0}^{t}\n A_{j}\n _{j}\right) 
\end{eqnarray*}
 Ale proto\v{z}e 
\[
\sum _{s=0}^{\infty }\n A_{s}\n _{s}<\infty \]


tak existuje U, \v{z}e \( U_{s}\to U \) v prostoru \( L^{\infty } \)

Tak\v{z}e shrneme-li, existuje posloupnost unit\'arn\'\i{}ch zobrazen\'\i{} \( U_{s} \)
, kter\'e maj\'\i{} v \( L^{\infty } \) limitu U. D\'ale pro ka\v{z}d\'e \( s\in \N  \)
plat\'\i{} 
\[
U_{s}\left( \omega \right) \Dom \left( \mathbf{K}\left( \omega \right) \right) =\Dom \left( \mathbf{K}\left( \omega \right) \right) ,\; kde\; \omega \in \Omega _{\infty }\]


Existuje i posloupnost \( \kappa _{s}V_{s} \), kter\'a m\'a v \( L^{\infty } \)
za limitu \textbf{V} a posloupnost \( B_{s}\in L^{\infty } \), jdouc\'\i{} v limit\v{e}
k B v \( L^{\infty } \), jeho\v{z} matice definovan\'a po slo\v{z}k\'ach vztahem
(\ref{def_slozek}) je diagon\'aln\'\i{}. 

Tyto posloupnosti jsou takov\'e, \v{z}e pro ka\v{z}d\'e \( \omega \in \Omega _{\infty } \)
, pro ka\v{z}d\'e \( v\in \Dom \mathbf{K}\left( \omega \right)  \) plat\'\i{} rovnost


\[
U_{s}\left( \omega \right) \left( \mathbf{K}\left( \omega \right) +\kappa _{s}V_{s}\right) U_{s}^{-1}\left( \omega \right) \, \left( v\right) =\mathbf{K}\left( \omega \right) v+B_{s}\left( \omega \right) v\]


\begin{lem}
Nech\q{t} \( B_{s},R_{s}\in L^{\infty } \) konverguj\'\i{} v \( L^{\infty } \)
\( B_{s}\to B,\; R_{s}\to R \) a nech\q{t} \( \mathbf{K}\left( \omega \right)  \) 

je uzav\v{r}en\'y oper\'ator na \( \KK  \) pro ka\v{z}d\'e \( \omega \in \Omega _{\infty } \). 

P\v{r}edpokl\'adejme, \v{z}e \( U_{s}\in L^{\infty } \) jsou unit\'arn\'\i{} zobrazen\'\i{},
kter\'e maj\'\i{} v \( L^{\infty } \) limitu U.

D\'ale po\v{z}adujeme pro ka\v{z}d\'e \( s\geq 0,\omega \in \Omega _{\infty } \)
\[
U_{s}\left( \omega \right) \Dom \left( \mathbf{K}\left( \omega \right) \right) =\Dom \left( \mathbf{K}\left( \omega \right) \right) \]
 

a pro ka\v{z}d\'e \( s\geq 0,\omega \in \Omega _{\infty },v\in \Dom \left( \mathbf{K}\left( \omega \right) \right)  \)
rovnost
\[
U_{s}\left( \omega \right) \left( \mathbf{K}\left( \omega \right) +R_{s}\left( \omega \right) \right) U_{s}^{-1}\left( \omega \right) v=\mathbf{K}\left( \omega \right) v+B_{s}\left( \omega \right) v.\]


Potom tvrd\'\i{}me: 
\[
U\Dom \left( \mathbf{K}\left( \omega \right) \right) =\Dom \left( \mathbf{K}\left( \omega \right) \right) ,\quad \forall \omega \in \Omega _{\infty }\]
 a pro ka\v{z}d\'e \( \omega \in \Omega _{\infty },v\in \Dom \left( \mathbf{K}\left( \omega \right) \right)  \)
plat\'\i{} rovnost
\[
U\left( \mathbf{K}\left( \omega \right) +R\left( \omega \right) \right) U^{-1}v=\mathbf{K}\left( \omega \right) v+B\left( \omega \right) v\]

\end{lem}
\begin{proof}
Je snadn\'ym d\r{u}sledkem uzav\v{r}enosti \textbf{K}. (Budeme pro p\v{r}ehlednost
v\v{s}ude vynech\'avat \( \omega  \)). 
\[
x\in \Dom \mathbf{K}\Rightarrow U_{s}^{-1}x\to U^{-1}x,\; a\; \mathbf{K}U_{s}^{-1}x\to U^{-1}\mathbf{K}x+U^{-1}Bx-RU^{-1}x\]
 z toho plyne, \v{z}e
\[
U^{-1}x\in \Dom \mathbf{K},\; \mathbf{K}U^{-1}x=U^{-1}\mathbf{K}x+U^{-1}Bx-RU^{-1}x\]
z \v{c}eho\v{z} ihned plyne druh\'e tvrzen\'\i{} v\v{e}ty. 

V\'\i{}me, \v{z}e pro ka\v{z}d\'e \( y\in \Dom \mathbf{K} \) je i \( U_{s}y\in \Dom \mathbf{K} \)
pro ka\v{z}d\'e \( s. \) Tedy polo\v{z}\'\i{}me-li \( v:=U_{s}y \) a vyu\v{z}ijeme
p\v{r}edpokladu, z\'\i{}sk\'ame:

Proto\v{z}e 
\[
U_{s}y\to Uy,\; a\; \mathbf{K}U_{s}y\to U\mathbf{K}y+URy-BUy\]


z uzav\v{r}enosti \textbf{K} plyne, \v{z}e \( Uy\in \Dom \mathbf{K} \) a

\[
\mathbf{K}Uy=U\mathbf{K}y+URy-BUy\]
 Spojen\'\i{}m obou 
\begin{eqnarray*}
x\in \Dom \mathbf{K}\Rightarrow U^{-1}x\in \Dom \mathbf{K} &  & \\
y\in \Dom \mathbf{K}\Rightarrow Uy\in \Dom \mathbf{K}\quad  &  & 
\end{eqnarray*}


tak\v{z}e
\[
U\Dom \mathbf{K}=\Dom \mathbf{K}\]

\end{proof}

\section{Hlavn\'\i{} v\v{e}ta}

\textbf{Teor\'em:}

Nech\q{t} H je oper\'ator na separabiln\'\i{}m Hilbertov\v{e} prostoru \( \HH  \)
a m\'a \v{c}ist\v{e} bodov\'e spektrum. P\v{r}i\v{c}em\v{z} ka\v{z}d\'a vlastn\'\i{}
hodnota \( H_{n} \) m\'a kone\v{c}nou degeneraci \( M_{n} \). P\v{r}edpkl\'adejme,
\v{z}e pro tyto hodnoty plat\'\i{} podm\'\i{}nka (\ref{1.gap}) a \v{z}e existuje kladn\'e
\v{c}\'\i{}slo \( r \), pro kter\'e plat\'\i{} i druh\'a podm\'\i{}nka (\ref{2.gap}).

Nech\q{t} je d\'ale d\'ana siln\v{e} spojit\'a mno\v{z}ina V(t) 2\( \pi  \) periodick\'ych
hermitovsk\'ych oper\'ator\r{u} na \( \HH  \). 

P\v{r}edpokl\'adejme, \v{z}e Floquet\r{u}v Hamiltoni\'an
\[
K_{F}\left( \omega \right) :=-i\partial _{t}+H+V\left( \omega t\right) \]
 na prostoru \( \KK :=L^{2}\left( \left( 0,T\right) ,\HH ,dt\right)  \) s periodickou
hrani\v{c}n\'\i{} podm\'\i{}nkou v t a s kruhovou frekvenc\'\i{} \( \omega \in \Omega _{0} \),
kde \( \Omega _{0} \) je kompaktn\'\i{} interval a \( T=\frac{2\pi }{\omega } \). 

Jestli\v{z}e V(t) spln\'\i{} podm\'\i{}nku (\ref{epsilon}) pro \( \sigma =\left[ r+\frac{3}{2}\right]  \)
a \v{c}\'\i{}slo \( \epsilon  \) je dostate\v{c}n\v{e} mal\'e,

tak existuje mno\v{z}ina \( \Omega _{\infty }\subset \Omega _{0} \) kladn\'e
Lebesgueovy m\'\i{}ry, takov\'a \v{z}e pro ka\v{z}d\'e \( \omega \in \Omega _{\infty } \)
m\'a oper\'ator \( K_{F}\left( \omega \right)  \) \v{c}ist\v{e} bodov\'e spektrum.


\section{Dodatky}


\subsection{Odhad m\'\i{}ry \protect\( \Omega _{\infty }\protect \).}

Nech\q{t} je spln\v{e}na podm\'\i{}nka (\ref{2.gap}) pro n\v{e}jak\'e \( r>0 \).
A pro 
\[
\sigma :=\left[ r+\frac{3}{2}\right] \]


kde \( \left[ \: \right]  \) zna\v{c}\'\i{} celou \v{c}\'ast, je spln\v{e}na podm\'\i{}nka
(\ref{epsilon}).

Jestli\v{z}e polo\v{z}\'\i{}me 
\[
\phi _{s}:=\frac{\epsilon R_{2}}{a^{sr}}s^{\alpha },\; E_{s}:=a^{s}R_{1}\]


kde \( R_{1},R_{2},a,\alpha  \) jsou zat\'\i{}m libovoln\'a re\'aln\'a \v{c}\'\i{}sla a \( \epsilon  \)
je definovan\'e vztahem (\ref{epsilon}).

Posloupnost \( W_{s} \) je definovan\'a pomoc\'\i{} induk\v{c}n\'\i{}ho kroku pro tyto
posloupnosti \( \phi _{s},E_{s} \). 

Tak potom existuje \( \epsilon _{0}>0 \) , \v{z}e plat\'\i{} li pro \( \epsilon  \)
definovan\'e (\ref{epsilon}), \v{z}e \( \epsilon >\epsilon _{0} \), lze konstanty
\( R_{1},R_{2},a,\alpha  \) volit tak, \v{z}e mno\v{z}ina
\[
\Omega _{\infty }:=\bigcap _{s=0}^{\infty }\Omega _{s}\]


je kladn\'e Lebesgueovy m\'\i{}ry. 

\begin{proof}
D\r{u}kaz pouze nazna\v{c}\'\i{}me, p\v{r}i\v{c}em\v{z} n\'am p\r{u}jde o zji\v{s}t\v{e}n\'\i{}
posta\v{c}uj\'\i{}c\'\i{} podm\'\i{}nky na stupe\v{n} diferencovatelnosti potenci\'alu \( V \),
aby cel\'y algoritmus dob\v{r}e konvergoval. Konkr\'etn\'\i{} po\v{c}\'\i{}t\'an\'\i{} t\v{e}chto
konstant nen\'\i{} ambic\'\i{} t\'eto pr\'ace. 

V dal\v{s}\'\i{}m tedy p\v{r}edpokl\'ad\'ame, \v{z}e konstanty tak \( R_{1},R_{2},a,\alpha  \)
lze volit tak, \v{z}e je spln\v{e}na podm\'\i{}nka (\ref{derivace_W}) a nav\'\i{}c
plat\'\i{} pro ka\v{z}d\'e \( s,m\in \N ,\omega \in \Omega _{s} \)


\begin{equation}
\label{podminkaW}
\left| \left( W_{s}\right) _{0mm}\left( \omega \right) \right| \leq \frac{1}{6}\Delta _{0}
\end{equation}
 

Ozna\v{c}me \( \Omega _{s}^{bad}:=\Omega _{0}\setminus \Omega _{s} \)

Pro \( \left( k,n,m\right) \in \Z \times \N \times \N  \) definujeme mno\v{z}inu
\( \Omega _{s+1}^{bad}\left( k,n,m\right)  \) n\'asleduj\'\i{}c\'\i{}m p\v{r}edpisem:

\begin{eqnarray*}
\omega \in \Omega _{s+1}^{bad}\left( k,n,m\right) \Leftrightarrow \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad  &  & \\
d_{s+1}=\dist (\Spec \left( k\omega -\Delta _{mn}+\left( W_{s}\right) _{0nn}\left( \omega \right) \right) ,\Spec \left( W_{s}\right) _{0mm}\left( \omega \right) )<\psi _{s+1}\left( k,n,m\right)  &  & 
\end{eqnarray*}
 kde \( \psi _{s+1} \) je definov\'ano p\v{r}edpisem jako v induk\v{c}n\'\i{}m kroku

\begin{eqnarray*}
\psi _{s+1} & := & \frac{\left| k\right| \Omega +\left| \Delta _{mn}\right| }{4}\; \; \; \; \; pro\, \frac{k\Omega }{\Delta _{mn}}\notin [\frac{1}{2},\, 2]\: a\: m\neq n\\
 & := & \phi _{s}\, e^{\frac{-\left| k\right| \rho _{s+1}}{2}\, }\sqrt{\left| k\right| }\, \left| \frac{\Delta _{mn}}{\delta }\right| \, \left( M_{m}\wedge M_{n}\right) ^{\frac{1}{2}}\; \; jinak
\end{eqnarray*}


Potom plat\'\i{}:

(1) \( \Omega _{s+1}^{bad}\left( k,n,m\right) =\Omega _{s+1}^{bad}\left( -k,m,n\right)  \)
co\v{z} je ihned vid\v{e}t z definic. 

(2) \( \Omega _{s+1}^{bad}\left( k,n,m\right) =\emptyset  \) pro \( \left( k,n,m\right)  \)
nekritick\'y index 


\begin{eqnarray*}
\frac{\left| k\right| \Omega +\left| \Delta _{mn}\right| }{4} & = & \psi _{s+1}\left( k,n,m\right) \\
 & > & \dist (\Spec \left( k\omega -\Delta _{mn}+\left( W_{s}\right) _{0nn}\left( \omega \right) \right) ,\Spec \left( W_{s}\right) _{0mm}\left( \omega \right) )\\
 & \geq  & |k|\omega +|\Delta _{mn}|-\frac{1}{3}\Delta _{0}\geq \frac{8}{9}|k|\Omega +\frac{1}{2}|\Delta _{mn}|+\frac{1}{2}\Delta _{0}-\frac{1}{3}\Delta _{0}\\
 & > & \frac{|k|\Omega +|\Delta _{mn}|}{2}
\end{eqnarray*}
 kde jsme vyu\v{z}ili: p\v{r}i prvn\'\i{} \'uprav\v{e} \( \dist \left( \Spec \left( \alpha I+A\right) ,\Spec \left( B\right) \right) \geq |\alpha |-\n A\n -\n B\n  \),
platn\'e pro ka\v{z}d\'e komlexn\'\i{} \v{c}\'\i{}slo \( \alpha  \), a libovoln\'e kone\v{c}n\'e
matice \( A,B \), d\'ale (\ref{podminkaW}) a definice \( \Omega _{0},\Delta _{0} \)

(3) \( \left| \Omega _{s+1}^{bad}\left( k,n,m\right) \right| \leq \frac{4\psi _{s+1}}{k}M_{m}M_{n} \),
pro \( \left( k,n,m\right)  \) kritick\'y index (z definice \( k\neq 0 \)),
\( M_{m} \) je degenerace vlastn\'\i{} hodnoty \( H_{m} \).

V kapitole 1. paragrafu 6.10 \cite{4} je dok\'az\'ano n\'asleduj\'\i{}c\'\i{} lemma:

Nech\q{t} \( A \) je zobrazen\'\i{} kter\'e \v{c}\'\i{}slu \( \omega \in \Omega \subset \R  \)
p\v{r}i\v{r}ad\'\i{} hermitovskou matici rozm\v{e}ru \( N \).

\( \lambda _{1}\left( \omega \right) <\lambda _{1}\left( \omega \right) <\dots <\lambda _{N}\left( \omega \right)  \)
jsou vlastn\'\i{} hodnoty A(\( \omega  \)). 

Potom jestli\v{z}e A, jako maticov\'a funkce \( \omega  \), je Lipschitzovsk\'a
s konstantou L, tak pro ka\v{z}d\'e \( i\in \widehat{N} \)

je \( \lambda _{i} \) jako\v{z}to funkce \( \omega \in \Omega  \) Lipschitzovsk\'a
s konstantou L.

Nyn\'\i{} uspo\v{r}\'ad\'ame v\v{s}echny r\r{u}zn\'e vlastn\'\i{} hodnoty \( \lambda _{i}^{n}\left( \omega \right)  \)
matice \( \left( W_{s}\right) _{onn}\left( \omega \right)  \), kde \( \omega \in \Omega _{s} \).

Potom 
\[
\Omega _{s+1}^{bad}\left( k,n,m\right) \subset \bigcup _{\substack {1\leq i\leq M_{m}\cr 1\leq j\leq M_{n}}}\Omega _{s+1}^{bad}\left( k,n,m,i,j\right) \]


kde 
\[
\Omega _{s+1}^{bad}\left( k,n,m,i,j\right) =\left\{ \omega \in \Omega _{s}|\: \left| \omega k-\Delta _{mn}+\lambda _{i}^{n}\left( \omega \right) -\lambda _{i}^{m}\left( \omega \right) \right| <\psi _{s+1}\right\} \]


co\v{z} se dok\'a\v{z}e rozeps\'an\'\i{}m. (St\'ale p\v{r}edpokl\'ad\'ame kritick\'y index).

Potom pro \( \omega ,\omega '\in \Omega _{s+1}^{bad}\left( k,n,m,i,j\right) ,\; \omega \neq \omega ' \)
plat\'\i{} 
\begin{eqnarray*}
\frac{2\psi _{s+1}}{\left| \omega -\omega '\right| } & \geq  & \inf _{k,m,n,i,j,\omega ,\omega '}\left| \frac{\omega k-\Delta _{mn}+\lambda _{i}^{n}\left( \omega \right) -\lambda _{i}^{m}\left( \omega \right) -\left( \omega 'k-\Delta _{mn}+\lambda _{i}^{n}\left( \omega '\right) -\lambda _{i}^{m}\left( \omega '\right) \right) }{\omega -\omega '}\right| \\
 & \geq  & \inf _{k,m,n,i,j,\omega ,\omega '}\left| k+\frac{\lambda _{i}^{n}\left( \omega \right) -\lambda _{i}^{n}\left( \omega '\right) }{\omega -\omega '}-\frac{\lambda _{i}^{m}\left( \omega \right) -\lambda _{i}^{m}\left( \omega '\right) }{\omega -\omega '}\right| 
\end{eqnarray*}
 A podle ocitovan\'eho lemmatu s vyu\v{z}it\'\i{}m (\ref{derivace_W}) dost\'av\'ame 
\[
\frac{2\psi _{s+1}}{\left| \omega -\omega '\right| }\geq |k|-\frac{1}{2}\geq \frac{1}{2}|k|\]


A odtud ji\v{z} plyne 
\[
\left| \Omega _{s+1}^{bad}\left( k,n,m\right) \right| \leq M_{m}M_{n}\frac{4\psi _{s+1}}{k}.\]
 Co\v{z} jsme cht\v{e}li uk\'azat.

(4) D\'\i{}ky bod\r{u}m (1), (2), (3) plat\'\i{} 
\[
\left| \Omega _{s+1}^{bad}\right| =\left| \bigcup _{k,n,m}\Omega _{s+1}^{bad}\left( k,n,m\right) \right| \leq 2\sum _{\substack {n,m\in \N ,k>0\cr \frac{k\Delta _{mn}}{\Omega }\in [\frac{1}{2}\Omega ,2\Omega ]}}4\phi _{s}k^{-\frac{1}{2}}e^{-\rho _{s+1}\frac{k}{2}}\frac{\Delta _{mn}}{\delta }\mu _{mn}\]
 \( \mu _{mn}:=M_{m}M_{n}\left( \min \left( M_{n},M_{m}\right) \right) ^{\frac{1}{2}} \)

V dal\v{s}\'\i{}ch \'uprav\'ach se budou p\v{r}ed sumou objevovat konstanty. Jak jsme
v\v{s}ak ji\v{z} \v{r}ekli nebudeme se o n\v{e} zaj\'\i{}mat a budeme je v\v{s}echny
ozna\v{c}ova shodn\v{e} symbolem \( C\left( \sigma ,\Omega ,\delta \right)  \),
v z\'avorce je z\'avislost na jin\'ych konstant\'ach.

Nyn\'\i{} odhadneme sumu p\v{r}es \( k \), odhadem po\v{c}tu \v{c}len\r{u} kr\'at
nejv\v{e}t\v{s}\'\i{} \v{c}len. Dost\'av\'ame 
\[
\left| \Omega _{s+1}^{bad}\right| \leq C\left( \sigma ,\Omega ,\delta \right) \phi _{s}\sum _{\substack {n,m\in \N ,\cr \frac{\Delta _{mn}}{\Omega }>\frac{1}{2}}}\mu _{mn}\left( \frac{\Delta _{mn}}{4\Omega }\right) ^{-r}\left( \frac{\Delta _{mn}}{4\Omega }\right) ^{r+\frac{3}{2}}e^{-\rho _{s+1}\frac{\Delta _{mn}}{4\Omega }}\]


Nyn\'\i{} vyu\v{z}ijeme \( \sup _{x>0}x^{\alpha }e^{-\beta x}=\left( \frac{\alpha }{e\beta }\right) ^{a} \)

\[
\left| \Omega _{s+1}^{bad}\right| \leq C\left( \sigma ,\Omega ,\delta \right) \phi _{s}\left( \frac{1}{\rho _{s+1}}\right) ^{r+\frac{3}{2}}\sum _{\substack {n,m\in \N ,\cr \frac{\Delta _{mn}}{\Omega }>\frac{1}{2}}}\mu _{mn}\left( \Delta _{mn}\right) ^{-r}\]
Tak\v{z}e pou\v{z}ijeme-li definici \( \mu _{mn} \) a (\ref{2.gap}) z\'\i{}sk\'ame
kone\v{c}n\v{e} odhad
\[
\left| \Omega _{s+1}^{bad}\right| \leq C\left( \sigma ,\Omega ,\delta \right) \phi _{s}\left( \frac{1}{\rho _{s+1}}\right) ^{r+\frac{3}{2}}\Delta _{r}^{-r}\left( \Omega \right) \]
 Nyn\'\i{} dosad\'\i{}me za na\v{s}e posloupnosti a budeme vy\v{s}et\v{r}ovat existenci
\[
\sum _{s=0}^{\infty }\phi _{s}\left( \frac{1}{\rho _{s+1}}\right) ^{r+\frac{3}{2}}=K\left( R_{1},R_{2},a,\alpha ,\sigma ,r,\epsilon \right) \sum _{s=0}^{\infty }s^{\alpha }\left( a^{-s+1}\right) ^{\sigma -\left( r+\frac{3}{2}\right) }\]
 \( K\left( R_{1},R_{2},a,\alpha ,\sigma ,r,\epsilon \right)  \) je konstanta
z\'avisej\'\i{}c\'\i{} na uveden\'ych parametrech.

Nyn\'\i{} jestli\v{z}e bude spln\v{e}na podm\'\i{}nka (a je to podm\'\i{}nka pro diferencovatelnost
\( V \))
\[
\sigma =\left[ r+\frac{3}{2}\right] \]


kde \( r \) je ur\v{c}eno podm\'\i{}nkou (\ref{2.gap}). a \( \left[ \: \right]  \)
zna\v{c}\'\i{} celou \v{c}\'ast, tak suma na prav\'e stran\v{e} v\'yrazu
\[
\left| \bigcup _{s=0}^{\infty }\Omega _{s+1}^{bad}\right| \leq C\left( \sigma ,\Omega ,\delta \right) \sum _{s=0}^{\infty }\phi _{s}\left( \frac{1}{\rho _{s+1}}\right) ^{r+\frac{3}{2}}\]


bude kone\v{c}n\'a a volbou dostate\v{c}n\v{e} mal\'eho \( \epsilon  \) lze
zajistit, \v{z}e nebude v\v{e}t\v{s}\'\i{} ne\v{z} d\'elka intervalu \( \Omega _{0} \).

Tak\v{z}e m\'\i{}ra
\[
\left| \Omega _{\infty }\right| =\left| \bigcap _{s=0}^{\infty }\Omega _{s}\right| \geq \left| \Omega _{0}\setminus \bigcup _{s=0}^{\infty }\Omega _{s+1}^{bad}\right| >0\]

\end{proof}

\subsection{D\r{u}kaz lemmatu 8. }

\( \,  \)

\textbf{Lemma 8}. \textit{Nech\q{t} \( K_{s},V_{s},\gX _{s} \) jsou definov\'any
jako v induk\v{c}n\'\i{}m kroku }

\textit{a} \textbf{\textit{K}} \textit{je definov\'ano vztahem (\ref{operator_K}).}

\textit{Jestli\v{z}e \( T_{s-1} \) je zobrazen\'\i{} \( \C K_{s}\oplus \gX _{s} \)
do sebe definovan\'e p\v{r}edpisem}
\[
T_{s-1}=e^{\Theta _{s-1}^{s-1}}\dots e^{\Theta _{s-1}^{0}}\]


\textit{kde
\[
\Theta _{u}^{s}\left( X\right) =\left[ \iota _{u+1,s+1}A_{s+1},X\right] \quad pro\; u\geq s\]
}

\textit{a \( A_{s+1} \) je antihermitovsk\'a z \( \gX _{s+1} \).}

\textit{Tak potom pro unit\'arn\'\i{} zobrazen\'\i{} \( U_{s} \) definovan\'e na \( \KK  \)}

\[
U_{s}\left( \omega \right) =e^{-i\kappa _{s+1}\left( iA_{s+1}\right) \left( \omega \right) }\dots e^{-i\kappa _{1}\left( iA_{1}\right) \left( \omega \right) }\]


\textit{plat\'\i{}, \v{z}e pro ka\v{z}d\'e \( \omega \in \Omega _{\infty } \) \( U_{s}\left( \omega \right)  \)
zachov\'av\'a defini\v{c}n\'\i{} obor} \textbf{\textit{\( \mathbf{K}\left( \omega \right)  \),}}
\textit{tedy
\[
U_{s}\left( \omega \right) \Dom \left( \mathbf{K}\left( \omega \right) \right) =\Dom \left( \mathbf{K}\left( \omega \right) \right) \]
}

\textit{a pro ka\v{z}d\'e v z defini\v{c}n\'\i{}ho oboru} \textbf{\textit{K\( \left( \omega \right)  \)}}
\textit{a \( \omega \in \Omega _{\infty } \) plat\'\i{}
\[
\left\{ \kappa _{s}T_{s-1}\left( K_{s}+V_{s}\right) \right\} \left( \omega \right) \: \left( v\right) =U_{s}\left( \omega \right) \left( \mathbf{K}\left( \omega \right) +\kappa _{s}V_{s}\right) U_{s}^{-1}\left( \omega \right) \: \left( v\right) \]
}

\begin{proof}
Zobrazen\'\i{} \( \kappa _{s} \) je pro \( iA_{s} \) dob\v{r}e definovan\'e, proto\v{z}e
\( A_{s} \) je antihermitovsk\'a.

\( -i\kappa _{s}\left( iA_{s}\right)  \) budeme ozna\v{c}ovat \( \mathbf{A}_{s} \)

Pro p\v{r}ehlednost budeme v\v{s}ude vynech\'avat \( \omega  \).

D\'\i{}ky (\ref{[theta,i]}) sta\v{c}\'\i{}, kdy\v{z} dok\'a\v{z}eme 
\[
\kappa _{s}\left( e^{\Theta _{s-1}^{s-1}}V_{s}\right) =e^{-i\kappa _{s}\left( iA_{s}\right) }\left( \kappa _{s}V_{s}\right) e^{i\kappa _{s}\left( iA_{s}\right) }\]


d\'ale
\[
e^{\Theta _{s-1}^{s-1}}\Dom \mathbf{K}=\Dom \mathbf{K}\]
 

a \v{z}e pro ka\v{z}d\'e \( v\in \Dom \mathbf{K} \) plat\'\i{}
\[
\kappa _{s}\left( e^{\Theta _{s-1}^{s-1}}K_{s}\right) \, \left( v\right) =e^{\mathbf{A}_{s}}\mathbf{K}e^{-\mathbf{A}_{s}}\, v\]


Nejprve pro \( V_{s}. \) 


\begin{eqnarray*}
\kappa _{s}\left( e^{\Theta _{s-1}^{s-1}}V_{s}\right)  & =\kappa _{s}\left( \sum _{n=0}^{\infty }\frac{\left( \Theta _{s-1}^{s-1}\right) ^{n}}{n!}V_{s}\right) =\kappa _{s}\left( \sum _{n=0}^{\infty }\frac{\left( ad_{A_{s}}\right) ^{n}}{n!}V_{s}\right) 
\end{eqnarray*}


\( \left( ad_{A_{s}}\right) ^{n}X=[A_{s},[A_{s},\dots [A_{s},X]\dots ]] \)
po\v{c}et komut\'ator\r{u} je n, a \( ad^{0}:=1 \). 

Potom proto\v{z}e \( \kappa _{s} \) je spojit\'e zobrazen\'\i{} a proto\v{z}e matici
n\'asobku dvou matic odpov\'\i{}d\'a p\v{r}i zobrazen\'\i{} \( \kappa _{s} \) oper\'ator slo\v{z}en\'\i{}
dost\'av\'ame
\begin{eqnarray*}
\kappa _{s}\left( e^{\Theta _{s-1}^{s-1}}V_{s}\right)  & = & \sum _{n=0}^{\infty }\kappa _{s}[A_{s},[A_{s},\dots [A_{s},V_{s}]\dots ]]=\sum _{n=0}^{\infty }[\mathbf{A}_{s},[\mathbf{A}_{s},\dots [\mathbf{A}_{s},\kappa _{s}V_{s}]\dots ]]\\
 & = & \sum _{n=0}^{\infty }\frac{\left( ad_{\mathbf{A}_{s}}\right) ^{n}}{n!}\kappa _{s}V_{s}
\end{eqnarray*}
 a tvrzen\'\i{} pro \( V_{s} \) plyne ji\v{z} ze zn\'am\'e formule platn\'e pro ka\v{z}d\'e
\( A,B\in \BB \left( \KK \right)  \) 
\[
e^{ad_{A}}B=e^{A}Be^{-A}\]


Nyn\'\i{} pro \textbf{K.} Nejprve budeme cht\'\i{}t uk\'azat, \v{z}e
\begin{equation}
\label{A_na_DomK}
\mathbf{A}_{s}\left( \Dom \mathbf{K}\right) \subset \Dom \mathbf{K}
\end{equation}


K tomu vyu\v{z}ijeme n\'asleduj\'\i{}c\'\i{}ho tvrzen\'\i{}, je\v{z} je vlastn\v{e} bezprost\v{r}edn\'\i{}m
d\r{u}sledkem uzav\v{r}enosti \textbf{K.}

\textbf{Tvrzen\'\i{}:} P\v{r}edpokl\'adejme, \v{z}e \textbf{A,B} jsou \textbf{}omezen\'e
oper\'atory na \( \KK  \), \textbf{K} je uzav\v{r}en\'y na \( \KK  \).

Nech\q{t} existuje nejv\'y\v{s}e spo\v{c}etn\'a mno\v{z}ina \( \left\{ e_{i}\right\} \subset \Dom \mathbf{K} \)
takov\'a \v{z}e 

(1) \( \forall v\in \Dom \mathbf{K},\quad \exists v_{n}=\sum _{i\in J}\xi _{i}^{n}e_{i}, \)
kde J je kone\v{c}n\'a a \( v_{n}\to v,\: \mathbf{K}v_{n}\to \mathbf{K}v \)

(2) \( \forall i\quad Ae_{i}\in \Dom \mathbf{K}\quad a\; \mathbf{K}Ae_{i}=A\mathbf{K}e_{i}+\mathbf{B}e_{i} \) 

potom 
\[
A\Dom \mathbf{K}\subset \Dom \mathbf{K},\quad a\; \forall v\in \Dom \mathbf{K},\; \mathbf{K}Av=A\mathbf{K}v+\mathbf{B}v\]


Odtud ji\v{z} snadno plyne (\ref{A_na_DomK}), sta\v{c}\'\i{} za \( \left\{ e_{i}\right\}  \)
zvolit vlastn\'\i{} b\'azi \textbf{K}, kter\'a zaru\v{c}uje spln\v{e}n\'\i{} prvn\'\i{}ho p\v{r}edpokladu,
\( \mathbf{A}:=\mathbf{A}_{s} \) a vyu\v{z}\'\i{}t (\ref{defcondTheta}). 

Tak\v{z}e \textbf{B} polo\v{z}\'\i{}me jako \( \mathbf{B}:=\kappa _{s}\left\{ -\iota _{s-1}\left( 1-\DD _{s-1}\right) \left( W_{s-1}-\iota _{s-2}W_{s-2}\right) -[A_{s,}\iota _{s-1}\DD _{s-1}W_{s-1}]\right\}  \) 

Je\v{s}t\v{e} budeme pot\v{r}ebovat identitu platnou pro \( A,B\in \BB \left( \KK \right)  \)
\begin{equation}
\label{help_identita}
\sum _{n=1}^{\infty }\frac{\left( -1\right) ^{n-1}}{n!}\sum _{j=0}^{n-1}A^{j}BA^{n-1-j}=e^{-A}\frac{e^{ad_{A}}-1}{ad_{A}}B
\end{equation}


V\'yraz na prav\'e stran\v{e} je op\v{e}t definov\'an mocninou \v{r}adou. D\r{u}kaz
se provede indukc\'\i{}.

Nyn\'\i{} nech\q{t} \( v\in \Dom \mathbf{K} \) z toho tedy podle ji\v{z} dok\'azan\'eho
(\ref{A_na_DomK}) plat\'\i{}
\[
v_{N}:=\sum _{n=0}^{N}\frac{\left( -1\right) ^{n}}{n!}\left( \mathbf{A}_{s}\right) ^{n}\in \Dom \mathbf{K},\quad \forall N\in \N \]
 

A plat\'\i{} 
\[
\mathbf{K}v_{N}=\sum _{n=0}^{N}\frac{\left( -1\right) ^{n}}{n!}\left( \mathbf{K}\left( \mathbf{A}_{s}\right) ^{n}v-\left( \mathbf{A}_{s}\right) ^{n}\mathbf{K}v\right) +\left( \sum _{n=0}^{N}\frac{\left( -1\right) ^{n}}{n!}\left( \mathbf{A}_{s}\right) ^{n}\right) \mathbf{K}v\]


\v{C}len pro n=0 je u prvn\'\i{} sumy nulov\'y.

Rozeps\'an\'\i{}m lze snadno ov\v{e}\v{r}it, \v{z}e pro \( n\geq 1 \) plat\'\i{}
\[
\mathbf{K}\left( \mathbf{A}_{s}\right) ^{n}v-\left( \mathbf{A}_{s}\right) ^{n}\mathbf{K}v=\sum _{j=0}^{n-1}\left( \mathbf{A}_{s}\right) ^{j}\left[ \mathbf{K},\mathbf{A}_{s}\right] \left( \mathbf{A}_{s}\right) ^{n-1-j}v\]
 Ozna\v{c}\'\i{}me-li \( \mathbf{B}:=[\mathbf{A}_{s},\mathbf{K}] \) co\v{z} jak
v\'\i{}me je podle p\v{r}edchoz\'\i{}ho tvrzen\'\i{} omezen\'y oper\'ator na \( \KK  \), z\'\i{}sk\'ame.
\[
\mathbf{K}v_{N}=\sum _{n=1}^{N}\frac{\left( -1\right) ^{n+1}}{n!}\sum _{j=0}^{n-1}\left( \mathbf{A}_{s}\right) ^{j}\mathbf{B}\left( \mathbf{A}_{s}\right) ^{n-1-j}+\left( \sum _{n=0}^{N}\frac{\left( -1\right) ^{n}}{n!}\left( \mathbf{A}_{s}\right) ^{n}\right) \mathbf{K}v\]
 Proto\v{z}e \( v_{N}\to e^{-\mathbf{A}_{s}}v \) a podle (\ref{help_identita })
plat\'\i{}
\[
\mathbf{K}v_{N}\to e^{-\mathbf{A}_{s}}\frac{e^{ad_{\mathbf{A}_{s}}}-1}{ad_{\mathbf{A}_{s}}}\mathbf{B}v+e^{-\mathbf{A}_{s}}\mathbf{K}v\]
 tak s vyu\v{z}it\'\i{}m uzav\v{r}enosti \textbf{K} dost\'av\'ame 
\[
e^{-\mathbf{A}_{s}}v\in \Dom \mathbf{K},\quad a\; \mathbf{K}e^{-\mathbf{A}_{s}}v=e^{-\mathbf{A}_{s}}\frac{e^{ad_{\mathbf{A}_{s}}}-1}{ad_{\mathbf{A}_{s}}}ad_{\mathbf{A}_{s}}\mathbf{K}v+e^{-\mathbf{A}_{s}}\mathbf{K}v\]


z toho plyne 

\( e^{-\mathbf{A}_{s}}\Dom \mathbf{K}\subset \Dom \mathbf{K}\Rightarrow \Dom \left( \mathbf{K}+\frac{e^{ad_{\mathbf{A}_{s}}}-1}{ad_{\mathbf{A}_{s}}}ad_{\mathbf{A}_{s}}\mathbf{K}\right) =\Dom \left( e^{\mathbf{A}_{s}}\mathbf{K}e^{-\mathbf{A}_{s}}\right) =\Dom \mathbf{K} \)

a pro ka\v{z}d\'e \( v\in \Dom \mathbf{K} \) rovnost 
\[
e^{\mathbf{A}_{s}}\mathbf{K}e^{-\mathbf{A}_{s}}v=e^{ad_{\mathbf{A}_{s}}}\mathbf{Kv}\]


Jestli\v{z}e ud\v{e}l\'ame limitu z druh\'e strany, tak podobn\v{e} dostaneme,
\v{z}e i 
\[
e^{-\mathbf{A}_{s}}\Dom \mathbf{K}\subset \Dom \mathbf{K}\]


Tak\v{z}e m\'ame 
\[
e^{\mathbf{A}_{s}}\Dom \mathbf{K}=\Dom \mathbf{K}.\]


K dokon\v{c}en\'\i{} d\r{u}kazu u\v{z}ijeme 
\begin{eqnarray*}
\kappa _{s}\left( e^{ad_{A_{s}}}K_{s}\right) v & = & \kappa _{s}\left( K_{s}+\sum _{n=1}^{\infty }\frac{\left( ad_{A_{s}}\right) ^{n}}{n!}K_{s}\right) v=\mathbf{K}v+\left( \sum _{n=1}^{\infty }\kappa _{s}\frac{\left( ad_{A_{s}}\right) ^{n}}{n!}K_{s}\right) v\\
 & = & \mathbf{K}v+\left( \sum _{n=1}^{\infty }\frac{\left( ad_{\mathbf{A}_{s}}\right) ^{n}}{n!}\mathbf{K}\right) v=\left( e^{ad_{\mathbf{A}_{s}}}\mathbf{K}\right) v
\end{eqnarray*}
 postup je podobn\'y jako u \( V_{s} \), nyn\'\i{} jsme v\v{s}ak vyu\v{z}ili faktu,
\v{z}e \( ad_{\mathbf{A}_{s}}\mathbf{K} \) jak u\v{z} v\'\i{}me je omezen\'y oper\'ator.

T\'\i{}m je d\r{u}kaz dokon\v{c}en.
\end{proof}
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\end{document}
