Terminado análisis funcional (tema 3)

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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\usepackage{commath}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
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+\use_package amsmath 1
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+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
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+\tracking_changes false
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+\html_math_output 0
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+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Algunos operadores acotados en espacios de Hilbert:
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ espacios prehilbertianos y 
+\begin_inset Formula $G$
+\end_inset
+
+ de dimensión finita con base 
+\begin_inset Formula $(e_{i})_{i}$
+\end_inset
+
+, todo homomorfismo 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ es acotado con
+\begin_inset Formula 
+\[
+\Vert T\Vert\leq\sqrt{\sum_{i}\Vert Te_{i}\Vert^{2}}.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con bases ortonormales 
+\begin_inset Formula $(e_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(f_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{a_{n}\}_{n}\subseteq\mathbb{K}$
+\end_inset
+
+ una sucesión acotada, el 
+\series bold
+operador diagonal
+\series default
+ 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+T(x)\coloneqq\sum_{n=1}^{\infty}a_{n}\langle x,e_{n}\rangle f_{n}
+\]
+
+\end_inset
+
+es acotado con 
+\begin_inset Formula $\Vert T\Vert=\sup_{n}|a_{n}|$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+, el 
+\series bold
+operador multiplicación por 
+\begin_inset Formula $g$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $T:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ dado por 
+\begin_inset Formula $Tf\coloneqq gf$
+\end_inset
+
+, es acotado con 
+\begin_inset Formula $\Vert T\Vert=\Vert g\Vert_{\infty}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con bases ortonormales respectivas 
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(v_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+ una matriz infinita con 
+\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<\infty$
+\end_inset
+
+, 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+T(x)\coloneqq\sum_{i,j}a_{ij}\langle x,u_{i}\rangle v_{j}
+\]
+
+\end_inset
+
+es un operador acotado con 
+\begin_inset Formula $\Vert T\Vert\leq\sqrt{\sum_{i,j}|a_{ij}|^{2}}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+, el 
+\series bold
+operador integral con núcleo 
+\begin_inset Formula $k$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $K:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+K(f)(t)\coloneqq\int_{a}^{b}k(t,s)f(s)\dif s,
+\]
+
+\end_inset
+
+es acotado con 
+\begin_inset Formula $\Vert K\Vert\leq\sqrt{\iint_{[a,b]\times[a,b]}|k|^{2}}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Una matriz infinita 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+ satisface el 
+\series bold
+test de Schur
+\series default
+ si existen 
+\begin_inset Formula $C,D\in\mathbb{R}$
+\end_inset
+
+ tales que
+\begin_inset Formula 
+\begin{align*}
+\forall i\in\mathbb{N},\sum_{j}|a_{ij}| & \leq C, & \forall j\in\mathbb{N}, & \sum_{i}|a_{ij}|\leq D.
+\end{align*}
+
+\end_inset
+
+Entonces, si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con bases ortonormales respectivas 
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(v_{n})_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+T(x)\coloneqq\sum_{i,j}a_{ij}\langle x,u_{i}\rangle v_{j}
+\]
+
+\end_inset
+
+es un operador acotado con 
+\begin_inset Formula $\Vert T\Vert\leq\sqrt{CD}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $k:[a,b]\times[a,b]\to\mathbb{K}$
+\end_inset
+
+ medible y 
+\begin_inset Formula $C,D\in\mathbb{R}$
+\end_inset
+
+ tales que
+\begin_inset Formula 
+\begin{align*}
+\forall t\in[a,b],\int_{a}^{b}|k(t,s)|\dif s & \leq C, & \forall s\in[a,b], & \int_{a}^{b}|k(t,s)|\dif t\leq D,
+\end{align*}
+
+\end_inset
+
+entonces 
+\begin_inset Formula $K:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ dada por
+\begin_inset Formula 
+\[
+K(f)(t)\coloneqq\int_{a}^{b}k(t,s)f(s)\dif s
+\]
+
+\end_inset
+
+es un operador acotado con 
+\begin_inset Formula $\Vert K\Vert\leq\sqrt{CD}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ con base ortonormal 
+\begin_inset Formula $(e_{n})_{n}$
+\end_inset
+
+, para 
+\begin_inset Formula $T\in L(H)$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in H$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+T(x)=\sum_{i,j}\langle x,e_{j}\rangle\langle Te_{j},e_{i}\rangle e_{i},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $T$
+\end_inset
+
+ admite una representación matricial 
+\begin_inset Formula $(\langle Te_{j},e_{i}\rangle)_{i,j}\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $T\in L(X,Y)$
+\end_inset
+
+ es 
+\series bold
+de rango finito
+\series default
+ si 
+\begin_inset Formula $\dim\text{Im}T<\infty$
+\end_inset
+
+.
+ Dados espacios de Hilbert 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $T\in L(G,H)$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es de rango finito si y sólo si viene dada por 
+\begin_inset Formula $T(x)=\sum_{i=1}^{n}\langle x,u_{i}\rangle v_{i}$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $u_{1},\dots,u_{n}\in G$
+\end_inset
+
+ y 
+\begin_inset Formula $v_{1},\dots,v_{n}\in H$
+\end_inset
+
+, en cuyo caso los 
+\begin_inset Formula $(v_{i})_{i}$
+\end_inset
+
+ pueden tomarse de forma que sean una base de 
+\begin_inset Formula $\text{Im}T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Inversión de operadores
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ son 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios normados, 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ y 
+\begin_inset Formula $S\in{\cal L}(Y,X)$
+\end_inset
+
+ cumplen 
+\begin_inset Formula $ST=1_{X}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $S$
+\end_inset
+
+ es el 
+\series bold
+inverso por la izquierda
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ y 
+\begin_inset Formula $T$
+\end_inset
+
+ es el 
+\series bold
+inverso por la derecha
+\series default
+ de 
+\begin_inset Formula $S$
+\end_inset
+
+, y 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ es 
+\series bold
+invertible
+\series default
+ si existe 
+\begin_inset Formula $T^{-1}\in{\cal L}(Y,X)$
+\end_inset
+
+ inverso de 
+\begin_inset Formula $T$
+\end_inset
+
+ por la izquierda y por la derecha.
+ Llamamos 
+\begin_inset Formula ${\cal L}(X)\coloneqq\text{End}_{\mathbb{K}}X={\cal L}(X,X)$
+\end_inset
+
+ e 
+\begin_inset Formula 
+\[
+\text{Isom}X\coloneqq\text{Isom}_{\mathbb{K}}(X)\coloneqq\{T\in{\cal L}(X)\mid T\text{ invertible}\}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de dimensión finita, 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+ tiene inverso por la izquierda si y sólo si lo tiene por la derecha, si
+ y sólo si es invertible.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+ Esto no es cierto en general en dimensión infinita; por ejemplo, el operador
+ 
+\series bold
+desplazamiento a derecha
+\series default
+, 
+\begin_inset Formula $S_{\text{r}}\in\ell^{2}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $S_{\text{r}}(x_{1},\dots,x_{n},\dots)\coloneqq(0,x_{1},\dots,x_{n},\dots)$
+\end_inset
+
+, tiene como inverso por la izquierda el 
+\series bold
+desplazamiento a izquierda
+\series default
+, 
+\begin_inset Formula $S_{\text{l}}\in\ell^{2}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $S_{\text{l}}(x_{1},\dots,x_{n},\dots)\coloneqq(x_{2},\dots,x_{n},\dots)$
+\end_inset
+
+, pero no tiene inverso por la derecha.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $T\in\text{End}_{\mathbb{K}}X$
+\end_inset
+
+, 
+\begin_inset Formula $\lambda\in\mathbb{K}$
+\end_inset
+
+ es un 
+\series bold
+valor regular
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ si 
+\begin_inset Formula $T-\lambda1_{X}$
+\end_inset
+
+ es invertible, un 
+\series bold
+valor espectral
+\series default
+ en otro caso, y un 
+\series bold
+valor propio
+\series default
+ si 
+\begin_inset Formula $\ker(T-\lambda1_{X})\neq0$
+\end_inset
+
+, en cuyo caso llamamos 
+\series bold
+subespacio propio
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ correspondiente al valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ a 
+\begin_inset Formula $\ker(T-\lambda1_{X})$
+\end_inset
+
+ y 
+\series bold
+valores propios
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ correspondientes al valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ a los elementos no nulos de este subespacio.
+ Llamamos 
+\series bold
+resolvente
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ al conjunto de sus valores regulares, 
+\series bold
+espectro
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma(T)$
+\end_inset
+
+, al conjunto de sus valores espectrales y 
+\series bold
+espectro puntual
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{\text{p}}(T)\subseteq\sigma(T)$
+\end_inset
+
+, al conjunto de sus valores propios.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $X$
+\end_inset
+
+ es de dimensión finita, 
+\begin_inset Formula $\sigma_{\text{p}}(T)=\sigma(T)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+ Sin embargo, 
+\begin_inset Formula $0\in\sigma(S_{\text{r}})$
+\end_inset
+
+ pero 
+\begin_inset Formula $\sigma_{\text{p}}(S_{\text{r}})=\emptyset$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach y 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+ cumple 
+\begin_inset Formula $\Vert T\Vert<1$
+\end_inset
+
+, 
+\begin_inset Formula $1_{X}-T$
+\end_inset
+
+ es invertible con inverso 
+\begin_inset Formula $\sum_{n\in\mathbb{N}}T^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert(1_{X}-T)^{-1}\Vert\leq\frac{1}{1-\Vert T\Vert}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{k=0}^{n}\Vert T^{k}\Vert\leq\sum_{k=0}^{n}\Vert T\Vert^{k}\leq\sum_{k\in\mathbb{N}}\Vert T\Vert^{n}=\frac{1}{1-\Vert T\Vert}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\sum_{n}\Vert T^{n}\Vert$
+\end_inset
+
+ converge y, por ser 
+\begin_inset Formula $X$
+\end_inset
+
+ de Banach, 
+\begin_inset Formula $S\coloneqq\sum_{n}T^{n}$
+\end_inset
+
+ también, pero 
+\begin_inset Formula $S(1_{X}-T)=S-ST=T^{0}=1_{X}$
+\end_inset
+
+ y análogamente 
+\begin_inset Formula $(1_{X}-T)S=1_{X}$
+\end_inset
+
+, luego 
+\begin_inset Formula $S=(1_{X}-T)^{-1}$
+\end_inset
+
+, y finalmente
+\begin_inset Formula 
+\[
+\Vert(1_{X}-T)^{-1}\Vert=\left\Vert \sum_{n}T^{n}\right\Vert \leq\sum_{n}\Vert T\Vert^{n}=\frac{1}{1-\Vert T\Vert}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de von Neumann:
+\series default
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach, 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+ invertible y 
+\begin_inset Formula $S\in{\cal L}(X)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\Vert T-S\Vert<\frac{1}{\Vert T^{-1}\Vert}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $S$
+\end_inset
+
+ es invertible con
+\begin_inset Formula 
+\begin{align*}
+S^{-1} & =\sum_{n\in\mathbb{N}}(T^{-1}(T-S))^{n}T^{-1}, & \left\Vert T^{-1}-S^{-1}\right\Vert  & \leq\frac{\Vert T^{-1}\Vert^{2}\Vert T-S\Vert}{1-\Vert T^{-1}\Vert\Vert T-S\Vert}.
+\end{align*}
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $\Vert T^{-1}(T-S)\Vert=\Vert T-S\Vert\Vert T^{-1}\Vert<1$
+\end_inset
+
+, luego por el teorema anterior 
+\begin_inset Formula $1_{X}-T^{-1}(T-S)=T^{-1}S$
+\end_inset
+
+ es invertible con
+\begin_inset Formula 
+\[
+(T^{-1}S)^{-1}=\sum_{n}(T^{-1}(T-S))^{n},
+\]
+
+\end_inset
+
+luego 
+\begin_inset Formula $S=T(T^{-1}S)$
+\end_inset
+
+ es invertible con inversa 
+\begin_inset Formula $(T^{-1}S)^{-1}T^{-1}$
+\end_inset
+
+ y
+\begin_inset Formula 
+\begin{align*}
+\Vert T^{-1}-S^{-1}\Vert & =\Vert T^{-1}-(T^{-1}S)^{-1}T^{-1}\Vert=\Vert(1_{X}-(T^{-1}S)^{-1})T^{-1}\Vert\leq\\
+ & \leq\left\Vert \left(1_{X}-\sum_{n}(T^{-1}(T-S))^{n}\right)T^{-1}\right\Vert =\left\Vert \sum_{n\geq1}(T^{-1}(T-S))^{n}T^{-1}\right\Vert \leq\\
+ & \leq\sum_{n\geq1}\Vert(T^{-1}(T-S))^{n}\Vert\Vert T^{-1}\Vert\leq\frac{\Vert T^{-1}\Vert^{2}\Vert T-S\Vert}{1-\Vert T^{-1}\Vert\Vert T-S\Vert}.
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $X$
+\end_inset
+
+ es un espacio de Banach, 
+\begin_inset Formula $\text{Isom}X$
+\end_inset
+
+ es un abierto de 
+\begin_inset Formula ${\cal L}(X)$
+\end_inset
+
+ y 
+\begin_inset Formula $\cdot^{-1}:\text{Isom}X\to\text{Isom}X$
+\end_inset
+
+ es continua con la norma de 
+\begin_inset Formula ${\cal L}(X)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{FVC}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Liouville:
+\series default
+ Toda función [...][compleja holomorfa y] acotada es constante.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Gelfand:
+\series default
+ Si 
+\begin_inset Formula $_{\mathbb{C}}X$
+\end_inset
+
+ es de Banach y 
+\begin_inset Formula $T\in{\cal L}(X)$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma(T)$
+\end_inset
+
+ es compacto no vacío contenido en 
+\begin_inset Formula $B(0,\Vert T\Vert)$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $\lambda\in\mathbb{C}\setminus B[0,\Vert T\Vert]$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{\Vert T\Vert}{|\lambda|}<1$
+\end_inset
+
+, luego 
+\begin_inset Formula $\lambda1_{X}-T=\lambda(1_{X}-\frac{T}{\lambda})$
+\end_inset
+
+ es invertible y 
+\begin_inset Formula $\lambda\notin\sigma(T)$
+\end_inset
+
+.
+ La función 
+\begin_inset Formula $\psi:\mathbb{C}\to{\cal L}(X)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\psi(\lambda)\coloneqq\lambda1_{X}-T$
+\end_inset
+
+ es continua y por tanto 
+\begin_inset Formula $\mathbb{C}\setminus\sigma(T)=\psi^{-1}(\text{Isom}X)$
+\end_inset
+
+ es abierto, con lo que 
+\begin_inset Formula $\sigma(T)$
+\end_inset
+
+ es cerrado acotado y por tanto compacto.
+ Si fuera vacío, podemos definir 
+\begin_inset Formula $\phi:\mathbb{C}\to\text{Isom}X$
+\end_inset
+
+ como 
+\begin_inset Formula $\phi(\lambda)\coloneqq(\lambda1_{X}-T)^{-1}$
+\end_inset
+
+, que es continua, pero para 
+\begin_inset Formula $\lambda,h\in\mathbb{C}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline*}
+\frac{\phi(\lambda+h)-\phi(\lambda)}{h}=\frac{((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1}((\lambda1_{X}-T)-((\lambda+h)1_{X}-T))}{h}=\\
+=-((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1},
+\end{multline*}
+
+\end_inset
+
+de donde
+\begin_inset Formula 
+\[
+\dot{\phi}(\lambda)=\lim_{h\to0}\frac{\phi(\lambda+h)-\phi(\lambda)}{h}=\lim_{h\to0}(-((\lambda+h)1_{X}-T)^{-1}(\lambda1_{X}-T)^{-1})=-((\lambda1_{X}-T)^{-1})^{2},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es holomorfa y 
+\begin_inset Formula $\dot{\phi}\neq0$
+\end_inset
+
+, pero
+\begin_inset Formula 
+\[
+\Vert\phi(\lambda)\Vert=\Vert(\lambda1_{X}-T)^{-1}\Vert=\frac{1}{|\lambda|}\left\Vert \left(1_{X}-\frac{T}{\lambda}\right)^{-1}\right\Vert =\frac{1}{|\lambda|}\left\Vert \sum_{n\in\mathbb{N}}\frac{T^{n}}{\lambda^{n}}\right\Vert \leq\frac{1}{|\lambda|}\frac{1}{1-\frac{\Vert T\Vert}{|\lambda|}}=\frac{1}{|\lambda|-\Vert T\Vert},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $\lim_{|\lambda|\to\infty}\Vert\phi(\lambda)\Vert=\infty$
+\end_inset
+
+ y por tanto, como 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es continua, es acotada y, por el teorema de Liouville
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Que todavía no hemos visto que se de para espacios vectoriales infinitos
+ pero suponemos que se cumple.
+\end_layout
+
+\end_inset
+
+, 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es constante y 
+\begin_inset Formula $\dot{\phi}=0\#$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<1$
+\end_inset
+
+ e 
+\begin_inset Formula $y\in\ell^{2}$
+\end_inset
+
+, el sistema
+\begin_inset Formula 
+\begin{align*}
+x_{k}-\sum_{j\in\mathbb{N}}a_{kj}x_{j} & =y_{k}, & k & \in\mathbb{N},
+\end{align*}
+
+\end_inset
+
+tiene solución única 
+\begin_inset Formula $z\in\ell^{2}$
+\end_inset
+
+, y para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, el sistema truncado
+\begin_inset Formula 
+\begin{align*}
+x_{k}-\sum_{j\in\mathbb{N}_{n}}a_{kj}x_{j} & =y_{k}, & k & \in\mathbb{N}_{n}
+\end{align*}
+
+\end_inset
+
+tiene una única solución 
+\begin_inset Formula $z_{n}\in\mathbb{K}^{n}$
+\end_inset
+
+ de modo que, si 
+\begin_inset Formula $J_{n}:\mathbb{K}^{n}\to\ell^{2}$
+\end_inset
+
+ es la inclusión canónica de 
+\begin_inset Formula $\mathbb{K}^{n}$
+\end_inset
+
+ en las 
+\begin_inset Formula $n$
+\end_inset
+
+ primeras coordenadas, 
+\begin_inset Formula $\lim_{n}J_{n}(z_{n})=z$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert k\Vert_{2}<1$
+\end_inset
+
+ y 
+\begin_inset Formula $g\in L^{2}([a,b])$
+\end_inset
+
+, la ecuación
+\begin_inset Formula 
+\begin{align*}
+f(t)-\int_{a}^{b}k(t,s)f(s)\dif s & =g(t), & t & \in[a,b],
+\end{align*}
+
+\end_inset
+
+tiene solución única que es de la forma
+\begin_inset Formula 
+\[
+g(t)+\int_{a}^{b}\tilde{k}(t,s)g(s)\dif s
+\]
+
+\end_inset
+
+para cierto 
+\begin_inset Formula $\tilde{k}\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es el operador integral con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert k\Vert_{2}<1$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+\forall t\in[a,b],\int_{a}^{b}|k(t,s)|^{2}\dif s\leq C,
+\]
+
+\end_inset
+
+para 
+\begin_inset Formula $g\in L^{2}([a,b])$
+\end_inset
+
+, la serie 
+\begin_inset Formula $\sum_{n}K^{n}g$
+\end_inset
+
+ converge en 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ y converge absoluta y uniformemente en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Con todo esto, para 
+\begin_inset Formula $g\in L^{2}([0,1])$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{R}\setminus\{1\}$
+\end_inset
+
+, la ecuación integral
+\begin_inset Formula 
+\[
+f(t)-\lambda\int_{0}^{1}\text{e}^{t-s}f(s)\dif s=g(t)
+\]
+
+\end_inset
+
+tiene solución única
+\begin_inset Formula 
+\[
+f(t)=g(t)+\frac{\lambda}{1-\lambda}\int_{0}^{1}\text{e}^{t-s}g(s)\dif s.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Operador adjunto
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son espacios de Hilbert y 
+\begin_inset Formula $T\in L(G,H)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+\Vert T\Vert=\sup_{x,y\in\overline{B_{G}}}|\langle Tx,y\rangle|=\sup_{x,y\in B_{G}}|\langle Tx,y\rangle|.
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Existe un único 
+\begin_inset Formula $T^{*}\in L(H,G)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in G,\forall y\in H,\langle Tx,y\rangle\equiv\langle x,T^{*}y\rangle$
+\end_inset
+
+, el 
+\series bold
+adjunto
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert T\Vert=\Vert T^{*}\Vert$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $J$
+\end_inset
+
+ 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert, 
+\begin_inset Formula $A,B\in L(G,H)$
+\end_inset
+
+, 
+\begin_inset Formula $C\in L(H,J)$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{K}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(A+B)^{*}=A^{*}+B^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(\alpha A)^{*}=\overline{\alpha}A^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A^{**}=A$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(AC)^{*}=C^{*}A^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es invertible, también lo es 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $(A^{*})^{-1}=(A^{-1})^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert AA^{*}\Vert=\Vert A^{*}A\Vert=\Vert A\Vert^{2}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\ker A=(\text{Im}A^{*})^{\bot}$
+\end_inset
+
+ y 
+\begin_inset Formula $\ker A^{*}=(\text{Im}A)^{\bot}.$
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(\ker A)^{\bot}=\overline{\text{Im}A^{*}}$
+\end_inset
+
+ y 
+\begin_inset Formula $(\ker A^{*})^{\bot}=\overline{\text{Im}A}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+En 
+\begin_inset Formula $\ell^{2}$
+\end_inset
+
+, el adjunto de 
+\begin_inset Formula $S_{\text{r}}$
+\end_inset
+
+ es 
+\begin_inset Formula $S_{\text{l}}$
+\end_inset
+
+ y viceversa.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert y 
+\begin_inset Formula $K\in{\cal L}(H)$
+\end_inset
+
+ es un operador de rango finito dado por 
+\begin_inset Formula $K(x)=\sum_{i=1}^{n}\langle x,u_{i}\rangle v_{i}$
+\end_inset
+
+, su adjunto es de rango finito dado por 
+\begin_inset Formula $K^{*}(x)=\sum_{i=1}^{n}\langle x,v_{i}\rangle u_{i}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert con base 
+\begin_inset Formula $(e_{i})_{i\in I}$
+\end_inset
+
+ y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es un operador diagonal con 
+\begin_inset Formula $A(e_{i})\coloneqq\lambda_{i}e_{i}$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $\lambda_{i}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ es un operador diagonal con 
+\begin_inset Formula $A^{*}(e_{i})=\overline{\lambda_{i}}e_{i}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\end_inset
+
+ es el operador multiplicación por 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ es el operador multiplicación por 
+\begin_inset Formula $\overline{g}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert separable con base hilbertiana 
+\begin_inset Formula $(e_{n})_{n\in I}$
+\end_inset
+
+ y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ se expresa en dicha base como 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{I\times I}$
+\end_inset
+
+, 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ se expresa en dicha base como 
+\begin_inset Formula $(\overline{a_{ji}})\in\mathbb{K}^{I\times I}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\end_inset
+
+ es el operador integral con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ es el operador integral con núcleo 
+\begin_inset Formula $k^{*}(t,s)\coloneqq\overline{k(s,t)}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert, 
+\begin_inset Formula $M\leq H$
+\end_inset
+
+ es cerrado e 
+\begin_inset Formula $\iota:M\hookrightarrow H$
+\end_inset
+
+ es la inclusión, 
+\begin_inset Formula $\iota^{*}:H\to M$
+\end_inset
+
+ es la proyección ortogonal.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En general el adjunto no existe en espacios prehilbertianos.
+ Por ejemplo, 
+\begin_inset Formula $T:c_{00}\to c_{00}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $T(x)\coloneqq\sum_{n\geq1}\frac{x_{n}}{n}(1,0,\dots)$
+\end_inset
+
+ no tiene adjunto en 
+\begin_inset Formula $(c_{00},\langle\cdot,\cdot\rangle_{2})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert, 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es 
+\series bold
+autoadjunto
+\series default
+ o 
+\series bold
+hermitiano
+\series default
+ si 
+\begin_inset Formula $A^{*}=A$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A,B\in{\cal L}(H)$
+\end_inset
+
+ son autoadjuntos:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert A\Vert=\sup_{x\in\overline{B_{H}}}|\langle Ax,x\rangle|=\sup_{x\in S_{H}}|\langle Ax,x\rangle|$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Los valores propios de 
+\begin_inset Formula $A$
+\end_inset
+
+ son reales.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall x\in H,\langle Ax,x\rangle=0\implies A=0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $H=\ker A\oplus\overline{\text{Im}A}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A+B$
+\end_inset
+
+ es autoadjunto, y 
+\begin_inset Formula $AB$
+\end_inset
+
+ lo es si y sólo si 
+\begin_inset Formula $AB=BA$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $_{\mathbb{C}}H$
+\end_inset
+
+ es un espacio de Hilbert y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $\forall x\in H,\langle Ax,x\rangle\in\mathbb{R}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\backslash
+Existen únicos 
+\begin_inset Formula $\text{Re}A,\text{Im}A\in{\cal L}(H)$
+\end_inset
+
+ autoadjuntos, la 
+\series bold
+parte real
+\series default
+ y la 
+\series bold
+imaginaria
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+, con 
+\begin_inset Formula $A=\text{Re}A+\text{i}\text{Im}A$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\llbracket A\rrbracket\coloneqq\sup_{x\in S_{H}}|\langle Ax,x\rangle|$
+\end_inset
+
+ es una norma en 
+\begin_inset Formula ${\cal L}(H)$
+\end_inset
+
+ equivalente a la usual.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert con base 
+\begin_inset Formula $(e_{i})_{i\in I}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+El operador diagonal 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ con 
+\begin_inset Formula $T(e_{i})\eqqcolon\lambda_{i}e_{i}$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $\{\lambda_{i}\}_{i\in I}\subseteq\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es separable y 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ se representa respecto a la base como la matriz 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{I\times I}$
+\end_inset
+
+, 
+\begin_inset Formula $A$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $\forall i,j\in I,a_{ij}=\overline{a_{ji}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+El operador multiplicación por 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+ en 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $g(t)$
+\end_inset
+
+ es real para casi todo 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+El operador integral con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ en 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ es autoadjunto si y sólo si 
+\begin_inset Formula $k(t,s)=\overline{k(s,t)}$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $(s,t)\in[a,b]\times[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Una proyección ortogonal 
+\begin_inset Formula $P:H\to H$
+\end_inset
+
+ sobre un subespacio cerrado es autoadjunto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert, 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es 
+\series bold
+normal
+\series default
+ si 
+\begin_inset Formula $AA^{*}=A^{*}A$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x,y\in H,\langle Ax,Ay\rangle=\langle A^{*}x,A^{*}y\rangle$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x\in H,\Vert Ax\Vert=\Vert A^{*}x\Vert$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert complejo, 
+\begin_inset Formula $A\in{\cal L}(H)$
+\end_inset
+
+ es normal si y sólo si 
+\begin_inset Formula $\text{Re}A\circ\text{Im}A=\text{Im}A\circ\text{Re}A$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Todo operador diagonal es normal.
+\end_layout
+
+\begin_layout Enumerate
+El operador integral sobre 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es normal si y sólo si
+\begin_inset Formula 
+\[
+\int_{a}^{b}\overline{k(s,t)}k(s,x)\dif s=\int_{a}^{b}k(t,s)\overline{k(x,s)}\dif s
+\]
+
+\end_inset
+
+para casi todo 
+\begin_inset Formula $(t,x)\in[a,b]\times[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+proyección
+\series default
+ en un espacio normado 
+\begin_inset Formula $X$
+\end_inset
+
+ es un operador 
+\begin_inset Formula $X\to X$
+\end_inset
+
+ idempotente.
+ Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un espacio de Hilbert y 
+\begin_inset Formula $P$
+\end_inset
+
+ es una proyección continua no nula en 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $P$
+\end_inset
+
+ es una proyección ortogonal si y sólo si 
+\begin_inset Formula $\Vert P\Vert=1$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\text{Im}P=(\ker P)^{\bot}$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\ker P=(\text{Im}P)^{\bot}$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $P$
+\end_inset
+
+ es autoadjunto, si y sólo si es normal, si y sólo si 
+\begin_inset Formula $\forall x\in H,\langle Px,x\rangle=\Vert Px\Vert^{2}$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x\in H,\langle Px,x\rangle\geq0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Existen proyecciones no ortogonales, como 
+\begin_inset Formula $p:\mathbb{R}^{2}\to\mathbb{R}^{2}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $p(x,y)\coloneqq(x+y,0)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{K}$
+\end_inset
+
+, 
+\begin_inset Formula $\lambda\in\sigma(T)\iff\overline{\lambda}\in\sigma(T^{*})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es normal:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\lambda\in\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\ker(T-\lambda1_{H})=\ker(T^{*}-\overline{\lambda}1_{H})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\lambda,\mu\in\mathbb{C},(\lambda\neq\mu\implies\ker(T-\lambda1_{H})\bot\ker(T-\mu1_{H}))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\ker(T-\lambda1_{H})$
+\end_inset
+
+ y 
+\begin_inset Formula $\ker(T-\lambda1_{H})^{\bot}$
+\end_inset
+
+ son 
+\begin_inset Formula $T$
+\end_inset
+
+-invariantes.
+\end_layout
+
+\begin_layout Section
+Operadores compactos
+\end_layout
+
+\begin_layout Standard
+Dado un espacio topológico 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $Y\subseteq X$
+\end_inset
+
+ es 
+\series bold
+relativamente compacto
+\series default
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ si su clausura en 
+\begin_inset Formula $X$
+\end_inset
+
+ es compacta.
+ Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios normados, una función lineal 
+\begin_inset Formula $T:X\to Y$
+\end_inset
+
+ es 
+\series bold
+compacta
+\series default
+ si 
+\begin_inset Formula $T(B_{X})$
+\end_inset
+
+ es relativamente compacta en 
+\begin_inset Formula $Y$
+\end_inset
+
+, si y sólo si para cada sucesión acotada 
+\begin_inset Formula $\{x_{n}\}_{n}\subseteq X$
+\end_inset
+
+, 
+\begin_inset Formula $(Tx_{n})_{n}$
+\end_inset
+
+ posee una subsucesión convergente, si y sólo si esto se cumple cuando cada
+ 
+\begin_inset Formula $\Vert x_{n}\Vert=1$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Los operadores de rango finito son compactos.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El operador identidad en un espacio de dimensión infinita nunca es compacto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula ${\cal K}(X,Y)$
+\end_inset
+
+ al subespacio vectorial de 
+\begin_inset Formula ${\cal L}(X,Y)$
+\end_inset
+
+ de los operadores compactos, que es cerrado si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es de Banach.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $A\in{\cal L}(X,Y)$
+\end_inset
+
+, 
+\begin_inset Formula $T\in{\cal K}(Y,Z)$
+\end_inset
+
+ y 
+\begin_inset Formula $B\in{\cal L}(Z,W)$
+\end_inset
+
+, 
+\begin_inset Formula $BTA\in{\cal K}(X,W)$
+\end_inset
+
+, y en particular 
+\begin_inset Formula ${\cal K}(X)\coloneqq{\cal K}(X,X)$
+\end_inset
+
+ es un ideal de 
+\begin_inset Formula ${\cal L}(X)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T\in{\cal K}(X,Y)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{Im}T$
+\end_inset
+
+ es un subespacio separable de 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es de Hilbert, 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+ es de dimensión infinita con base hilbertiana 
+\begin_inset Formula $(e_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ y, para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $P_{n}\in{\cal L}(Y)$
+\end_inset
+
+ es la proyección ortogonal sobre 
+\begin_inset Formula $\text{span}\{e_{i}\}_{i\leq n}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $T=\lim_{n}P_{n}T\in{\cal L}(X,Y)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $Y$
+\end_inset
+
+ es de Hilbert, 
+\begin_inset Formula ${\cal K}(X,Y)$
+\end_inset
+
+ es la clausura en 
+\begin_inset Formula ${\cal L}(X,Y)$
+\end_inset
+
+ del conjunto de operadores acotados de rango finito.
+ Esto no es cierto cuando 
+\begin_inset Formula $Y$
+\end_inset
+
+ es un espacio de Banach arbitrario.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son espacios de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(G,H)$
+\end_inset
+
+ es compacto si y sólo si lo es 
+\begin_inset Formula $T^{*}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Standard
+Con esto:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(e_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ y 
+\begin_inset Formula $(f_{n})_{n\in\mathbb{N}}$
+\end_inset
+
+ son bases hilbertianas respectivas de 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $T:G\to H$
+\end_inset
+
+ es un operador diagonal dado por 
+\begin_inset Formula $Te_{n}\coloneqq\lambda_{n}f_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es compacto si y sólo si 
+\begin_inset Formula $\lim_{n}\lambda_{n}=0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El operador multiplicación por 
+\begin_inset Formula $g\in L^{\infty}([a,b])$
+\end_inset
+
+ es compacto si y sólo si 
+\begin_inset Formula $g=0$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son espacios de Hilbert de dimensión 
+\begin_inset Formula $\aleph_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $T\in{\cal L}(G,H)$
+\end_inset
+
+ se representa en ciertas bases de 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ como 
+\begin_inset Formula $(a_{ij})\in\mathbb{K}^{\mathbb{N}\times\mathbb{N}}$
+\end_inset
+
+, si 
+\begin_inset Formula $\sum_{i,j}|a_{ij}|^{2}<\infty$
+\end_inset
+
+, 
+\begin_inset Formula $T$
+\end_inset
+
+ es compacto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+El operador integral 
+\begin_inset Formula $K\in{\cal L}(L^{2}([a,b]))$
+\end_inset
+
+ con núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es compacto, 
+\begin_inset Formula ${\cal C}([a,b])$
+\end_inset
+
+ es 
+\begin_inset Formula $K$
+\end_inset
+
+-invariante y 
+\begin_inset Formula $K|_{{\cal C}([a,b])}:({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})\to({\cal C}([a,b]),\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ es compacto.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Teorema espectral
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert de dimensión finita y 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es autoadjunto:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lambda_{1},\dots,\lambda_{m}$
+\end_inset
+
+ son los distintos valores propios de 
+\begin_inset Formula $T$
+\end_inset
+
+, 
+\begin_inset Formula $H=\bigoplus_{k=1}^{m}\ker(T-\lambda_{k}I_{H})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Existe una base ortonormal 
+\begin_inset Formula $(e_{k})_{k}$
+\end_inset
+
+ de 
+\begin_inset Formula $H$
+\end_inset
+
+ formada por vectores propios de 
+\begin_inset Formula $T$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, 
+\begin_inset Formula $Tx=\sum_{k}\mu_{k}\langle x,e_{k}\rangle e_{k}$
+\end_inset
+
+, donde 
+\begin_inset Formula $\mu_{k}$
+\end_inset
+
+ es el valor propio asociado a 
+\begin_inset Formula $e_{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T$
+\end_inset
+
+ es un operador compacto autoadjunto en el espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert T\Vert$
+\end_inset
+
+ o 
+\begin_inset Formula $-\Vert T\Vert$
+\end_inset
+
+ es valor propio de 
+\begin_inset Formula $T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Todo operador normal compacto en un 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+-espacio de Hilbert tiene algún valor propio.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es compacto en el 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{K}\setminus0$
+\end_inset
+
+, 
+\begin_inset Formula $\ker(T-\lambda1_{H})$
+\end_inset
+
+ es de dimensión finita.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $X$
+\end_inset
+
+ e 
+\begin_inset Formula $Y$
+\end_inset
+
+ espacios de Banach y 
+\begin_inset Formula $T\in{\cal L}(X,Y)$
+\end_inset
+
+ compacto, 
+\begin_inset Formula $\sigma_{\text{p}}(T)$
+\end_inset
+
+ es contable, contiene a 
+\begin_inset Formula $\sigma(T)\setminus\{0\}$
+\end_inset
+
+ y, si es infinito, es una sucesión acotada con a lo sumo un punto de acumulació
+n, el 0, y si 
+\begin_inset Formula $T$
+\end_inset
+
+ es normal el 0 es punto de acumulación.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema espectral para operadores compactos autoadjuntos:
+\series default
+ Sean 
+\begin_inset Formula $H$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert y 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ compacto normal:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\sigma_{\text{p}}(T)\setminus\{0\}$
+\end_inset
+
+ es contable.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $P_{\lambda}\in{\cal L}(H)$
+\end_inset
+
+ es la proyección ortogonal sobre 
+\begin_inset Formula $\ker(T-\lambda1_{H})$
+\end_inset
+
+, 
+\begin_inset Formula $T=\sum_{\lambda\in\sigma_{\text{p}}(T)}\lambda P_{\lambda}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overline{\text{Im}T}=\bigoplus_{\lambda\in\sigma_{\text{p}}(T)\setminus\{0\}}\ker(T-\lambda1_{H})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $H=\ker T\oplus\overline{\text{Im}T}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Existe una base ortonormal 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ de 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{C}$
+\end_inset
+
+ tales que, para 
+\begin_inset Formula $x\in H$
+\end_inset
+
+, 
+\begin_inset Formula $(\mu_{n}\langle x,e_{n}\rangle e_{n})_{n\in J}$
+\end_inset
+
+ es sumable con suma 
+\begin_inset Formula $Tx$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\sigma_{\text{p}}(T)\setminus\{0\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall\lambda\in\sigma_{\text{p}}(T)\setminus\{0\},|\{n\in J\mid\mu_{n}=\lambda\}|=\dim\ker(T-\lambda1_{H})$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $P_{0}$
+\end_inset
+
+ es la proyección ortogonal sobre 
+\begin_inset Formula $\ker T$
+\end_inset
+
+, 
+\begin_inset Formula $\forall x\in H,x=P_{0}x+\sum_{n\in J}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es compacto autoadjunto si y sólo si hay una familia ortonormal contable
+ 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ de modo que 
+\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+ y 0 es el único punto de acumulación de 
+\begin_inset Formula $(\mu_{n})_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de alternativa de Fredholm:
+\series default
+ Sean 
+\begin_inset Formula $H$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ compacto autoadjunto, 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ una base ortonormal de 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+ de modo que 
+\begin_inset Formula $Tx\eqqcolon\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $\mu_{n}\in\mathbb{K}$
+\end_inset
+
+ e 
+\begin_inset Formula $y\in H$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $\lambda\in\mathbb{K}\setminus\{\sigma_{\text{p}}(T)\cup\{0\})$
+\end_inset
+
+, la ecuación 
+\begin_inset Formula $(\lambda1_{H}-T)x=y$
+\end_inset
+
+ tiene como única solución
+\begin_inset Formula 
+\[
+x=\frac{1}{\lambda}\left(y+\sum_{n\in J}\frac{\mu_{n}}{\lambda-\mu_{n}}\langle y,e_{n}\rangle e_{n}\right).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si existe solución 
+\begin_inset Formula $x\in H$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+(\lambda1_{H}-T)x=y\iff\lambda x=Tx+y\iff x=\frac{1}{\lambda}\left(\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}+y\right),
+\]
+
+\end_inset
+
+pero entonces 
+\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\lambda}(\mu_{n}\langle x,e_{n}\rangle+\langle y,e_{n}\rangle)$
+\end_inset
+
+ y 
+\begin_inset Formula $(\lambda-\mu_{n})\langle x,e_{n}\rangle=\langle y,e_{n}\rangle$
+\end_inset
+
+, y como 
+\begin_inset Formula $\lambda-\mu_{n}\neq0$
+\end_inset
+
+, podemos sustituir 
+\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\lambda-\mu_{n}}\langle y,e_{n}\rangle$
+\end_inset
+
+ en lo anterior y queda la solución del enunciado.
+ Queda ver que la serie converge, pero si 
+\begin_inset Formula $\sigma_{\text{p}}(T)$
+\end_inset
+
+ es infinito, 
+\begin_inset Formula $\{\mu_{n}\}_{n}\subseteq\sigma_{\text{p}}(T)$
+\end_inset
+
+ es acotado y por tanto lo es 
+\begin_inset Formula $\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\sum_{n\in J}\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|^{2}|\langle y,e_{n}\rangle|^{2}\leq\sup_{n\in J}\left|\frac{\mu_{n}}{\lambda-\mu_{n}}\right|^{2}\sum_{n\in J}|\langle y,e_{n}\rangle|^{2}<\infty.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $\lambda\in\sigma_{\text{p}}(T)\setminus\{0\}$
+\end_inset
+
+, la ecuación 
+\begin_inset Formula $(\lambda1_{H}-T)x=y$
+\end_inset
+
+ tiene solución si y sólo si 
+\begin_inset Formula $y\bot\ker(\lambda1_{H}-T)$
+\end_inset
+
+, en cuyo caso las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x & =\frac{1}{\lambda}\left(y+\sum_{\begin{subarray}{c}
+n\in J\\
+\mu_{n}\neq\lambda
+\end{subarray}}\frac{\mu_{n}}{\lambda-\mu_{n}}\langle y,e_{n}\rangle e_{n}\right)+z, & z & \in\ker(\lambda1_{H}-T).
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si la ecuación tiene solución 
+\begin_inset Formula $x$
+\end_inset
+
+, entonces 
+\begin_inset Formula $y=(\lambda1_{H}-T)x\in\text{Im}(\lambda1_{H}-T)\subseteq\overline{\text{Im}(\lambda1_{H}-T)}=\ker((\lambda1_{H}-T)^{*})^{\bot}=\ker(\lambda1_{H}-T)^{\bot}$
+\end_inset
+
+ por ser 
+\begin_inset Formula $1_{H}$
+\end_inset
+
+ y 
+\begin_inset Formula $T$
+\end_inset
+
+ autoadjuntos, y claramente dos soluciones difieren en un vector de 
+\begin_inset Formula $\ker(\lambda1_{H}-T)$
+\end_inset
+
+.
+ Queda ver que, si 
+\begin_inset Formula $y\in\ker(\lambda1_{H}-T)^{\bot}$
+\end_inset
+
+, la 
+\begin_inset Formula $x$
+\end_inset
+
+ del enunciado es solución, para lo cual hacemos la misma sustitución que
+ al principio del primer apartado pero, cuando 
+\begin_inset Formula $\lambda=\mu_{n}$
+\end_inset
+
+, en su lugar vemos que 
+\begin_inset Formula $(\lambda-\mu_{n})\langle x,e_{n}\rangle=\langle y,e_{n}\rangle$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\langle y,e_{n}\rangle=0$
+\end_inset
+
+, por lo que excluimos dicho factor de la serie, la cual converge por el
+ mismo motivo que en el primer apartado y resulta en la solución del enunciado.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $y=0$
+\end_inset
+
+, 
+\begin_inset Formula $Tx=y$
+\end_inset
+
+ tiene solución si y sólo si 
+\begin_inset Formula $y\bot\ker T$
+\end_inset
+
+ y 
+\begin_inset Formula $\sum_{n\in J}\left|\frac{\langle y,e_{n}\rangle}{\mu_{n}}\right|^{2}<\infty$
+\end_inset
+
+, en cuyo caso las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x & =\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+z, & z & \in\ker T.
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si la ecuación tiene solución 
+\begin_inset Formula $x$
+\end_inset
+
+, 
+\begin_inset Formula $y\in\text{Im}T\subseteq(\ker T)^{\bot}$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}=Tx=y=\sum_{n\in J}\langle y,e_{n}\rangle e_{n},
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $\langle x,e_{n}\rangle=\frac{1}{\mu_{n}}\langle y,e_{n}\rangle$
+\end_inset
+
+ para cada 
+\begin_inset Formula $n$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\sum_{n\in J}\left|\frac{\langle y,e_{n}\rangle}{\mu_{n}}\right|^{2}=\Vert x\Vert^{2}<\infty$
+\end_inset
+
+, y como 
+\begin_inset Formula $(e_{n})_{n}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $\overline{\text{Im}T}$
+\end_inset
+
+, 
+\begin_inset Formula $x\in\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+\overline{\text{Im}T}^{\bot}$
+\end_inset
+
+ con 
+\begin_inset Formula $\overline{\text{Im}T}^{\bot}=\ker T$
+\end_inset
+
+.
+ Finalmente, si esta condición se cumple, 
+\begin_inset Formula $y\in\overline{\text{Im}T}$
+\end_inset
+
+, la serie del enunciado converge y
+\begin_inset Formula 
+\[
+T\left(\sum_{n\in J}\frac{1}{\mu_{n}}\langle y,e_{n}\rangle e_{n}+z\right)=\sum_{n\in J}\langle y,e_{n}\rangle e_{n}+0=y.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Sea 
+\begin_inset Formula $A$
+\end_inset
+
+ un operador en un espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A$
+\end_inset
+
+ es una isometría si y sólo si 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ es inverso por la izquierda de 
+\begin_inset Formula $A$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall x,y\in H,\langle Ax,Ay\rangle=\langle x,y\rangle$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A$
+\end_inset
+
+ es un isomorfismo isométrico, si y sólo si es una isometría suprayectiva,
+ si y sólo si 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ es inverso de 
+\begin_inset Formula $A$
+\end_inset
+
+, y entonces decimos que 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\series bold
+unitario
+\series default
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $H$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert y 
+\begin_inset Formula $S,T\in{\cal L}(H)$
+\end_inset
+
+ compactos autoadjuntos, 
+\begin_inset Formula $\forall\lambda\in\mathbb{K},\dim\ker(T-\lambda1_{H})=\dim\ker(S-\lambda1_{H})$
+\end_inset
+
+ si y sólo si existe 
+\begin_inset Formula $U\in{\cal L}(H)$
+\end_inset
+
+ unitario con 
+\begin_inset Formula $U^{*}SU=T$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $S,T\in{\cal L}(H)$
+\end_inset
+
+ en el 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacio de Hilbert 
+\begin_inset Formula $H$
+\end_inset
+
+ son 
+\series bold
+simultáneamente diagonalizables
+\series default
+ si existe una familia ortonormal 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\alpha_{n}\}_{n\in J},\{\beta_{n}\}_{n\in J}\subseteq\mathbb{K}$
+\end_inset
+
+ tal que
+\begin_inset Formula 
+\[
+\forall x\in H,\left(Sx=\sum_{n\in J}\alpha_{n}\langle x,e_{n}\rangle e_{n}\land Tx=\sum_{n\in J}\beta_{n}\langle x,e_{n}\rangle e_{n}\right).
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $S$
+\end_inset
+
+ y 
+\begin_inset Formula $T$
+\end_inset
+
+ son compactos y autoadjuntos esto equivale a que 
+\begin_inset Formula $ST=TS$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema espectral para operadores compactos normales:
+\series default
+ Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+-espacio de Hilbert y 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ compacto normal, ocurre lo mismo que en el anterior teorema espectral.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $H$
+\end_inset
+
+ es un 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+-espacio de Hilbert, 
+\begin_inset Formula $T\in{\cal L}(H)$
+\end_inset
+
+ es compacto normal si y sólo si hay una familia ortonormal contable 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{C}$
+\end_inset
+
+ con 0 como único punto de acumulación de modo que 
+\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un operador entre 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+-espacios de Hilbert 
+\begin_inset Formula $T\in{\cal L}(G,H)$
+\end_inset
+
+ es compacto si y sólo si hay una familia contable 
+\begin_inset Formula $\{\nu_{n}\}_{n\in J}\subseteq\mathbb{R}^{+}$
+\end_inset
+
+ con 0 como punto de acumulación, 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq G$
+\end_inset
+
+ y 
+\begin_inset Formula $\{f_{n}\}_{n\in J}\subseteq H$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\forall x\in H,Tx=\sum_{n\in J}\nu_{n}\langle x,e_{n}\rangle f_{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Ecuaciones integrales de Fredholm
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+ecuación integral de Fredholm
+\series default
+ es una de la forma
+\begin_inset Formula 
+\[
+x(t)-\mu\int_{a}^{b}k(t,s)x(s)\dif s=g(t),
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $x,g\in L^{2}([a,b])$
+\end_inset
+
+, 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ y la incógnita es 
+\begin_inset Formula $x$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Un núcleo 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es 
+\series bold
+simétrico
+\series default
+ si 
+\begin_inset Formula $k(t,s)=\overline{k(s,t)}$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $s,t\in[a,b]$
+\end_inset
+
+.
+ 
+\series bold
+Teorema de alternativa de Fredholm:
+\series default
+ Sean 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ un núcleo simétrico, 
+\begin_inset Formula $K$
+\end_inset
+
+ el operador integral asociado y 
+\begin_inset Formula $g\in L^{2}([a,b])$
+\end_inset
+
+, si 
+\begin_inset Formula $Kx=\sum_{n\in J}\mu_{j}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+ para cierta base hilbertiana contable 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ de 
+\begin_inset Formula $\overline{\text{Im}K}$
+\end_inset
+
+, ciertos 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ y todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, considerando la ecuación integral de Fredholm de arriba, 
+\begin_inset Formula $x-Kx=g$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\mu=0$
+\end_inset
+
+, la ecuación tiene como única solución 
+\begin_inset Formula $x=g$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\frac{1}{\mu}\notin\{\mu_{n}\}_{n}$
+\end_inset
+
+, la ecuación tiene como única solución
+\begin_inset Formula 
+\[
+x(t)=g(t)+\mu\left(\sum_{n}\frac{\mu_{n}}{1-\mu\mu_{n}}\left(\int_{a}^{b}g\overline{e_{n}}\right)e_{n}(t)\right),
+\]
+
+\end_inset
+
+y existe 
+\begin_inset Formula $\alpha>0$
+\end_inset
+
+ que depende solo de 
+\begin_inset Formula $k$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\Vert x\Vert_{2}\leq\alpha\Vert g\Vert_{2}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si existe 
+\begin_inset Formula $n\in J$
+\end_inset
+
+ con 
+\begin_inset Formula $\mu_{n}=\frac{1}{\mu}$
+\end_inset
+
+, la ecuación tiene solución si y sólo si 
+\begin_inset Formula $g\bot\ker(\frac{1_{L^{2}([a,b])}}{\mu}-K)$
+\end_inset
+
+, y entonces las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x(t) & =g(t)+\mu\sum_{\begin{subarray}{c}
+n\in J\\
+\mu_{n}\neq\frac{1}{\mu}
+\end{subarray}}\frac{\mu_{n}}{1-\mu\mu_{n}}\left(\int g\overline{e_{n}}\right)e_{j}+u, & u & \in\ker(\tfrac{1_{L^{2}([a,b])}}{\mu}-K).
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La convergencia de las series es de media cuadrática, pero en ciertos casos
+ puede ser uniforme.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $k\in L^{2}([a,b]\times[a,b])$
+\end_inset
+
+ es un núcleo simétrico con
+\begin_inset Formula 
+\[
+\sup_{t\in[a,b]}\int_{a}^{b}|k(t,s)|^{2}\dif s<\infty,
+\]
+
+\end_inset
+
+
+\begin_inset Formula $K$
+\end_inset
+
+ es el operador integral asociado y hay una base hilbertiana 
+\begin_inset Formula $(e_{n})_{n\in J}$
+\end_inset
+
+ de 
+\begin_inset Formula $\overline{\text{Im}K}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ y tales que 
+\begin_inset Formula $Kx=\sum_{n}\mu_{n}\langle x,e_{n}\rangle e_{n}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema de Hilbert-Schmidt:
+\series default
+ Para 
+\begin_inset Formula $x\in L^{2}([a,b])$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int_{a}^{b}k(t,s)x(s)\dif s=\sum_{n\in J}\mu_{n}\left(\int_{a}^{b}x\overline{e_{n}}\right)e_{n}(t)
+\]
+
+\end_inset
+
+para casi todo 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+, y si 
+\begin_inset Formula $J$
+\end_inset
+
+ es numerable la serie converge absoluta y uniformemente en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para la primera parte basta tomar en el teorema anterior un 
+\begin_inset Formula $\mu\neq0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\frac{1}{\mu}$
+\end_inset
+
+ no sea valor propio y despejar.
+ Para la segunda podemos suponer 
+\begin_inset Formula $J=(\mathbb{N},\geq)$
+\end_inset
+
+, y queremos ver que
+\begin_inset Formula 
+\[
+\sum_{n}\left|\mu_{n}\left(\int_{a}^{b}x\overline{e_{n}}\right)e_{n}(t)\right|=\sum_{n}|\mu_{n}\langle x,e_{n}\rangle e_{n}(t)|
+\]
+
+\end_inset
+
+es uniformemente de Cauchy en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+ Por la desigualdad de Cauchy-Schwartz,
+\begin_inset Formula 
+\[
+\sum_{n=p}^{q}|\mu_{n}e_{n}(t)||\langle x,e_{n}\rangle|\leq\sqrt{\sum_{n=p}^{q}|\mu_{n}e_{n}(t)|^{2}\sum_{n=p}^{q}|\langle x,e_{n}\rangle|^{2}},
+\]
+
+\end_inset
+
+pero para 
+\begin_inset Formula $n\in J$
+\end_inset
+
+ y 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\mu_{n}e_{n}(t)=K(e_{n})(t)=\int_{a}^{b}k(t,s)e_{k}(s)\dif s=\langle e_{k},\overline{k_{t}}\rangle,
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $k_{t}(s)\coloneqq k(t,s)$
+\end_inset
+
+, luego
+\begin_inset Formula 
+\[
+\sqrt{\sum_{n=p}^{q}|\mu_{n}e_{n}(t)|^{2}}=\sqrt{\sum_{n=p}^{q}|\langle e_{n},\overline{k_{t}}\rangle|^{2}}\leq\Vert k_{t}\Vert_{2}\leq\sup_{t\in[a,b]}\Vert k_{t}\Vert_{2}<\infty,
+\]
+
+\end_inset
+
+con lo que esto está acotado superiormente por un valor independiente de
+ 
+\begin_inset Formula $t$
+\end_inset
+
+ y el resultado sale de que 
+\begin_inset Formula $|\langle x,e_{n}\rangle|^{2}$
+\end_inset
+
+ tampoco depende de 
+\begin_inset Formula $t$
+\end_inset
+
+ y 
+\begin_inset Formula $\lim_{p,q}\sum_{n=p}^{q}|\langle x,e_{n}\rangle|^{2}=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Las series del teorema de alternativa de Fredholm convergen absoluta y uniformem
+ente en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $k\in{\cal C}([a,b]\times[a,b])$
+\end_inset
+
+ es un núcleo simétrico, existen una familia ortonormal contable 
+\begin_inset Formula $\{e_{n}\}_{n\in J}\subseteq({\cal C}([a,b]),\Vert\cdot\Vert_{2})$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\mu_{n}\}_{n\in J}\subseteq\mathbb{R}$
+\end_inset
+
+ tales que, si 
+\begin_inset Formula $K$
+\end_inset
+
+ es el operador integral asociado a 
+\begin_inset Formula $k$
+\end_inset
+
+ y 
+\begin_inset Formula $f\in{\cal C}([a,b])$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+Kf(t)=\sum_{n\in J}\mu_{n}\left(\int_{a}^{b}f\overline{e_{n}}\right)e_{n}(t)
+\]
+
+\end_inset
+
+para todo 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+ y la convergencia de la serie es absoluta y uniforme.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Problemas de Sturm-Liouville
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+problema regular de Sturm-Liouville
+\series default
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+La forma general del problema tiene como ecuación 
+\begin_inset Formula $\od{}{x}(p\dot{x})+qx+\lambda\sigma x+y=0$
+\end_inset
+
+ con 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ continuas y estrictamente positivas.
+ Aquí tomamos 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $q$
+\end_inset
+
+ constantes en 1.
+\end_layout
+
+\end_inset
+
+ es uno de la forma
+\begin_inset Formula 
+\begin{align*}
+-\ddot{x}+qx-\lambda x & =y, & \alpha x(a)+\beta\dot{x}(a) & =0, & \gamma x(b)+\delta\dot{x}(b) & =0,
+\end{align*}
+
+\end_inset
+
+donde 
+\begin_inset Formula $q\in{\cal C}([a,b],\mathbb{R})$
+\end_inset
+
+, 
+\begin_inset Formula $y\in{\cal C}([a,b],\mathbb{C})$
+\end_inset
+
+, 
+\begin_inset Formula $\lambda\in\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha,\beta,\gamma,\delta\in\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $|\alpha|+|\beta|,|\gamma|+|\delta|\neq0$
+\end_inset
+
+ y la incógnita 
+\begin_inset Formula $x\in{\cal C}^{2}([a,b],\mathbb{C})$
+\end_inset
+
+.
+ Su 
+\series bold
+operador de Sturm-Liouville
+\series default
+ asociado es 
+\begin_inset Formula $S\in{\cal L}(D_{S},{\cal C}([a,b],\mathbb{C}))$
+\end_inset
+
+ dado por 
+\begin_inset Formula $S(x)\coloneqq-\ddot{x}+qx$
+\end_inset
+
+, donde
+\begin_inset Formula 
+\[
+D_{S}\coloneqq\{x\in{\cal C}^{2}([a,b],\mathbb{C})\mid\alpha x(a)+\beta\dot{x}(a)=\gamma x(b)+\delta\dot{x}(b)=0\},
+\]
+
+\end_inset
+
+y entonces el problema anterior es 
+\begin_inset Formula $(S-\mu1_{D_{S}})x=y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $q\in{\cal C}([a,b],\mathbb{R})$
+\end_inset
+
+ e 
+\begin_inset Formula $y_{0},y_{1}\in\mathbb{R}$
+\end_inset
+
+, el problema de Cauchy
+\begin_inset Formula 
+\begin{align*}
+-\ddot{x}+qx & =0, & x(a) & =y_{0}, & \dot{x}(a) & =y_{1}
+\end{align*}
+
+\end_inset
+
+tiene una única solución real, y para 
+\begin_inset Formula $\alpha,\beta\in\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $|\alpha|+|\beta|\neq0$
+\end_inset
+
+, si 
+\begin_inset Formula $(y_{0},y_{1})\in\mathbb{R}^{2}$
+\end_inset
+
+ recorre la recta 
+\begin_inset Formula $\alpha y_{0}+\beta y_{1}=0$
+\end_inset
+
+, la correspondiente solución del problema recorre una recta (subespacio
+ de dimensión 1) de 
+\begin_inset Formula ${\cal C}^{2}([a,b])$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+determinante wronskiano
+\series default
+ de 
+\begin_inset Formula $x_{1},\dots,x_{n}\in{\cal C}^{n-1}([a,b],\mathbb{K})$
+\end_inset
+
+ es 
+\begin_inset Formula $W(x_{1},\dots,x_{n}):[a,b]\to\mathbb{K}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $t\mapsto\det(x_{j}^{(i)}(t))_{0\leq i<n}^{1\leq j\leq n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $S:D_{S}\to{\cal C}([a,b],\mathbb{C})$
+\end_inset
+
+ es un operador de Sturm-Liouville asociado al problema con parámetros 
+\begin_inset Formula $q,y,\lambda,\alpha,\beta,\gamma,\delta$
+\end_inset
+
+, existen 
+\begin_inset Formula $u,v\in{\cal C}([a,b],\mathbb{R})$
+\end_inset
+
+ con 
+\begin_inset Formula $-\ddot{u}+qu=0$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha x(a)+\beta\dot{x}(a)=0$
+\end_inset
+
+, 
+\begin_inset Formula $-\ddot{v}+qv=0$
+\end_inset
+
+ y 
+\begin_inset Formula $\gamma x(b)+\delta\dot{x}(b)=0$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $W(u,v)(t)$
+\end_inset
+
+ es constante en 
+\begin_inset Formula $t$
+\end_inset
+
+ y, si 
+\begin_inset Formula $S$
+\end_inset
+
+ es inyectivo, 
+\begin_inset Formula $W(u,v)(t)\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $u$
+\end_inset
+
+ y 
+\begin_inset Formula $v$
+\end_inset
+
+ son linealmente independientes, y llamamos 
+\series bold
+función de Green
+\series default
+ asociada a 
+\begin_inset Formula $S$
+\end_inset
+
+ al núcleo simétrico 
+\begin_inset Formula $k\in{\cal C}([a,b]\times[a,b])$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+k(t,s)\coloneqq-\frac{u(\min\{t,s\})v(\max\{t,s\})}{W(u,v)(a)},
+\]
+
+\end_inset
+
+que no depende de 
+\begin_inset Formula $u$
+\end_inset
+
+ y 
+\begin_inset Formula $v$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $S:D_{S}\to{\cal C}([a,b])$
+\end_inset
+
+ es un operador de Sturm-Liouville inyectivo con función de Green 
+\begin_inset Formula $k$
+\end_inset
+
+, llamamos 
+\series bold
+operador de Green
+\series default
+ asociado a 
+\begin_inset Formula $S$
+\end_inset
+
+ al operador integral 
+\begin_inset Formula $G:L^{2}([a,b])\to L^{2}([a,b])$
+\end_inset
+
+ asociado al núcleo 
+\begin_inset Formula $k$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $G|_{{\cal C}([a,b])}$
+\end_inset
+
+ es el inverso de 
+\begin_inset Formula $S$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Así, 
+\begin_inset Formula $(S-\mu1_{D_{S}})x=y$
+\end_inset
+
+ tiene solución única 
+\begin_inset Formula $x\in D_{S}$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $(1_{{\cal C}([a,b])}-\mu G)x=Gy$
+\end_inset
+
+ tiene solución única 
+\begin_inset Formula $x\in{\cal C}([a,b])$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $S:D_{S}\to{\cal C}([a,b],\mathbb{C})$
+\end_inset
+
+ es el operador de Sturm-Liouville asociado al problema con parámetros 
+\begin_inset Formula $q,y,\lambda,\alpha,\beta,\gamma,\delta$
+\end_inset
+
+, existe una sucesión 
+\begin_inset Formula $(\nu_{n})_{n}$
+\end_inset
+
+ de reales distintos con 
+\begin_inset Formula $\sum_{n}\frac{1}{\nu_{n}^{2}}<\infty$
+\end_inset
+
+ y una base hilbertiana numerable 
+\begin_inset Formula $(u_{n})_{n}$
+\end_inset
+
+ de 
+\begin_inset Formula $L^{2}([a,b])$
+\end_inset
+
+ tales que:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall n\in\mathbb{N},Su_{n}=\nu_{n}u_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+\forall x\in D_{S},\forall t\in[a,b],x(t)=\sum_{n}\left(\int_{a}^{b}xu_{n}\right)u_{n}(t),
+\]
+
+\end_inset
+
+donde la serie converge absoluta y uniformemente para 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lambda\notin\{\nu_{n}\}_{n}$
+\end_inset
+
+, el problema tiene como única solución
+\begin_inset Formula 
+\[
+x(t)=\sum_{n}\frac{1}{\nu_{n}-\lambda}\left(\int_{a}^{b}yu_{n}\right)u_{n}(t),
+\]
+
+\end_inset
+
+donde la serie converge absoluta y uniformemente para 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lambda=\nu_{k}$
+\end_inset
+
+ para algún 
+\begin_inset Formula $k$
+\end_inset
+
+, el problema tiene solución si y sólo si 
+\begin_inset Formula $y\bot u_{k}$
+\end_inset
+
+, y entonces las soluciones son
+\begin_inset Formula 
+\begin{align*}
+x(t) & =\alpha u_{k}+\sum_{n\in\mathbb{N}\setminus\{k\}}\frac{1}{\nu_{n}-\lambda}\left(\int_{a}^{b}yu_{n}\right)u_{n}(t), & \alpha & \in\mathbb{C},
+\end{align*}
+
+\end_inset
+
+donde la serie converge absoluta y uniformemente para 
+\begin_inset Formula $t\in[a,b]$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+nproof
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document