\chapter{Operator theory}




\begin{definition}[Operator]
An \emph{operator} is map between two TLSs with the same field $T : X \to Y$.
\end{definition}


\begin{example}[Fourier transform]
	\[\mathcal{F} : L^1(\mathbb{R}) \to L^{\infty}(\mathbb{R})\]
\end{example}


A type of frequently occuring operator is a bounded and continuous operator on normed linear spaces; indeed the Fourier transform 
This gives rise to an equivalent definition for a bounded linear operator when considering operators between normed linear spaces.

\begin{definition}[Bounded linear operator (normed linear space)]
A \emph{bounded linear operator} is a linear operator $T : X \to Y$ such that there exists some $c$ such that for all $\mathbf{x}\in X$ we have $\|T\mathbf{x}\|_Y \leq c \|\mathbf{x}\|_X$
In other words, it is Lipschitz continuous.
\end{definition}

Futhermore, one can equivalently prove that BLOs are precisely the linear operators that map bounded sets to bounded sets; under this condition we can generalize BLOs to any TLS.

\begin{definition}[Bounded linear operator (BLO)]
	A \emph{bounded linear operator (BLO)} is a linear operator $T : X \to Y$ such that $T(U)$ is bounded in $(Y,\mathcal{T}_Y)$ when $U$ is bounded in $(X,\mathcal{T}_X)$
\end{definition}



\begin{proposition}
Let $T : X \to Y$ be an linear operator on normed linear spaces, then $T$ is a BLO iff it is continuous.
\end{proposition}

\begin{corollary}
Let $T$ be a bijective BLO, then $T^{-1}$ is linear.
Let $T : X \to Y$ be a bijective operator, then $T^{-1}$ is continuous if $\|T\mathbf{x}\|_Y \leq c \|\mathbf{x}\|_X$.
\end{corollary}




\begin{corollary}
	Let $T : X \to Y$ be a BLO between normed linear spacs, then 
$T$ preserves linear spaces
$T$ preserves convex sets 
\end{corollary}







\begin{definition}[Functional]
A \emph{functional} is a BLO between normed linear space and its field $T : X \to F$.
%A \emph{functional} is a bounded linear operator from a linear space to its field $T : X \to F$.
\end{definition}



Extends the idea of projection matrixes to operators.
\begin{definition}[Projection]
A \emph{projection} is an idempotent operator.
An operation is idempotent if subsequent compositions of the operation do not change its result.
\[P^2=P\]
\end{definition}


Projections only have $0$ and $1$ as eigenvalues.



\section{Operator norms}


The "Lipschitz condition" that all BLOs between normed linear spaces obey implies that the quantity $\frac{\|T\mathbf{x}\|_Y}{\|\mathbf{x}\|_X}$ is bounded above by some Lipschitz constant; the Dedekind completeness of $\mathbb{R}$ means that $\{ \frac{\|T\mathbf{x}\|_Y}{\|\mathbf{x}\|_X} \in \mathbb{R} : x \in X \}$ has a supremum; this value can be used as the smallest possible Lipschitz constant, and we call this constant the \emph{operator norm}.

\begin{definition}[Operator norm]
	Let $T$ be an operator, then its \emph{operator norm} is defined by the following expression
	\[\|T\| = \sup_{\mathbf{x} \neq \mathbf{0}} \frac{\|T\mathbf{x}\|_Y}{\|\mathbf{x}\|_X }\]
\end{definition}

As it turns out (and I not-so-suavely alluded to by calling it the operator NORM) operator norms are indeed a norm! We denote these operator spaces by B(X,Y)

When the codomain of the operator is a complete space, the operator space is also complete.

Compositions of operators are operators

This follows directly from the fact that continuity and linearity are preserved under composition. What's more, however, is the following inequality.

\begin{proposition}
\[\|ST\| \leq \|S\|\|T\|\]
\end{proposition}


\begin{proposition}
	Let $T : X \to Y$
\[\|ST\| \leq \|S\|\|T\|\]
\end{proposition}


\begin{proposition}
If $T$ is a bijective BLO then $T^{-1}$ is a BLO, and $\|T^{-1}\| \geq \|T\|^{-1}$
\end{proposition}


\subsection{Matrix norms}

Linear maps between finite dimensional linear spaces $T : F^n \to F^m$ are examples of BLOs between normed linear spaces (one can always equip some $p$-norms on spaces of the form $F^n$, and of these the most common would be the $2$-norm). This class of operators can always be represented by a matrix, where its indexes are elements of $F$.

Operator norms can be difficult to calculate, however for linear maps between finite dimensional linear spaces, $\|\cdot\|_{1,1}$ and $\|\cdot\|_{\infty,\infty}$ are easy to calculate and they give rise to upper and lower bounds for $\|\cdot\|_{2,2}$ giving rise to approximations of the operator norm based on matrix entries such as the \emph{Frobenius norm}.


\begin{definition}[Frobenius norm]
	Let $\mathbf{A}$ be an $n \times m$ matrix, then the \emph{Frobenius norm} is the following expression.
\[ \|\mathbf{A}\|_{\mathrm{Frob}} = \sqrt{\sum^{n}_{i=1} \sum_{j=1}^{m} |[\mathbf{A}]_{ij}|^2} \]
\end{definition}

Although it is easy to state the definition of the Frobenius norm, it is not immediately apparent in which context it is useful. As previously stated, the $2$-norm is often of the most interest (especially since it is the norm resulting from the dot product on $F^n$, therefore being the norm of the unique Hilbert space on $F^n$); the Frobenius norm will eventually be found to be an upper bound of the operator norm when both spaces use a $2$-norm.

\[\|\mathbf{A}\|_{1,1} = \max\{ \sum^{n}_{i=1} |[\mathbf{A}_{ij}]|  \in \mathbb{R} : j \in [1,m] \cap \mathbb{N} \}\]
\[\|\mathbf{A}\|_{\infty,\infty} = \max\{ \sum^{m}_{j=1} |[\mathbf{A}_{ij}]|  \in \mathbb{R} : i \in [1,n] \cap \mathbb{N} \}\]
\[\|\mathbf{A}\|_{2,2} \leq \|\mathbf{A}\|_{\mathrm{Frob}} \]

- differentiation operator
- differential operator
- adjoint operator
- bounded linear operator
- operator norm

\section{Uniform bounded principle}
\section{Spectral theorem}
\section{Hahn-Banach theorem}
\section{Open mapping theorem}
\section{Closed graph theorem}
\section{Uniform bounded principle}