self-adjoint operator SelfAdjointOperator 2003-07-28 12:59:30 2006-06-15 19:31:03 Definition Hermitian operator symmetric operator essentially self-adjoint self-adjoint % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Per}{\operatorname{Per}} \PMlinkescapeword{symmetric} \PMlinkescapeword{Hermitian} A densely defined linear operator $A\colon\mathscr{D}(A)\subset \mathscr{H}\to\mathscr{H}$ on a Hilbert space $\mathscr{H}$ is a \emph{Hermitian} or \emph{symmetric} operator if $(Ax,y) = (x,Ay)$ for all $x,y\in \mathscr{D}(A)$. This means that the adjoint $A^*$ of $A$ is defined at least on $\mathscr{D}(A)$ and that its restriction to that set coincides with $A$. This fact is often denoted by $A\subset A^*$. The operator $A$ is \emph{self-adjoint} if it coincides with its adjoint, i.e. if $A=A^*$. If $A$ is closable and its closure coincides with its adjoint (i.e. $\overline{A}=A^*$), then $A$ is said to be \emph{essentially self-adjoint}.