# Contents

## Definition

Let $A:H\to H$ be an unbounded operator on a Hilbert space $H$. An unbounded operator ${A}^{*}$ is its adjoint if

• $\left(\mathrm{Ax}\mid y\right)=\left(x\mid {A}^{*}y\right)$ for all $x\in \mathrm{dom}\left(A\right)$, $y\in \mathrm{dom}\left({A}^{*}\right)$
• every $B$ satisfying the above property for ${A}^{*}$ is a restriction of $A$

An adjoint does not need to exist in general.

An unbounded operator is symmetric if $\mathrm{dom}\left(A\right)\subset \mathrm{dom}\left({A}^{*}\right)$ and $Ax={A}^{*}x$ for all $x\in \mathrm{dom}\left(A\right)$ (one also writes $A\subset {A}^{*}$).

The domain of ${A}^{*}$ is the set of all vectors $y\in H$ such that the linear functional $x↦\left(\mathrm{Ax}\mid y\right)$ is bounded on $\mathrm{dom}\left(A\right)$.

The graph ${\Gamma }_{A}\subset H\oplus H$ satisfies ${\Gamma }_{{A}^{*}}=\tau \left({\Gamma }_{A}{\right)}^{\perp }$ where $\perp$ denotes the orthogonal complement and $\tau$ denotes the transposition of the direct summands changing the sign of one of the factors, i.e. $x\oplus y↦-y\oplus x$. An unbounded operator $A$ is closed if ${\Gamma }_{A}$ is closed subspace of $H\oplus H$. An operator $B$ is a closure of an operator $A$ if ${\Gamma }_{B}$ is a closure of operator ${\Gamma }_{A}$. It is said that $B$ is an extension of $A$ and one writes $B\supset A$ if ${\Gamma }_{B}\supset {\Gamma }_{A}$. The closure of an unbounded operator does not need to exist.

For any unbounded operator $A$ with a dense $\mathrm{dom}\left(A\right)susbetH$, if the adjoint operator ${A}^{*}$ exists, ${A}^{*}$ is closed, and if $\left({A}^{*}{\right)}^{*}$ exists then it coincides with a closure of $A$.

An unbounded operator $A:H\to H$ on a Hilbert space $H$ is self-adjoint if

• it has a densely defined domain $\mathrm{dom}\left(A\right)\subset H$
• $A={A}^{*}$, i.e. $\mathrm{dom}\left({A}^{*}\right)=\mathrm{dom}\left(A\right)$ and $\mathrm{Ax}={A}^{*}x$ for all $x\in \mathrm{dom}\left(A\right)$

An (unbounded) operator is essentially self-adjoint if it is symmetric and its spectrum is a subset of the real line. Alternatively, it is symmetric if its closure is self-adjoint.

A Hermitean (or hermitian) operator is the same as a self-adjoint operator, though some authors prefer that terminology for bounded self-adjoint operators.

For a bounded operator $A:H\to K$ between Hilbert spaces define the Hermitean conjugate operator ${A}^{*}:K\to H$ by $\left(\mathrm{Ax}\mid y{\right)}_{H}=\left(x\mid {A}^{*}y{\right)}_{K}$, for all $x\in K$, $y\in H$. Distinguish it from the concept of the transposed operator? ${A}^{T}:{K}^{*}\to {H}^{*}$ between the dual spaces.

## References

• A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988
• A. B. Antonevič, Ja. B. Radyno, Funkcional’nij analiz i integral’nye uravnenija, Minsk 1984
• S. Kurepa, Funkcionalna analiza, elementi teorije operatora, Školska knjiga, Zagreb 1981.
• Reed, M.; Simon, B.: Methods of modern mathematical physics. Volume 1, Functional Analysis
• Walter Rudin, Functional analysis

Revised on May 17, 2013 02:45:12 by Urs Schreiber (82.169.65.155)