nLab
self-adjoint operator

Definition

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

• $(\mathrm{Ax}\mid y)=(x\mid A^{*}y)$ for all $x\in \mathrm{dom}(A)$, $y\in \mathrm{dom}(A^{*})$
• 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 $\mathrm{dom}(A)\subset \mathrm{dom}(A^{*})$ and $Ax=A^{*}x$ for all $x\in \mathrm{dom}(A)$ (one also writes $A\subset A^{*}$).

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

The graph ${\Gamma }_A\subset H\oplus H$ satisfies ${\Gamma }_{A^{*}}=\tau ({\Gamma }_A{)}^{\perp }$ 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. $x\oplus y\mapsto -y\oplus x$. 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 $\mathrm{dom}(A)susbetH$, if the adjoint operator $A^{*}$ exists, $A^{*}$ is closed, and if $(A^{*}{)}^{*}$ 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 $\mathrm{dom}(A)\subset H$
• $A=A^{*}$, i.e. $\mathrm{dom}(A^{*})=\mathrm{dom}(A)$ and $\mathrm{Ax}=A^{*}x$ for all $x\in \mathrm{dom}(A)$

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 $(\mathrm{Ax}\mid y{)}_H=(x\mid A^{*}y{)}_K$, 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.

Related concepts

• self-adjoint extension

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