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

• $(A x|y) = (x|A^*y)$ for all $x\in dom(A)$ and $y\in dom(A^*)$; and
• 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 $dom(A)\subset dom(A^*)$ and $A x = A^*x$ for all $x\in dom(A)$ (one also writes $A\subset A^*$).

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

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 $dom(A)\subset H$, if the adjoint operator $A^*$ exists, then $A^*$ 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 $dom(A)\subset H$
• $A = A^*$, i.e. $dom(A^*)= dom(A)$ and $A x = A^* x$ for all $x\in dom(A)$

An (unbounded) operator is essentially self-adjoint if it is symmetric and its spectrum (as a subspace of the complex plane) is contained in the real line. Alternatively, it is symmetric if its closure is self-adjoint.

A Hermitean (or hermitian) operator is a bounded symmetric operator (which is necessarily self-adjoint), although some authors use the term for any self-adjoint operator.

For a bounded operator $A: H\to K$ between Hilbert spaces, define the Hermitean conjugate operator $A^*: K\to H$ by $(Ax|y)_H = (x|A^*y)_K$, 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.

In an arbitrary $*$-algebra, a self-adjoint or hermitian element is any element $A$ such that $A^* = A$.

Related concepts

• self-adjoint extension

• unitary operator

• Hermitian form, Hermitian matrix

• quantum observable, daseinisation

• positive operator

References

Original articles:

• John von Neumann, §I in: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1930) 49–131 [doi:10.1007/BF01782338]

See also:

• 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

• Konrad Schmuedgen, Unbounded self-adjoint operators on Hilbert space, Springer GTM 265