PlanetMath: self-adjoint operator

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self-adjoint operator

(Definition)

A densely defined linear operator $A\colon\mathscr{D}(A)\subset \mathscr{H}\to\mathscr{H}$ on a Hilbert space $\mathscr{H}$ is a Hermitian or symmetric operator if for all $x,y\in \mathscr{D}(A)$ . This means that the adjoint of is defined at least on $\mathscr{D}(A)$ and that its restriction to that set coincides with . This fact is often denoted by $A\subset A^*$ .

The operator is self-adjoint if it coincides with its adjoint, i.e. if . If is closable and its closure coincides with its adjoint (i.e. $\overline{A}=A^*$ ), then is said to be essentially self-adjoint.

"self-adjoint operator" is owned by Koro.

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See Also: Hermitian matrix

Also defines:

Hermitian operator, symmetric operator, essentially self-adjoint, self-adjoint

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Cross-references: closure, closable, restriction, adjoint, operator, Hilbert space, linear operator, densely defined
There are 17 references to this entry.

This is version 5 of self-adjoint operator, born on 2003-07-28, modified 2006-06-15.
Object id is 4527, canonical name is SelfAdjointOperator.
Accessed 14042 times total.

Classification:

AMS MSC:	47B15 (Operator theory :: Special classes of linear operators :: Hermitian and normal operators )
	47B25 (Operator theory :: Special classes of linear operators :: Symmetric and selfadjoint operators )

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