self-adjoint operator

A densely defined linear operatorMathworldPlanetmath A:𝒟(A) on a Hilbert space is a Hermitian or symmetricPlanetmathPlanetmathPlanetmathPlanetmath operatorMathworldPlanetmath if (Ax,y)=(x,Ay) for all x,y𝒟(A). This means that the adjointPlanetmathPlanetmathPlanetmath A* of A is defined at least on 𝒟(A) and that its restrictionPlanetmathPlanetmath to that set coincides with A. This fact is often denoted by AA*.

The operator A is self-adjointPlanetmathPlanetmath if it coincides with its adjoint, i.e. if A=A*. If A is closable and its closureMathworldPlanetmathPlanetmath coincides with its adjoint (i.e. A¯=A*), then A is said to be essentially self-adjoint.

Title self-adjoint operator
Canonical name SelfadjointOperator
Date of creation 2013-03-22 13:48:23
Last modified on 2013-03-22 13:48:23
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Definition
Classification msc 47B15
Classification msc 47B25
Related topic HermitianMatrix
Defines Hermitian operator
Defines symmetric operator
Defines essentially self-adjoint
Defines self-adjoint