Self-adjoint operator

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Hermitian operator

A linear operator defined on a linear everywhere-dense set in a Hilbert space and coinciding with its adjoint operator , that is, such that and


for every . Every self-adjoint operator is closed and cannot be extended with the preservation of (*) to a linear manifold wider than ; in view of this a self-adjoint operator is also called hypermaximal. Therefore, if is a bounded self-adjoint operator, then it is defined on the whole of .

Every self-adjoint operator uniquely determines a resolution of the identity , ; the following integral representation holds:

where the integral is understood as the strong limit of the integral sums for each , and

The spectrum of a self-adjoint operator is non-empty and lies on the real line. The quadratic form generated by a self-adjoint operator is real, and this enables one to introduce the concept of a positive operator.

Many boundary value problems of mathematical physics are described by means of self-adjoint operators.


[1] L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Wiley (1974) (Translated from Russian)
[2] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[3] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)


Cf. also Hermitian operator; Symmetric operator; Self-adjoint linear transformation.

How to Cite This Entry:
Self-adjoint operator. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098