Self-adjoint operator

Hermitian operator

A linear operator $ A $ defined on a linear everywhere-dense set $ D ( A) $ in a Hilbert space $ H $ and coinciding with its adjoint operator $ A ^ {*} $, that is, such that $ D ( A) = D ( A ^ {*} ) $ and

$$ \tag{* } \langle Ax, y \rangle = \langle x, Ay \rangle $$

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

Every self-adjoint operator uniquely determines a resolution of the identity $ E _ \lambda $, $ - \infty < \lambda < \infty $; the following integral representation holds:

$$ Ax = \int\limits _ {- \infty } ^ \infty \lambda dE _ \lambda x, $$

where the integral is understood as the strong limit of the integral sums for each $ x \in D ( A) $, and

$$ D ( A) = \ \left \{ {x } : { \int\limits _ {- \infty } ^ \infty \lambda ^ {2} d \langle E _ \lambda x, x \rangle < \infty } \right \} . $$

The spectrum of a self-adjoint operator is non-empty and lies on the real line. The quadratic form $ K ( A) = \langle Ax, x \rangle $ generated by a self-adjoint operator $ A $ 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.

References

[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)

Comments

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