# Self-adjoint operator

*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.

#### 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.

**How to Cite This Entry:**

Self-adjoint operator. V.I. Sobolev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Self-adjoint_operator&oldid=16222