A linear operator on a Hilbert space with a dense domain of definition and such that for any . This condition is equivalent to: 1) ; and 2) for all , where is the adjoint of , that is, to . A bounded Hermitian operator is either defined on the whole of or can be so extended by continuity, and then , that is, is a self-adjoint operator. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a Hermitian matrix.
|||N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)|
|||F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)|
Hermitian operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_operator&oldid=13375