# Hermitian Operator

## Hermitian operator

[er′mish·ən ′äp·ə‚rād·ər]*A*on vectors in a Hilbert space, such that if

*x*and

*y*are in the range of

*A*then the inner products (

*Ax,y*) and (

*x,Ay*) are equal.

## Hermitian Operator

an infinite-dimensional analogue of the Hermitian linear transformation. A bounded linear operator *A* in a complex Hilbert space *H* is said to be Hermitian if for any two vectors *x* and *y* in the space the relation (*Ax, y) = (x, Ay)* holds, where (*x, y)* is the scalar product of *H.* Examples of Hermitian operators are integral operators (*see*INTEGRAL EQUATIONS) for which the kernel *K(x, y)* is given in a bounded region and is a continuous function such that ; in this case, *K(x, y)* is called a Hermitian kernel. The concept of Hermitian operators may be extended to unbounded linear operators in a Hilbert space.

Hermitian operators play an important role in quantum mechanics, providing a convenient means of describing mathematically the observable quantities that characterize a physical system.