Hermitian Operator

Hermitian operator [er′mish·ən ′äp·ə‚rād·ər]

(mathematics)

A linear operatorAon vectors in a Hilbert space, such that ifxandyare in the range ofAthen 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 (seeINTEGRAL 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.