Adjoint Operator (redirected from Hermitian adjoint)

adjoint operator [′aj‚ȯint ′äp·ə‚rād·ər]

(mathematics)

An operatorBsuch that the inner products (Ax,y) and (x,By) are equal for a given operatorAand for all elementsxandyof a Hilbert space. Also known as associate operator; Hermitian conjugate operator.

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Adjoint Operator 

a concept of operator theory. Two bounded linear operators T and T* on a Hilbert space H are said to be adjoint if, for all vectors x and y in H,

(Tx, y) = (x, T*y)

For example, if

then the adjoint of the operator

is

where is the complex conjugate of the function K(x, y).

If T is not bounded and if its domain of definition D_m is everywhere dense (seeDENSE AND NONDENSE SETS), then the adjoint of T is defined on the set of vectors y for which a vector y* can be found such that the equality (Tx, y) = (x, y*) holds for all x ∊ D_m. It is assumed here that T*y = y*.

The concept of adjoint operator can be extended to operators in other spaces.