adjoint operator[′aj‚ȯint ′äp·ə‚rād·ər]
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
where is the complex conjugate of the function K(x, y).
If T is not bounded and if its domain of definition Dm 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 ∊ Dm. It is assumed here that T*y = y*.
The concept of adjoint operator can be extended to operators in other spaces.