# Adjoint Operator

## adjoint operator

[′aj‚ȯint ′äp·ə‚rād·ər] (mathematics)

An operator

*B*such that the inner products (*Ax,y*) and (*x,By*) are equal for a given operator*A*and for all elements*x*and*y*of a Hilbert space. Also known as associate operator; Hermitian conjugate operator.## 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 (*see*DENSE 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.