Adjoint Operator

adjoint operator

[′aj‚ȯint ′äp·ə‚rād·ər]
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


where Adjoint Operator 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 xDm. It is assumed here that T*y = y*.

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

References in periodicals archive ?
2] be a bounded linear operator with its adjoint operator [A.
We use the same notation H to denote this self adjoint operator.
is the adjoint operator of P with respect to [<x, x>.
continuous, T is the adjoint operator of some L : X [right arrow] Y.
In the theory of inverse problems, we are asked to find the solution of the operator equation Bf = g where B: H [right arrow] H is a self adjoint operator on a Hilbert space H, and g e H is the exact datum.
c] is a linear, closed, unbounded, self adjoint operator defined on a dense subspace of [L.
Additional Key Words and Phrases: Adjoint model, adjoint operator, automatic differentiation, computational differentiation, data assimilation, differentiation of algorithms, implicit functions, inverse modeling, optimization, reverse mode
For instance, the spectrum of the adjoint operator of the equation linearized around a pulse has not been analysed carefully in this case.
dagger]] be the adjoint operator and the Moore-Penrose generalized inverse of linear operator T, respectively.
His topics include the delta function, the eigenfunction method, the adjoint operator, principal solutions, and Green's function method for the wave operator.
T] = V[absolute value of T] consists of an isometry V and an unbounded self adjoint operator [absolute value of T]: = [(T x [bar.
x] belongs either to the range of the Banach space adjoint operator [A.