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.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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 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.

Mentioned in

References in periodicals archive

where [mathematical expression not reproducible] denotes the orthogonal projection in [L.sup.2]([OMEGA]) onto the admissible set of the control and [B.sup.*] is the adjoint operator of B.

Finite Volume Element Approximation for the Elliptic Equation with Distributed Control

The matrix-vector multiplication should be replaced with the specifically designed function to perform the equivalent of the forward operator [g.sub.A] and adjoint operator [g.sup.H.sub.A].

Group Sparse Basis Pursuit Denoising Reconstruction Algorithm for Polarimetric Through-the-Wall Radar Imaging

The main properties of the adjoint operator were stated in Ch.

On Some Inequalities with Operators in Hilbert Spaces

Throughout the paper, we will assume that A(t) is invertible, A(t) = [A.sup.*](t) for all [mathematical expression not reproducible] for all t [member of] [q.sup.N], where * denotes the adjoint operator. Note that we can also define the operator L using the infinite matrix

SPECTRAL ANALYSIS OF A MATRIX-VALUED QUANTUM-DIFFERENCE OPERATOR

We use the same notation H to denote this self adjoint operator. The operator admits the partial wave expansion.

Aharonov-Bohm effect in resonances for scattering by three solenoids

PA = A[P.sup.*] where [P.sup.*] is the adjoint operator of P with respect to [<x, x>.sub.H], defined by [P.sup.*]x = x - U[<U, AU>.sup.-1.sub.H][<AU,x>.sub.H].

Preconditioned recycling Krylov subspace methods for self-adjoint problems

(i) [??] (ii) Let X and Y be left Banach G-modules and suppose that T : X [right arrow] Y is a bounded linear G-module map and [T.sup.*] : [Y.sup.*] [right arrow] [X.sup.*] is its adjoint operator. For [mu] [member of] M(G), x [member of] X and [phi] [member of] [Y.sup.*],

Module maps and invariant subsets of Banach modules of locally compact groups

It is claimed that the mapping f is contraction with constant [beta] = 1/2, A is a bounded linear operator on R with adjoint operator [A.sup.*] and [parallel]A[parallel] = [parallel][A.sup.*] [parallel] = 1/2, and B is a strongly positive bounded linear self-adjoint operator with constant [bar.[gamma]] = 1 on R.

Explicit scheme for fixed point problem for nonexpansive semigroup and split equilibrium problem in Hilbert space

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.

A modified regularization scheme for classification problems

Recall that the adjoint operator [T.sup.*] is then defined by [T.sup.*][phi] = [[phi].sup.*] for all [phi] [member of] D([T.sup.*]).

Approximation of bandlimited functions on a non-compact manifold by bandlimited functions on compact submanifolds