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 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 ?
is the adjoint operator of P with respect to [<x, x>.
1] show up in the definition of the operator P or its adjoint operator [P.
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.
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.
x] belongs either to the range of the Banach space adjoint operator [A.
The coefficiens are evaluated by inner product with a set of functions related to the orthogonal basis through the adjoint operator of the linear operator.
1] is a Hilbert-Schmidt operator (we remind that a bounded self adjoint operator is a Hilbert-Schmidt operator if a sum of its squared matrix elements in an orthonormal basis is finite; the spectrum of such an operator is discrete, with a single accumulation point at 0).
Organization is in eight chapters covering linear equations, systems of linear first order equations, power series solutions, adjoint operators and nonhomogeneous boundary value problems, green functions, eigenfunction expansions, long time behavior of systems of differential equations, and existence and uniqueness theorems.
Self adjoint operators can be partially ordered by declaring S [less than or equal to]T if and only if [less than]Sx/x[greater than] [less than or equal to] [less than]Tx/x[greater than] for all x in H.
Self adjoint operators can be partially ordered by declaring S [less than or equal to] T if and only if <Sx | x> [less than or equal to] <Tx | x> for all x in H.