Self-adjoint Operator - Self-adjoint Operators

Self-adjoint Operators

Given a densely defined linear operator A on H, its adjoint A* is defined as follows:

• The domain of A* consists of vectors x in H such that

(which is a densely defined linear map) is a continuous linear functional. By continuity and density of the domain of A, it extends to a unique continuous linear functional on all of H.

• By the Riesz representation theorem for linear functionals, if x is in the domain of A*, there is a unique vector z in H such that

This vector z is defined to be A* x. It can be shown that the dependence of z on x is linear.

Notice that it is the denseness of the domain of the operator, along with the uniqueness part of Riesz representation, that ensures the adjoint operator is well defined.

A result of Hellinger-Toeplitz type says that an operator having an everywhere defined bounded adjoint is bounded.

The condition for a linear operator on a Hilbert space to be self-adjoint is stronger than to be symmetric.

For any densely defined operator A on Hilbert space one can define its adjoint operator A*. For a symmetric operator A, the domain of the operator A* contains the domain of the operator A, and the restriction of the operator A* on the domain of A coincides with the operator A, i.e., in other words A* is extension of A. For a self-adjoint operator A the domain of A* is the same as the domain of A, and A=A*. See also Extensions of symmetric operators and unbounded operator.