Hilbert Space - Operators On Hilbert Spaces - Unbounded Operators

Unbounded Operators

Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics. An unbounded operator T on a Hilbert space H is defined as a linear operator whose domain D(T) is a linear subspace of H. Often the domain D(T) is a dense subspace of H, in which case T is known as a densely defined operator.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space L2(R) are:

• A suitable extension of the differential operator

where i is the imaginary unit and f is a differentiable function of compact support.

• The multiplication-by-x operator:

These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L2(R).