Extensions of Symmetric Operators - Symmetric Operators

Symmetric Operators

Let H be a Hilbert space. A linear operator A acting on H with dense domain Dom(A) is symmetric if

<Ax, y> = <x, Ay>, for all x, y in Dom(A).

If Dom(A) = H, the Hellinger-Toeplitz theorem says that A is a bounded operator, in which case A is self-adjoint and the extension problem is trivial. In general, a symmetric operator is self-adjoint if the domain of its adjoint, Dom(A*), lies in Dom(A).

When dealing with unbounded operators, it is often desirable to be able to assume that the operator in question is closed. In the present context, it is a convenient fact that every symmetric operator A is closable. That is, A has a smallest closed extension, called the closure of A. This can be shown by invoking the symmetric assumption and Riesz representation theorem. Since A and its closure have the same closed extensions, it can always be assumed that the symmetric operator of interest is closed.

In the sequel, a symmetric operator will be assumed to be densely defined and closed.