Self-adjoint Operator - Symmetric Operators

Symmetric Operators

A partially defined linear operator A on a Hilbert space H is called symmetric if

for all elements x and y in the domain of A. More generally, a partially defined linear operator A from a topological vector space E into its continuous dual space E∗ is said to be symmetric if

for all elements x and y in the domain of A. This usage is fairly standard in the functional analysis literature.

A symmetric everywhere defined operator is self-adjoint. By the Hellinger-Toeplitz theorem, a symmetric everywhere defined operator is also bounded.

Bounded symmetric operators are also called Hermitian.

The previous definition agrees with the one for matrices given in the introduction to this article, if we take as H the Hilbert space Cn with the standard dot product and interpret a square matrix as a linear operator on this Hilbert space. It is however much more general as there are important infinite-dimensional Hilbert spaces.

The spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may have no eigenvalues.

A general version of the spectral theorem which also applies to bounded symmetric operators (see Reed and Simon, vol. 1, chapter VII, or other books cited) is stated below. If the set of eigenvalues for a symmetric operator is non empty, and the eigenvalues are nondegenerate, then it follows from the definition that eigenvectors corresponding to distinct eigenvalues are orthogonal. Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all (although the spectrum of any self-adjoint operator is nonempty). The example below illustrates the special case when an (unbounded) symmetric operator does have a set of eigenvectors which constitute a Hilbert space basis. The operator A below can be seen to have a compact inverse, meaning that the corresponding differential equation A f = g is solved by some integral, therefore compact, operator G. The compact symmetric operator G then has a countable family of eigenvectors which are complete in . The same can then be said for A.

Example. Consider the complex Hilbert space L2 and the differential operator

defined on the subspace consisting of all complex-valued infinitely differentiable functions f on with the boundary conditions:

Then integration by parts shows that A is symmetric. Its eigenfunctions are the sinusoids

with the real eigenvalues n2π2; the well-known orthogonality of the sine functions follows as a consequence of the property of being symmetric.

We consider generalizations of this operator below.