Symmetric operator - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space. A densely defined linear operator $A:\mathcal D(A)\subseteq H\to H$ is symmetric when

\langle Ax,y\rangle=\langle x,Ay\rangle

for every $x,y\in\mathcal D(A)$ . Equivalently, $A\subseteq A^*$ : the domain $\mathcal D(A)$ is contained in $\mathcal D(A^*)$ , and $A^*x=Ax$ there, where $A^*$ denotes the adjoint of a densely defined operator. Symmetry is thus an inclusion of domain-sensitive operators. It does not by itself give equality with the adjoint.

Basic properties

Every symmetric operator is closable, and its closure is again symmetric. Its eigenvalues are real, and eigenvectors belonging to distinct eigenvalues are orthogonal. These facts resemble the bounded Hermitian theory, but the domain can obstruct both self-adjointness and the existence of a spectral resolution.

Relationship to self-adjointness

A self-adjoint unbounded operator is symmetric, but a symmetric operator can have a strictly larger adjoint domain. Self-adjoint extensions are controlled by the two deficiency subspaces $\ker(A^*-iI)$ and $\ker(A^*+iI)$ ; equal deficiency dimensions permit extensions, while both dimensions zero force self-adjointness Schmüdgen, chapter 1 and part VI.

Example and warning

On $L^2(0,1)$ , the operator $-i\,d/dx$ with domain $C_c^\infty(0,1)$ is symmetric. Its adjoint has a larger Sobolev domain, so it is not self-adjoint.

References

Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012. DOI record. Relevant: chapter 1 and part VI on symmetric operators and self-adjoint extensions.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised edition, Academic Press, 1980. Publisher record. Relevant: chapter VIII on unbounded operators.