Self-adjoint unbounded operator - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space and let $A:\mathcal D(A)\subseteq H\to H$ be densely defined. In the unbounded-operator setting, $A$ is self-adjoint when

A=A^*,

meaning both $\mathcal D(A)=\mathcal D(A^*)$ and $Ax=A^*x$ for every vector in that common domain, where $A^*$ is the adjoint of a densely defined operator. No boundedness is assumed. Equality of the operator formulas without equality of their domains is insufficient. Every self-adjoint operator is a closed linear operator and, in particular, has a dense domain.

Equivalent characterizations

A densely defined symmetric operator $A$ is self-adjoint if and only if

\operatorname{Ran}(A+iI)=H=\operatorname{Ran}(A-iI).

Equivalently, its deficiency subspaces $\ker(A^*-iI)$ and $\ker(A^*+iI)$ both vanish. These criteria are developed in Schmüdgen, chapter 1.

Spectral consequences

The spectrum of a self-adjoint operator is real, and $A-zI$ has a bounded inverse for every nonreal $z$ . The spectral theorem then supplies projection-valued functional calculus, making self-adjoint operators the appropriate mathematical model for possibly unbounded observables.

Example and boundary case

On $L^2(\mathbb R)$ , multiplication by the coordinate,

(Af)(x)=xf(x),\qquad \mathcal D(A)=\{f\in L^2(\mathbb R):xf\in L^2(\mathbb R)\},

is self-adjoint and unbounded. Restricting the same formula to a smaller dense domain may produce only a symmetric operator; the domain is therefore part of the operator, not bookkeeping.

References

Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012. DOI record. Relevant: chapter 1 on closed, adjoint, symmetric, and self-adjoint operators.
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.