Densely defined operator - Imported Knowlpedia Content

Definition

Let $H$ and $K$ be Hilbert spaces. A densely defined operator from $H$ to $K$ is a pair $(D(T),T)$ , where $D(T)$ is a linear subspace of $H$ , dense in $H$ , and

T:D(T)\longrightarrow K

is a linear map. The domain $D(T)$ is part of the operator: two maps given by the same formula on different domains are different operators. When $H=K$ , one says that $T$ is an operator in $H$ . No boundedness, continuity, or closedness is assumed.

The adjoint

Density is precisely what makes an adjoint single-valued. Using inner products linear in the first variable, $y\in K$ belongs to $D(T^*)$ when there is $z\in H$ such that

\langle Tx,y\rangle_K=\langle x,z\rangle_H

for every $x\in D(T)$ ; then $T^*y=z$ . The vector $z$ is unique because $D(T)$ is dense. The adjoint is always closed, although its own domain need not be dense.

Graph, closure, and examples

The graph of $T$ is the linear subspace $\{(x,Tx):x\in D(T)\}\subseteq H\oplus K$ . The operator is closed when this graph is closed, and closable when its graph closure is again the graph of an operator. On an $L^2$ -space, differentiation on smooth compactly supported functions and multiplication by an unbounded measurable function are standard densely defined operators. Their domains cannot be discarded without changing their adjoints and closures.

Conventions and scope

References

K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012. Springer DOI record. Relevant: Chapter 1, “Basics of Closed Operators.”