Core of a closed operator - Imported Knowlpedia Content

Definition

Let $T:\operatorname{Dom}(T)\subseteq X\to X$ be a closed linear operator on a Banach space. A linear subspace $D\subseteq\operatorname{Dom}(T)$ is a core for $T$ if the closure of the restricted operator $T|_D$ is $T$ . Equivalently, $D$ is dense in $\operatorname{Dom}(T)$ for the graph norm

\|x\|_T=\|x\|+\|Tx\|.

Thus every $x\in\operatorname{Dom}(T)$ admits $x_n\in D$ with both $x_n\to x$ and $Tx_n\to Tx$ in $X$ . Density of $D$ in $X$ alone is not enough.

Graph interpretation

The graph of $T|_D$ is

\{(x,Tx):x\in D\}\subseteq X\times X.

The core condition says that its closure is exactly the graph of $T$ . Consequently, values of $T$ on the smaller test domain determine the closed operator uniquely. This is why differential operators are often first computed on smooth compactly supported functions and then recovered by graph closure.

How cores are used

Suppose $S$ is a closable operator with $\operatorname{Dom}(S)=D\subseteq\operatorname{Dom}(T)$ and $Sx=Tx$ on $D$ . Then $D$ is a core for $T$ exactly when $\overline S=T$ . In particular, proving that a proposed test domain is a core justifies checking identities and approximation arguments there before passing to the full domain Schmüdgen, Chapter 1.

Warning: ambient density versus graph density

For an unbounded operator, convergence $x_n\to x$ does not control $Tx_n$ . A dense invariant subspace can therefore fail to be a core. Likewise, “essential domain” is sometimes used as a synonym for a core, but in discussions of symmetric operators the nearby phrase “essentially self-adjoint” adds a separate assertion about the closure being self-adjoint.

References

Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1995. Publisher record. Relevant: Chapter III on closability, graph norms, and cores.
Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012. Publisher record. Relevant: Chapter 1 on closed operators and core domains.