Closed linear operator - Imported Knowlpedia Content

Definition

Let $X$ and $Y$ be normed vector spaces, and let $T:D(T)\subseteq X\to Y$ be linear on the linear subspace $D(T)$ . The operator $T$ is closed if its graph

\Gamma(T)=\{(x,Tx):x\in D(T)\}

is a closed set in $X\times Y$ . Equivalently, whenever $x_n\in D(T)$ , $x_n\to x$ in $X$ , and $Tx_n\to y$ in $Y$ , one has $x\in D(T)$ and $Tx=y$ . Closedness constrains simultaneous convergence of inputs and outputs; it neither says that $D(T)$ is closed in $X$ nor implies that $T$ is bounded on $D(T)$ with the norm inherited from $X$ .

Closed versus bounded

The closed graph theorem says that a closed linear operator $T:X\to Y$ defined on all of a Banach space $X$ , with $Y$ Banach, is bounded. Properly defined closed operators can be unbounded. Differential operators on $L^p$ - or Hilbert spaces are principal examples: their domains encode the regularity and boundary conditions needed for the graph to be closed.

Closure and closability

An operator is closable if the closure of $\Gamma(T)$ is itself the graph of an operator. This happens exactly when

x_n\to0,\quad Tx_n\to y \quad\Longrightarrow\quad y=0.

The operator whose graph is $\overline{\Gamma(T)}$ is the closure $\overline T$ , the smallest closed extension of $T$ . Hence “closed” and “closable” are not synonyms.

Domain-sensitive operations

For unbounded operators, algebraic expressions have domain conditions. The product $ST$ is defined only on those $x\in D(T)$ for which $Tx\in D(S)$ , and a sum requires a common domain. Closed operators are therefore not automatically preserved by sums or products. Statements about adjoints, resolvents, or self-adjointness must likewise include density and domain hypotheses.

References

Tosio Kato, Perturbation Theory for Linear Operators, Springer, 1995 reprint of the 2nd ed. Springer DOI record. Relevant: Chapter III.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1980. Elsevier book record. Relevant: Chapter VIII.