PlanetMath: closed operator

(more info)

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closed operator

(Definition)

Let be a Banach space. A linear operator $A\colon\mathscr{D}(A)\subset B\to B$ is said to be closed if for every sequence $\{x_n\}_{n\in \mathbb{N}}$ in $\mathscr{D}(A)$ converging to $x\in B$ such that $Ax_n\xrightarrow[n\to\infty]{} y\in B$ , it holds $x\in\mathscr{D}(A)$ and . Equivalently, is closed if its graph is closed in $B\oplus B$ .

Given an operator , not necessarily closed, if the closure of its graph in $B\oplus B$ happens to be the graph of some operator, we call that operator the closure of , and we say that is closable. We denote the closure of by $\overline{A}$ . It follows easily that is the restriction of $\overline{A}$ to $\mathscr{D}(A)$ .

A core of a closable operator is a subset $\mathscr{C}$ of $\mathscr{D}(A)$ such that the closure of the restriction of to $\mathscr{C}$ is $\overline{A}$ .

The following properties are easily checked:

Any bounded linear operator defined on the whole space is closed;
If is closed then $A-\lambda I$ is closed;
If is closed and it has an inverse, then $A^{-1}$ is also closed;
An operator admits a closure if and only if for every pair of sequences $\{x_n\}$ and $\{y_n\}$ in $\mathscr{D}(A)$ , both converging to $z\in B$ , and such that both $\{Ax_n\}$ and $\{Ay_n\}$ converge, it holds $\lim_n Ax_n = \lim_n Ay_n$ .