closed operator

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

Given an operator $A$ , 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 $A$ , and we say that $A$ is closable. We denote the closure of $A$ by $\overline{A}$ . It follows easily that $A$ is the restriction of $\overline{A}$ to .

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

The following properties are easily checked:

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