closed operator

Let $B$ be a Banach space. A linear operator $A\colon\mathscr{D}(A)\subset B\to B$ is said to be 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 $Ax=y$. 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 $\mathscr{D}(A)$.

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

The following properties are easily checked:

1. 1.

Any bounded linear operator defined on the whole space $B$ is closed;

2. 2.

If $A$ is closed then $A-\lambda I$ is closed;

3. 3.

If $A$ is closed and it has an inverse, then $A^{-1}$ is also closed;

4. 4.

An operator $A$ 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 $\lim_{n}Ax_{n}=\lim_{n}Ay_{n}$.

Title closed operator ClosedOperator 2013-03-22 13:48:20 2013-03-22 13:48:20 Koro (127) Koro (127) 9 Koro (127) Definition msc 47A05 closed closure closable core