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.

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 $\mathscr{D}(A)$ , 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
Canonical name	ClosedOperator
Date of creation	2013-03-22 13:48:20
Last modified on	2013-03-22 13:48:20
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	9
Author	Koro (127)
Entry type	Definition
Classification	msc 47A05
Synonym	closed
Defines	closure
Defines	closable
Defines	core