closed operator
Let $B$ be a Banach space^{}. A linear operator^{} $A:\mathcal{D}(A)\subset B\to B$ is said to be if for every sequence ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ in $\mathcal{D}(A)$ converging to $x\in B$ such that $A{x}_{n}\underset{n\to \mathrm{\infty}}{\overset{}{\to}}y\in B$, it holds $x\in \mathcal{D}(A)$ 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 $\mathcal{D}(A)$.
A core of a closable operator is a subset $\mathcal{C}$ of $\mathcal{D}(A)$ such that the closure of the restriction of $A$ to $\mathcal{C}$ 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 $\mathcal{D}(A)$, both converging to $z\in B$, and such that both $\{A{x}_{n}\}$ and $\{A{y}_{n}\}$ converge, it holds ${lim}_{n}A{x}_{n}={lim}_{n}A{y}_{n}$.
Title  closed operator 

Canonical name  ClosedOperator 
Date of creation  20130322 13:48:20 
Last modified on  20130322 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 