closed operator

Let $B$ be a Banach space.A linear operator $A:𝒟(A)\subset B\to B$ is said to be iffor every sequence ${\{x_n\}}_{n\in \mathbb{N}}$ in $𝒟(A)$ converging to $x\in B$ such that $A{x}_n\underset{n\to \mathrm{\infty }}{\overset{}{\to }}y\in B$, it holds $x\in 𝒟(A)$ 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 $𝒟(A)$.

A core of a closable operator is a subset $𝒞$ of $𝒟(A)$ such that the closure of the restriction of $A$ to $𝒞$ 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 $𝒟(A)$, both converging to $z\in B$, and such that both $\{A{x}_n\}$ and $\{A{y}_n\}$ converge, it holds ${lim}_{n}A{x}_n={lim}_{n}A{y}_n$.