Let $B$ be a Banach space. A linear operator $A\colon\mathscr{D}(A)\subset B\to B$ is said to be closed if for every sequence $\{x_n\}_{n\in \N}$ 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 $\lim_n Ax_n = \lim_n Ay_n$