closed operator
ClosedOperator
2003-07-28 12:29:33
2006-06-08 18:50:43
Definition
closure
closable
core
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Let $B$ be a Banach space.
A linear operator $A\colon\mathscr{D}(A)\subset B\to B$ is said to be \PMlinkescapetext{\textbf{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 \textbf{closure} of $A$, and we say that $A$ is \textbf{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 \textbf{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:
\begin{enumerate}
\item Any bounded linear operator defined on the whole space $B$ is closed;
\item If $A$ is closed then $A-\lambda I$ is closed;
\item If $A$ is closed and it has an inverse, then $A^{-1}$ is also closed;
\item 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 $\lim_n Ax_n = \lim_n Ay_n$.
\end{enumerate}