Closed linear operator

The closed graph theorem states that a linear operator ${\displaystyle f:X\to Y}$ between Banach spaces with full domain ${\displaystyle X}$ is a closed operator if and only if it is a bounded operator. In practice, many operators are unbounded, but it is still desirable to make them have a closed graph. Hence, they cannot be defined on all of ${\displaystyle X}$. To stay useful, they are instead defined on a proper but dense subspace, which still allows approximating any vector and keeps key tools (closures, adjoints, spectral theory) available.

Definition

It is common in functional analysis to consider partial functions, which are functions defined on a subset of some space ${\displaystyle X.}$ A partial function ${\displaystyle f}$ is declared with the notation ${\displaystyle f:D\subseteq X\to Y,}$ which indicates that ${\displaystyle f}$ has prototype ${\displaystyle f:D\to Y}$ (that is, its domain is ${\displaystyle D}$ and its codomain is ${\displaystyle Y}$)

Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the graph of a partial function ${\displaystyle f}$ is the set ${\displaystyle \mathrm{graph}(f)=\{(x,f(x)):x\in \mathrm{dom}f\}.}$ However, one exception to this is the definition of "closed graph". A partial function ${\displaystyle f:D\subseteq X\to Y}$ is said to have a closed graph if ${\displaystyle \mathrm{graph}f}$ is a closed subset of ${\displaystyle X\times Y}$ in the product topology; importantly, note that the product space is ${\displaystyle X\times Y}$ and not ${\displaystyle D\times Y=\mathrm{dom}f\times Y}$ as it was defined above for ordinary functions. In contrast, when ${\displaystyle f:D\to Y}$ is considered as an ordinary function (rather than as the partial function ${\displaystyle f:D\subseteq X\to Y}$), then "having a closed graph" would instead mean that ${\displaystyle \mathrm{graph}f}$ is a closed subset of ${\displaystyle D\times Y.}$ If ${\displaystyle \mathrm{graph}f}$ is a closed subset of ${\displaystyle X\times Y}$ then it is also a closed subset of ${\displaystyle \mathrm{dom}(f)\times Y}$ although the converse is not guaranteed in general.

Definition: If X and Y are topological vector spaces (TVSs) then we call a linear map f : D(f) ⊆ X → Y a closed linear operator if its graph is closed in X × Y.

The antonym of "closed" is "unclosed", that is, an unclosed linear operator is a linear operator whose graph is strictly smaller than its closure.

Closable maps and closures

A linear operator ${\displaystyle f:D\subseteq X\to Y}$ is closable in ${\displaystyle X\times Y}$ if there exists a vector subspace ${\displaystyle E\subseteq X}$ containing ${\displaystyle D}$ and a function (resp. multifunction) ${\displaystyle F:E\to Y}$ whose graph is equal to the closure of the set ${\displaystyle \mathrm{graph}f}$ in ${\displaystyle X\times Y.}$ Such an ${\displaystyle F}$ is called a closure of ${\displaystyle f}$ in ${\displaystyle X\times Y}$, is denoted by ${\displaystyle \overline{f},}$ and necessarily extends ${\displaystyle f.}$

If ${\displaystyle f:D\subseteq X\to Y}$ is a closable linear operator then a core or an essential domain of ${\displaystyle f}$ is a subset ${\displaystyle C\subseteq D}$ such that the closure in ${\displaystyle X\times Y}$ of the graph of the restriction ${\displaystyle f{|}_C:C\to Y}$ of ${\displaystyle f}$ to ${\displaystyle C}$ is equal to the closure of the graph of ${\displaystyle f}$ in ${\displaystyle X\times Y}$ (i.e. the closure of ${\displaystyle \mathrm{graph}f}$ in ${\displaystyle X\times Y}$ is equal to the closure of ${\displaystyle \mathrm{graph}f{|}_C}$ in ${\displaystyle X\times Y}$).