Closed linear operator

In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator.

The closed graph theorem says a linear operator between Banach spaces is a closed operator if and only if it is a bounded operator. Hence, a closed linear operator that is used in practice is typically only defined on a dense subspace of a Banach space.

Definition

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

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

Definition: If Template:Mvar and Template:Mvar are topological vector spaces (TVSs) then we call a linear map Template:Math a closed linear operator if its graph is closed in Template:Math.

Closable maps and closures

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

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

Examples

A bounded operator is a closed operator. Here are examples of closed operators that are not bounded.

• If $(X,\tau )$ is a Hausdorff TVS and $\nu$ is a vector topology on $X$ that is strictly finer than $\tau ,$ then the identity map $\mathrm{Id}:(X,\tau )\to (X,\nu )$ a closed discontinuous linear operator.Template:Sfn
• Consider the derivative operator $A=\frac{d}{dx}$ where $X=Y=C([a,b])$ is the Banach space of all continuous functions on an interval $[a,b].$ If one takes its domain $D(f)$ to be $C^1([a,b]),$ then $f$ is a closed operator, which is not bounded. On the other hand, if $D(f)$ is the space $C^{\mathrm{\infty }}([a,b])$ of smooth functions scalar valued functions then $f$ will no longer be closed, but it will be closable, with the closure being its extension defined on $C^1([a,b]).$

Basic properties

The following properties are easily checked for a linear operator Template:Math between Banach spaces:

• If Template:Mvar is closed then Template:Math is closed where Template:Mvar is a scalar and Template:Math is the identity function;
• If Template:Mvar is closed, then its kernel (or nullspace) is a closed vector subspace of Template:Mvar;
• If Template:Mvar is closed and injective then its inverse Template:Math is also closed;
• A linear operator Template:Mvar admits a closure if and only if for every Template:Math and every pair of sequences Template:Math and Template:Math in Template:Math both converging to Template:Mvar in Template:Mvar, such that both Template:Math and Template:Math converge in Template:Mvar, one has Template:Math.

References

• Template:Dolecki Mynard Convergence Foundations Of Topology
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Rudin Walter Functional Analysis