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 $\operatorname {graph} {\!(f)}=\{(x,f(x)):x\in \operatorname {dom} f\}.$ However, one exception to this is the definition of "closed graph". A partial function $f:D\subseteq X\to Y$ is said to have a closed graph if $\operatorname {graph} f$ is a closed subset of $X\times Y$ in the product topology; importantly, note that the product space is $X\times Y$ and not $D\times Y=\operatorname {dom} f\times Y$ 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 $\operatorname {graph} f$ is a closed subset of $D\times Y.$ If $\operatorname {graph} f$ is a closed subset of $X\times Y$ then it is also a closed subset of $\operatorname {dom} (f)\times Y$ 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.

Closable maps and closures

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

If $f:D\subseteq X\to Y$ is a closable linear operator then a core or an essential domain of $f$ is a subset $C\subseteq D$ such that the closure in $X\times Y$ of the graph of the restriction $f{\big \vert }_{C}:C\to Y$ of $f$ to $C$ is equal to the closure of the graph of $f$ in $X\times Y$ (i.e. the closure of $\operatorname {graph} f$ in $X\times Y$ is equal to the closure of $\operatorname {graph} f{\big \vert }_{C}$ 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 $\operatorname {Id} :(X,\tau )\to (X,\nu )$ a closed discontinuous linear operator.
• 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 $C^{\infty }([a,b])$ 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 f : D(f) ⊆ X → Y between Banach spaces:

• If A is closed then A − λId_D(f) is closed where λ is a scalar and Id_D(f) is the identity function;
• If f is closed, then its kernel (or nullspace) is a closed vector subspace of X;
• If f is closed and injective then its inverse f ⁻¹ is also closed;
• A linear operator f admits a closure if and only if for every x ∈ X and every pair of sequences x_• = (x_i)^∞
  _i=1 and y_• = (y_i)^∞
  _i=1 in D(f) both converging to x in X, such that both f(x_•) = (f(x_i))^∞
  _i=1 and f(y_•) = (f(y_i))^∞
  _i=1 converge in Y, one has lim_{i → ∞} fx_i = lim_{i → ∞} fy_i.

References

• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.