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 ${\displaystyle X.}$ 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 ${\displaystyle \operatorname {graph} {\!(f)}=\{(x,f(x)):x\in \operatorname {dom} f\}.}$ However, one exception to this is the definition of "closed graph". A partial function ${\displaystyle f:D\subseteq X\to Y}$ is said to have a closed graph if ${\displaystyle \operatorname {graph} f}$ 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 ${\displaystyle D\times Y=\operatorname {dom} f\times Y}$ 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 ${\displaystyle \operatorname {graph} f}$ is a closed subset of ${\displaystyle D\times Y.}$ If ${\displaystyle \operatorname {graph} f}$ is a closed subset of ${\displaystyle X\times Y}$ then it is also a closed subset of ${\displaystyle \operatorname {dom} (f)\times Y}$ 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 ${\displaystyle f:D\subseteq X\to Y}$ 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 ${\displaystyle \operatorname {graph} f}$ 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 ${\displaystyle f{\big \vert }_{C}:C\to Y}$ 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 ${\displaystyle \operatorname {graph} f}$ in ${\displaystyle X\times Y}$ is equal to the closure of ${\displaystyle \operatorname {graph} f{\big \vert }_{C}}$ in ${\displaystyle X\times Y}$).

Examples

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

• If ${\displaystyle (X,\tau )}$ is a Hausdorff TVS and ${\displaystyle \nu }$ is a vector topology on ${\displaystyle X}$ that is strictly finer than ${\displaystyle \tau ,}$ then the identity map ${\displaystyle \operatorname {Id} :(X,\tau )\to (X,\nu )}$ a closed discontinuous linear operator.^[1]
• Consider the derivative operator ${\displaystyle A={\frac {d}{dx}}}$ where ${\displaystyle X=Y=C([a,b])}$ is the Banach space of all continuous functions on an interval ${\displaystyle [a,b].}$

If one takes its domain ${\displaystyle D(f)}$ to be ${\displaystyle C^{1}([a,b]),}$ then ${\displaystyle f}$ is a closed operator, which is not bounded.^[2] On the other hand, if ${\displaystyle D(f)}$ is the space ${\displaystyle C^{\infty }([a,b])}$ of smooth functions scalar valued functions then ${\displaystyle f}$ will no longer be closed, but it will be closable, with the closure being its extension defined on ${\displaystyle C^{1}([a,b]).}$

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

1. Narici & Beckenstein 2011, p. 480.
2. Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc. p. 294. ISBN 0-471-50731-8.

• 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.