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 $f : X \to Y$ between Banach spaces is a closed operator if and only if it is a bounded operator and the domain of the operator is $X$ . 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 {{em|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 {{em|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 {{mvar|X}} and {{mvar|Y}} are topological vector spaces (TVSs) then we call a linear map {{math|f : D(f) ⊆ X → Y}} a closed linear operator if its graph is closed in {{math|X × Y}}.

= Closable maps and closures =

A linear operator $f : D \subseteq X \to Y$ is {{visible anchor|closable}} in $X \times Y$ if there exists a {{em|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 {{visible anchor|core}} or an {{visible anchor|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.{{sfn|Narici|Beckenstein|2011|p=480}}
Consider the derivative operator $A = \frac{d}{d x}$ 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.{{Cite book|title=Introductory Functional Analysis With Applications|last=Kreyszig|first=Erwin|publisher=John Wiley & Sons. Inc.|year=1978|isbn=0-471-50731-8|location=USA|pages=294}} 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 {{math|f : D(f) ⊆ X → Y}} between Banach spaces:

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

References

{{Dolecki Mynard Convergence Foundations Of Topology}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Rudin Walter Functional Analysis|edition=2}}

Category:Linear operators