View source for 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 (functional analysis)|closed graph theorem]] says a linear operator <math>f : X \to Y</math> between [[Banach space]]s is a closed operator if and only if it is a [[bounded operator]] and the domain of the operator is <math>X</math>. Hence, a closed linear operator that is used in practice is typically only [[densely defined operator|defined on a dense subspace]] of a Banach space.

== Definition ==
It is common in functional analysis to consider [[partial function]]s, which are functions defined on a [[subset]] of some space <math>X.</math>
A partial function <math>f</math> is declared with the notation <math>f : D \subseteq X \to Y,</math> which indicates that <math>f</math> has prototype <math>f : D \to Y</math> (that is, its [[Domain of a function|domain]] is <math>D</math> and its [[codomain]] is <math>Y</math>)

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

'''Definition''': If {{mvar|X}} and {{mvar|Y}} are [[topological vector space]]s (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'' ×&thinsp;''Y''}}.

=== Closable maps and closures ===

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

If <math>f : D \subseteq X \to Y</math> is a closable linear operator then a '''{{visible anchor|core}}''' or an '''{{visible anchor|essential domain}}''' of <math>f</math> is a subset <math>C \subseteq D</math> such that the closure in <math>X \times Y</math> of the graph of the restriction <math>f\big\vert_C : C \to Y</math> of <math>f</math> to <math>C</math> is equal to the closure of the graph of <math>f</math> in <math>X \times Y</math> (i.e. the closure of <math>\operatorname{graph} f</math> in <math>X \times Y</math> is equal to the closure of <math>\operatorname{graph} f\big\vert_C</math> in <math>X \times Y</math>).

== Examples ==

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

* If <math>(X, \tau)</math> is a Hausdorff TVS and <math>\nu</math> is a vector topology on <math>X</math> that is strictly finer than <math>\tau,</math> then the identity map <math>\operatorname{Id} : (X, \tau) \to (X, \nu)</math> a closed discontinuous linear operator.{{sfn|Narici|Beckenstein|2011|p=480}}
* Consider the [[derivative]] operator <math>A = \frac{d}{d x}</math> where <math>X = Y = C([a, b])</math> is the Banach space of all [[continuous function]]s on an [[Interval (mathematics)|interval]] <math>[a, b].</math> If one takes its domain <math>D(f)</math> to be <math>C^1([a, b]),</math> then <math>f</math> is a closed operator, which is not bounded.<ref>{{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}}</ref> On the other hand, if <math>D(f)</math> is the space <math>C^\infty([a, b])</math> of [[smooth function]]s scalar valued functions then <math>f</math> will no longer be closed, but it will be closable, with the closure being its extension defined on <math>C^1([a, b]).</math>

== 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'' − ''λ''Id<sub>''D''(''f'')</sub>}} is closed where {{mvar|λ}} is a scalar and {{math|Id<sub>''D''(''f'')</sub>}} is the [[identity function]];
* If {{mvar|f}} is closed, then its [[Kernel (linear operator)|kernel]] (or nullspace) is a closed vector subspace of {{mvar|X}};
* If {{mvar|f}} is closed and [[injective function|injective]] then its [[inverse function|inverse]] {{math|''f''&thinsp;<sup>−1</sup>}} 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''<sub>•</sub> = (''x''<sub>''i''</sub>){{su|p=∞|b=''i''=1}}}} and {{math|1=''y''<sub>•</sub> = (''y''<sub>''i''</sub>){{su|p=∞|b=''i''=1}}}} in {{math|''D''(''f'')}} both converging to {{mvar|x}} in {{mvar|X}}, such that both {{math|1=''f''(''x''<sub>•</sub>) = (''f''(''x''<sub>''i''</sub>)){{su|p=∞|b=''i''=1}}}} and {{math|1=''f''(''y''<sub>•</sub>) = (''f''(''y''<sub>''i''</sub>)){{su|p=∞|b=''i''=1}}}} converge in {{mvar|Y}}, one has {{math|1=lim<sub>''i'' → ∞</sub> ''fx''<sub>''i''</sub> = lim<sub>''i'' → ∞</sub> ''fy''<sub>''i''</sub>}}.

== References ==
{{reflist|group=note}}

{{reflist}}

* {{Dolecki Mynard Convergence Foundations Of Topology}} <!-- {{sfn|Dolecki|2016|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->

[[Category:Linear operators]]