Closed linear operator

{{Short description|Linear operator whose graph is closed}} In functional analysis, a branch of mathematics, a '''closed linear operator''' or often a '''closed operator''' is a partially defined 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 <math>f : X \to Y</math> between Banach spaces is a closed operator if and only if it is a bounded operator and the domain of the operator is <math>X</math>. In practice, many operators are unbounded, but it is still desirable to make them have closed graph. Hence, they cannot be defined on all of <math>X</math>. To stay useful, they are instead defined on a proper but dense subspace, which still allows approximating any vector and keeps key tools (closures, adjoints, spectral theory) available.

== Definition == It is common in functional analysis to consider partial functions, 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 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 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 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''}}.

The antonym of "closed" is "unclosed", that is, an '''unclosed linear operator''' is a linear operator whose graph is strictly smaller than its closure.

=== 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 closed operator between Banach spaces, is bounded, by the closed graph theorem. More interesting examples of closed operators are unbounded.

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>f = \frac{d}{d x}</math> where <math>X = Y = C([a, b])</math> is the Banach space (with supremum norm) of all continuous functions on an 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 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> To show that <math>f</math> is not closed when restricted to <math>C^\infty([a, b]) \to C^\infty([a, b])</math>, take a function <math>u</math> that is <math>C^1</math> but not smooth, such as <math>u(x) = x^{3/2}</math>. Then mollify it to a sequence of smooth functions <math>(u_n)_{n \in \N}</math> such that <math>\|u_n - u \|_\infty \to 0</math>, then <math>\|f(u_n) - u' \|_\infty \to 0</math>, but <math>(u, u')</math> is not in the graph of <math>f|_{C^\infty([a, b])}</math>.

== Basic properties ==

The following properties are easily checked for a linear operator <math>f : \operatorname{D}(f) \subseteq X \to Y</math> between Banach spaces:

* If <math>f</math> is defined on the entire domain <math>X</math>, then <math>f</math> is closed iff it is bounded. * If <math>A</math> is closed then <math>A - \lambda \mathrm{Id}_{\operatorname{D}(f)}</math> is closed where <math>\lambda</math> is a scalar and <math>\mathrm{Id}_{\operatorname{D}(f)}</math> is the identity function; * If <math>f</math> is closed, then its kernel (or nullspace) is a closed vector subspace of <math>X</math>; * If <math>f</math> is closed and injective then its inverse <math>f^{-1}</math> is also closed; * A linear operator <math>f</math> admits a closure if and only if for every <math>x \in X</math> and every pair of sequences <math>x_{\bullet} = (x_i)_{i=1}^{\infty}</math> and <math>y_{\bullet} = (y_i)_{i=1}^{\infty}</math> in <math>\operatorname{D}(f)</math> both converging to <math>x</math> in <math>X</math>, such that both <math>f(x_{\bullet}) = (f(x_i))_{i=1}^{\infty}</math> and <math>f(y_{\bullet}) = (f(y_i))_{i=1}^{\infty}</math> converge in <math>Y</math>, one has <math>\lim_{i \to \infty} f(x_i) = \lim_{i \to \infty} f(y_i)</math>.

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

* {{Dolecki Mynard Convergence Foundations Of Topology}}  * {{Citation |last=Mortad |first=Mohammed Hichem |title=Closedness |date=2022 |work=Counterexamples in Operator Theory |pages=307–344 |url=https://link.springer.com/10.1007/978-3-030-97814-3_19|place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-97814-3_19 |isbn=978-3-030-97813-6|url-access=subscription }}{{sfn|Mortad|2022|p=}} * {{Narici Beckenstein Topological Vector Spaces|edition=2}}  * {{Rudin Walter Functional Analysis|edition=2}}

Category:Linear operators