View source for Integral linear operator

{{Short description|Mathematical function}}

In mathematical analysis, an '''integral linear operator''' is a linear operator ''T'' given by integration; i.e.,
:<math>(Tf)(x) = \int f(y) K(x, y) \, dy</math>
where <math>K(x, y)</math> is called an integration kernel.

More generally, an '''integral bilinear form''' is a [[bilinear map|bilinear functional]] that belongs to the continuous dual space of <math>X \widehat{\otimes}_{\epsilon} Y</math>, the [[injective tensor product]] of the locally convex [[topological vector space]]s (TVSs) ''X'' and ''Y''. An '''integral linear operator''' is a continuous linear operator that arises in a canonical way from an integral bilinear form. 

These maps play an important role in the theory of [[nuclear space]]s and [[nuclear map]]s.

== Definition - Integral forms as the dual of the injective tensor product ==
{{See also|Injective tensor product|Projective tensor product}}

Let ''X'' and ''Y'' be locally convex TVSs, let <math>X \otimes_{\pi} Y</math> denote the [[projective tensor product]], <math>X \widehat{\otimes}_{\pi} Y</math> denote its completion, let <math>X \otimes_{\epsilon} Y</math> denote the [[injective tensor product]], and <math>X \widehat{\otimes}_{\epsilon} Y</math> denote its completion. 
Suppose that <math>\operatorname{In} : X \otimes_{\epsilon} Y \to X \widehat{\otimes}_{\epsilon} Y</math> denotes the TVS-embedding of <math>X \otimes_{\epsilon} Y</math> into its completion and let <math>{}^{t}\operatorname{In} : \left( X \widehat{\otimes}_{\epsilon} Y \right)^{\prime}_b \to \left( X \otimes_{\epsilon} Y \right)^{\prime}_b</math> be its [[transpose]], which is a vector space-isomorphism. This identifies the continuous dual space of <math>X \otimes_{\epsilon} Y</math> as being identical to the continuous dual space of <math>X \widehat{\otimes}_{\epsilon} Y</math>. 

Let <math>\operatorname{Id} : X \otimes_{\pi} Y \to X \otimes_{\epsilon} Y</math> denote the identity map and <math>{}^{t}\operatorname{Id} : \left( X \otimes_{\epsilon} Y \right)^{\prime}_b \to \left( X \otimes_{\pi} Y \right)^{\prime}_b</math> denote its [[transpose]], which is a continuous injection. Recall that <math>\left( X \otimes_{\pi} Y \right)^{\prime}</math> is canonically identified with <math>B(X, Y)</math>, the space of continuous bilinear maps on <math>X \times Y</math>. In this way, the continuous dual space of <math>X \otimes_{\epsilon} Y</math> can be canonically identified as a vector subspace of <math>B(X, Y)</math>, denoted by <math>J(X, Y)</math>. The elements of <math>J(X, Y)</math> are called '''integral (bilinear) forms''' on <math>X \times Y</math>. The following theorem justifies the word <em>integral</em>. 

{{math theorem | name = Theorem{{sfn | Schaefer|Wolff| 1999 | p=168}}{{sfn | Trèves | 2006 | pp=500-502}} | math_statement =
The dual {{math|''J''(''X'', ''Y'')}} of <math>X \widehat{\otimes}_{\epsilon} Y</math> consists of exactly of the continuous bilinear forms {{mvar|u}} on <math>X \times Y</math> of the form 

:<math> u(x,y) = \int_{S \times T} \langle x, x'\rangle \langle y, y' \rangle\; d \mu\!\left( x', y' \right),</math>

where {{mvar|S}} and {{mvar|T}} are respectively some weakly closed and equicontinuous (hence weakly compact) subsets of the duals <math>X^{\prime}</math> and <math>Y^{\prime}</math>, and <math>\mu</math> is a (necessarily bounded) positive [[Radon measure]] on the (compact) set <math>S \times T</math>. 
}}

There is also a closely related formulation {{sfn | Grothendieck | 1955 | pp=124-126}} of the theorem above that can also be used to explain the terminology ''integral'' bilinear form: a continuous bilinear form <math>u</math> on the product <math>X\times Y</math> of locally convex spaces is integral if and only if there is a ''compact'' topological space <math>\Omega</math> equipped with a (necessarily bounded) positive Radon measure <math>\mu</math> and continuous linear maps <math>\alpha</math> and <math>\beta</math> from <math>X</math> and <math>Y</math> to the Banach space <math>L^{\infty}(\Omega,\mu)</math> such that

:<math>u(x,y) = \langle\alpha(x),\beta(y)\rangle = \int_{\Omega}\alpha(x)\beta(y)\;d\mu</math>,

i.e., the form <math>u</math> can be realised by integrating (essentially bounded) functions on a compact space.

== Integral linear maps ==

A continuous linear map <math>\kappa : X \to Y'</math> is called '''integral''' if its associated bilinear form is an integral bilinear form, where this form is defined by <math>(x, y) \in X \times Y \mapsto (\kappa x)(y)</math>.{{sfn | Schaefer|Wolff| 1999 | p=169}} It follows that an integral map <math>\kappa : X \to Y'</math> is of the form:{{sfn | Schaefer|Wolff| 1999 | p=169}} 
: <math>x \in X \mapsto \kappa(x) = \int_{S \times T} \left\langle x', x \right\rangle y' \mathrm{d} \mu\! \left( x', y' \right)</math>
for suitable weakly closed and equicontinuous subsets ''S'' and ''T'' of <math>X'</math> and <math>Y'</math>, respectively, and some positive Radon measure <math>\mu</math> of total mass ≤ 1. 
The above integral is the [[weak integral]], so the equality holds if and only if for every <math>y \in Y</math>, <math display="inline">\left\langle \kappa(x), y \right\rangle = \int_{S \times T} \left\langle x', x \right\rangle \left\langle y', y \right\rangle \mathrm{d} \mu\! \left( x', y' \right)</math>. 

Given a linear map <math>\Lambda : X \to Y</math>, one can define a canonical bilinear form <math>B_{\Lambda} \in Bi\left(X, Y' \right)</math>, called the '''associated bilinear form''' on <math>X \times Y'</math>, by <math>B_{\Lambda}\left( x, y' \right) := \left(  y' \circ \Lambda \right) \left( x \right)</math>. 
A continuous map <math>\Lambda : X \to Y</math> is called '''integral''' if its associated bilinear form is an integral bilinear form.{{sfn | Trèves | 2006 | pp=502-505}} An integral map <math>\Lambda: X \to Y</math> is of the form, for every <math>x \in X</math> and <math>y' \in Y'</math>:
: <math>\left\langle y', \Lambda(x) \right\rangle = \int_{A' \times B''} \left\langle x', x \right\rangle \left\langle y'', y' \right\rangle \mathrm{d} \mu\! \left( x', y'' \right)</math>
for suitable weakly closed and equicontinuous aubsets <math>A'</math> and <math>B''</math> of <math>X'</math> and <math>Y''</math>, respectively, and some positive Radon measure <math>\mu</math> of total mass <math>\leq 1</math>.

=== Relation to Hilbert spaces ===

The following result shows that integral maps "factor through" Hilbert spaces.

Proposition:{{sfn | Trèves | 2006 | pp=506-508}}  Suppose that <math>u : X \to Y</math> is an integral map between locally convex TVS with ''Y'' Hausdorff and complete. There exists a Hilbert space ''H'' and two continuous linear mappings <math>\alpha : X \to H</math> and <math>\beta : H \to Y</math> such that <math>u = \beta \circ \alpha</math>. 

Furthermore, every integral operator between two [[Hilbert space]]s is [[nuclear operator|nuclear]].{{sfn | Trèves | 2006 | pp=506-508}} Thus a continuous linear operator between two [[Hilbert space]]s is [[nuclear operator|nuclear]] if and only if it is integral.

=== Sufficient conditions ===

Every [[nuclear map]] is integral.{{sfn | Trèves | 2006 | pp=502-505}}  An important partial converse is that every integral operator between two [[Hilbert space]]s is [[nuclear operator|nuclear]].{{sfn | Trèves | 2006 | pp=506-508}}

Suppose that ''A'', ''B'', ''C'', and ''D'' are Hausdorff locally convex TVSs and that <math>\alpha : A \to B</math>, <math>\beta : B \to C</math>, and <math>\gamma: C \to D</math> are all continuous linear operators. If <math>\beta : B \to C</math> is an integral operator then so is the composition <math>\gamma \circ \beta \circ \alpha : A \to D</math>.{{sfn | Trèves | 2006 | pp=506-508}}  

If <math>u : X \to Y</math> is a continuous linear operator between two normed space then <math>u : X \to Y</math>  is integral if and only if <math>{}^{t}u : Y' \to X'</math> is integral.{{sfn | Trèves | 2006 | pp=505}}   

Suppose that <math>u : X \to Y</math> is a continuous linear map between locally convex TVSs.  
If <math>u : X \to Y</math> is integral then so is its [[transpose]] <math>{}^{t}u : Y^{\prime}_b \to X^{\prime}_b</math>.{{sfn | Trèves | 2006 | pp=502-505}}  Now suppose that the transpose  <math>{}^{t}u : Y^{\prime}_b \to X^{\prime}_b</math> of the continuous linear map <math>u : X \to Y</math> is integral. Then <math>u : X \to Y</math> is integral if the canonical injections <math>\operatorname{In}_X : X \to X''</math> (defined by <math>x \mapsto </math> value at {{mvar|x}}) and <math>\operatorname{In}_Y : Y \to Y''</math> are [[TVS-embedding]]s (which happens if, for instance, <math>X</math> and <math>Y^{\prime}_b</math> are barreled or metrizable).{{sfn | Trèves | 2006 | pp=502-505}} 

=== Properties ===

Suppose that ''A'', ''B'', ''C'', and ''D'' are Hausdorff locally convex TVSs with ''B'' and ''D'' [[complete metric space|complete]]. If <math>\alpha : A \to B</math>, <math>\beta : B \to C</math>, and <math>\gamma: C \to D</math> are all integral linear maps then their composition <math>\gamma \circ \beta \circ \alpha : A \to D</math> is [[nuclear operator|nuclear]].{{sfn | Trèves | 2006 | pp=506-508}}  
Thus, in particular, if {{mvar|X}} is an infinite-dimensional [[Fréchet space]] then a continuous linear surjection <math>u : X \to X</math> cannot be an integral operator.

== See also ==

* [[Auxiliary normed spaces]]
* [[Final topology]]
* [[Injective tensor product]]
* [[Nuclear operator]]s
* [[Nuclear space]]s
* [[Projective tensor product]]
* [[Topological tensor product]]

== References ==
{{Reflist}}

== Bibliography ==

* {{Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited}} <!-- {{sfn | Diestel | 2008 | p=}} -->
* {{Dubinsky The Structure of Nuclear Fréchet Spaces}} <!-- {{sfn | Dubinsky | 1979 | p=}} -->
* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} <!-- {{sfn | Grothendieck | 1955 | p=}} -->
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!-- {{sfn | Husain | Khaleelulla | 1978 | p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | 1982 | p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!-- {{sfn | Hogbe-Nlend | 1977 | p=}} -->
* {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}} <!-- {{sfn | Hogbe-Nlend | Moscatelli | 1981 | p=}} -->
* {{Pietsch Nuclear Locally Convex Spaces|edition=2}} <!-- {{sfn | Pietsch | 1979 | p=}} -->
* {{Robertson Topological Vector Spaces}} <!-- {{sfn | Robertson | Robertson | 1980 | p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} --> 
* {{Ryan Introduction to Tensor Products of Banach Spaces|edition=1}} <!-- {{sfn | Ryan | 2002 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer|Wolff| 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->
* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!-- {{sfn | Wong | 1979 | p=}} -->

== External links ==

* [https://ncatlab.org/nlab/show/nuclear+space Nuclear space at ncatlab]

{{Functional analysis}}
{{TopologicalTensorProductsAndNuclearSpaces}}

<!--- Categories --->

[[Category:Topological vector spaces]]
[[Category:Topological tensor products]]
[[Category:Linear operators]]