Integral linear operator

In mathematical analysis, an integral linear operator is a linear operator T given by integration; i.e.,

: $(Tf)(x) = \int f(y) K(x, y) \, dy$

where $K(x, y)$ is called an integration kernel.

More generally, an integral bilinear form is a bilinear functional that belongs to the continuous dual space of $X \widehat{\otimes}_{\epsilon} Y$ , the injective tensor product of the locally convex topological vector spaces (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 spaces and nuclear maps.

Definition - Integral forms as the dual of the injective tensor product

Let X and Y be locally convex TVSs, let $X \otimes_{\pi} Y$ denote the projective tensor product, $X \widehat{\otimes}_{\pi} Y$ denote its completion, let $X \otimes_{\epsilon} Y$ denote the injective tensor product, and $X \widehat{\otimes}_{\epsilon} Y$ denote its completion.

Suppose that $\operatorname{In} : X \otimes_{\epsilon} Y \to X \widehat{\otimes}_{\epsilon} Y$ denotes the TVS-embedding of $X \otimes_{\epsilon} Y$ into its completion and let ${}^{t}\operatorname{In} : \left( X \widehat{\otimes}_{\epsilon} Y \right)^{\prime}_b \to \left( X \otimes_{\epsilon} Y \right)^{\prime}_b$ be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of $X \otimes_{\epsilon} Y$ as being identical to the continuous dual space of $X \widehat{\otimes}_{\epsilon} Y$ .

Let $\operatorname{Id} : X \otimes_{\pi} Y \to X \otimes_{\epsilon} Y$ denote the identity map and ${}^{t}\operatorname{Id} : \left( X \otimes_{\epsilon} Y \right)^{\prime}_b \to \left( X \otimes_{\pi} Y \right)^{\prime}_b$ denote its transpose, which is a continuous injection. Recall that $\left( X \otimes_{\pi} Y \right)^{\prime}$ is canonically identified with $B(X, Y)$ , the space of continuous bilinear maps on $X \times Y$ . In this way, the continuous dual space of $X \otimes_{\epsilon} Y$ can be canonically identified as a vector subspace of $B(X, Y)$ , denoted by $J(X, Y)$ . The elements of $J(X, Y)$ are called integral (bilinear) forms on $X \times Y$ . The following theorem justifies the word integral.

{{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 $X \widehat{\otimes}_{\epsilon} Y$ consists of exactly of the continuous bilinear forms {{mvar|u}} on $X \times Y$ of the form

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

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

}}

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 $u$ on the product $X\times Y$ of locally convex spaces is integral if and only if there is a compact topological space $\Omega$ equipped with a (necessarily bounded) positive Radon measure $\mu$ and continuous linear maps $\alpha$ and $\beta$ from $X$ and $Y$ to the Banach space $L^{\infty}(\Omega,\mu)$ such that

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

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

Integral linear maps

A continuous linear map $\kappa : X \to Y'$ is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by $(x, y) \in X \times Y \mapsto (\kappa x)(y)$ .{{sfn | Schaefer|Wolff| 1999 | p=169}} It follows that an integral map $\kappa : X \to Y'$ is of the form:{{sfn | Schaefer|Wolff| 1999 | p=169}}

: $x \in X \mapsto \kappa(x) = \int_{S \times T} \left\langle x', x \right\rangle y' \mathrm{d} \mu\! \left( x', y' \right)$

for suitable weakly closed and equicontinuous subsets S and T of $X'$ and $Y'$ , respectively, and some positive Radon measure $\mu$ of total mass ≤ 1.

The above integral is the weak integral, so the equality holds if and only if for every $y \in Y$ , $\left\langle \kappa(x), y \right\rangle = \int_{S \times T} \left\langle x', x \right\rangle \left\langle y', y \right\rangle \mathrm{d} \mu\! \left( x', y' \right)$ .

Given a linear map $\Lambda : X \to Y$ , one can define a canonical bilinear form $B_{\Lambda} \in Bi\left(X, Y' \right)$ , called the associated bilinear form on $X \times Y'$ , by $B_{\Lambda}\left( x, y' \right) := \left( y' \circ \Lambda \right) \left( x \right)$ .

A continuous map $\Lambda : X \to Y$ is called integral if its associated bilinear form is an integral bilinear form.{{sfn | Trèves | 2006 | pp=502-505}} An integral map $\Lambda: X \to Y$ is of the form, for every $x \in X$ and $y' \in Y'$ :

: $\left\langle y', \Lambda(x) \right\rangle = \int_{A' \times B$ } \left\langle x', x \right\rangle \left\langle y, y' \right\rangle \mathrm{d} \mu\! \left( x', y'' \right)

for suitable weakly closed and equicontinuous aubsets $A'$ and $B$ of $X'$ and $Y$ , respectively, and some positive Radon measure $\mu$ of total mass $\leq 1$ .

= Relation to Hilbert spaces =

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

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

Furthermore, every integral operator between two Hilbert spaces is nuclear.{{sfn | Trèves | 2006 | pp=506-508}} Thus a continuous linear operator between two Hilbert spaces is 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 spaces is nuclear.{{sfn | Trèves | 2006 | pp=506-508}}

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

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

Suppose that $u : X \to Y$ is a continuous linear map between locally convex TVSs.

If $u : X \to Y$ is integral then so is its transpose ${}^{t}u : Y^{\prime}_b \to X^{\prime}_b$ .{{sfn | Trèves | 2006 | pp=502-505}} Now suppose that the transpose ${}^{t}u : Y^{\prime}_b \to X^{\prime}_b$ of the continuous linear map $u : X \to Y$ is integral. Then $u : X \to Y$ is integral if the canonical injections $\operatorname{In}_X : X \to X$ (defined by $x \mapsto$ value at {{mvar|x}}) and $\operatorname{In}_Y : Y \to Y$ are TVS-embeddings (which happens if, for instance, $X$ and $Y^{\prime}_b$ 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. If $\alpha : A \to B$ , $\beta : B \to C$ , and $\gamma: C \to D$ are all integral linear maps then their composition $\gamma \circ \beta \circ \alpha : A \to D$ is nuclear.{{sfn | Trèves | 2006 | pp=506-508}}

Thus, in particular, if {{mvar|X}} is an infinite-dimensional Fréchet space then a continuous linear surjection $u : X \to X$ cannot be an integral operator.

References

Bibliography

{{Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited}}
{{Dubinsky The Structure of Nuclear Fréchet Spaces}}
{{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}}
{{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}
{{Khaleelulla Counterexamples in Topological Vector Spaces}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Hogbe-Nlend Bornologies and Functional Analysis}}
{{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}}
{{Pietsch Nuclear Locally Convex Spaces|edition=2}}
{{Robertson Topological Vector Spaces}}
{{Rudin Walter Functional Analysis|edition=2}}
{{Ryan Introduction to Tensor Products of Banach Spaces|edition=1}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Trèves François Topological vector spaces, distributions and kernels}}
{{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}}

External links

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

Category:Topological vector spaces

Category:Topological tensor products

Category:Linear operators