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

Let $\mathrm{Id}:X{\otimes }_{\pi }Y\to X{\otimes }_{ϵ}Y$ denote the identity map and ${}^t\mathrm{Id}:{\left(X{\otimes }_{ϵ}Y\right)}_b^{\prime }\to {\left(X{\otimes }_{\pi }Y\right)}_b^{\prime }$ 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 $X{\otimes }_{ϵ}Y$ 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.

There is also a closely related formulation Template:Sfn 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 $\mathrm{\Omega }$ equipped with a (necessarily bounded) positive Radon measure $\mu$ and continuous linear maps $\alpha$ and $\beta$ from $X$ and $Y$ to the Banach space $L^{\mathrm{\infty }}(\mathrm{\Omega },\mu )$ such that

$u(x,y)=⟨\alpha (x),\beta (y)⟩=\underset{\mathrm{\Omega }}{\int }\alpha (x)\beta (y)d\mu$,

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

Integral linear maps

A continuous linear map $\kappa :X\to Y^{\prime }$ is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by $(x,y)\in X\times Y\mapsto (\kappa x)(y)$.Template:Sfn It follows that an integral map $\kappa :X\to Y^{\prime }$ is of the form:Template:Sfn

$x\in X\mapsto \kappa (x)=\underset{S\times T}{\int }⟨x^{\prime },x⟩y^{\prime }\mathrm{d}\mu (x^{\prime },y^{\prime })$

for suitable weakly closed and equicontinuous subsets S and T of $X^{\prime }$ and $Y^{\prime }$, 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 $y\in Y$, $⟨\kappa (x),y⟩=\underset{S\times T}{\int }⟨x^{\prime },x⟩⟨y^{\prime },y⟩\mathrm{d}\mu (x^{\prime },y^{\prime })$.

Given a linear map $\mathrm{\Lambda }:X\to Y$, one can define a canonical bilinear form $B_{\mathrm{\Lambda }}\in Bi(X,Y^{\prime })$, called the associated bilinear form on $X\times Y^{\prime }$, by $B_{\mathrm{\Lambda }}(x,y^{\prime }):=(y^{\prime }\circ \mathrm{\Lambda })\left(x\right)$. A continuous map $\mathrm{\Lambda }:X\to Y$ is called integral if its associated bilinear form is an integral bilinear form.Template:Sfn An integral map $\mathrm{\Lambda }:X\to Y$ is of the form, for every $x\in X$ and $y^{\prime }\in Y^{\prime }$:

$⟨y^{\prime },\mathrm{\Lambda }(x)⟩=\underset{A^{\prime }\times B^{″}}{\int }⟨x^{\prime },x⟩⟨y^{″},y^{\prime }⟩\mathrm{d}\mu (x^{\prime },y^{″})$

for suitable weakly closed and equicontinuous aubsets $A^{\prime }$ and $B^{″}$ of $X^{\prime }$ and $Y^{″}$, respectively, and some positive Radon measure $\mu$ of total mass $\le 1$.

Relation to Hilbert spaces

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

Proposition:Template:Sfn 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.Template:Sfn 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.Template:Sfn An important partial converse is that every integral operator between two Hilbert spaces is nuclear.Template:Sfn

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 $\gamma \circ \beta \circ \alpha :A\to D$.Template:Sfn

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 ${}^{t}u:Y^{\prime }\to X^{\prime }$ is integral.Template:Sfn

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 ${}^{t}u:Y_b^{\prime }\to X_b^{\prime }$.Template:Sfn Now suppose that the transpose ${}^{t}u:Y_b^{\prime }\to X_b^{\prime }$ of the continuous linear map $u:X\to Y$ is integral. Then $u:X\to Y$ is integral if the canonical injections ${\mathrm{In}}_X:X\to X^{″}$ (defined by $x\mapsto$ value at Template:Mvar) and ${\mathrm{In}}_Y:Y\to Y^{″}$ are TVS-embeddings (which happens if, for instance, $X$ and $Y_b^{\prime }$ are barreled or metrizable).Template:Sfn

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

See also

• Auxiliary normed spaces
• Final topology
• Injective tensor product
• Nuclear operators
• Nuclear spaces
• Projective tensor product
• Topological tensor product

Bibliography

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

External links

• Nuclear space at ncatlab