Integral linear operator

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 ${}^{t}\operatorname {In$ 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 ${}^{t}\operatorname {Id$ 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.

Theorem[1][2] — The dual $J (X, Y)$ of $X{\widehat {\otimes }}_{\epsilon }Y$ consists of exactly those continuous bilinear forms $c$ on $X\times Y$ that can be represented in the form of a map

b\in B(X,Y)\mapsto v(b)=\int _{S\times T}b{\big \vert }_{S\times T}\left(x^{\prime },y^{\prime }\right)\operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)

where $S$ and $T$ are some closed, equicontinuous subsets of $X_{\sigma }^{\prime }$ and $Y_{\sigma }^{\prime }$ , respectively, and $\mu$ is a positive Radon measure on the compact set $S\times T$ with total mass $\leq 1.$ Furthermore, if $A$ is an equicontinuous subset of $J (X, Y)$ then the elements $v\in A$ can be represented with $S\times T$ fixed and $\mu$ running through a norm bounded subset of the space of Radon measures on $S\times T.$

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 $(x,y)\in X\times Y\mapsto (\kappa x)(y)$ .[3] It follows that an integral map $\kappa :X\to Y^{\prime }$ is of the form:[3]

x\in X\mapsto \kappa (x)=\int _{S\times T}\left\langle x^{\prime },x\right\rangle y^{\prime }\operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)

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 $y\in Y$ , $\left\langle \kappa (x),y\right\rangle =\int _{S\times T}\left\langle x^{\prime },x\right\rangle \left\langle y^{\prime },y\right\rangle \operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)$ .

Given a linear map $\Lambda :X\to Y$ , one can define a canonical bilinear form $B_{\Lambda }\in Bi\left(X,Y^{\prime }\right)$ , called the associated bilinear form on $X\times Y^{\prime }$ , by $B_{\Lambda }\left(x,y^{\prime }\right):=\left(y^{\prime }\circ \Lambda \right)\left(x\right)$ . A continuous map $\Lambda :X\to Y$ is called integral if its associated bilinear form is an integral bilinear form.[4] An integral map $\Lambda :X\to Y$ is of the form, for every $x\in X$ and $y^{\prime }\in Y^{\prime }$ :

\left\langle y^{\prime },\Lambda (x)\right\rangle =\int _{A^{\prime }\times B^{\prime \prime }}\left\langle x^{\prime },x\right\rangle \left\langle y^{\prime \prime },y^{\prime }\right\rangle \operatorname {d} \mu \left(x^{\prime },y^{\prime \prime }\right)

for suitable weakly closed and equicontinuous aubsets $A^{\prime }$ and $B^{\prime \prime }$ of $X^{\prime }$ and $Y^{\prime \prime }$ , 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:[5] 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.[5] 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.[4] An important partial converse is that every integral operator between two Hilbert spaces is nuclear.[5]

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 $\gamma \circ \beta \circ \alpha :A\to D$ .[5]

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.[6]

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 ${}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }$ .[4] 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 $\operatorname {In} _{X}:X\to X^{\prime \prime }$ (defined by $x\mapsto$ value at $x$ ) and $\operatorname {In} _{Y}:Y\to Y^{\prime \prime }$ are TVS-embeddings (which happens if, for instance, $X$ and $Y_{b}^{\prime }$ are barreled or metrizable).[4]

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

References

Schaefer & Wolff 1999, p. 168.
Trèves 2006, pp. 500-502.
Schaefer & Wolff 1999, p. 169.
Trèves 2006, pp. 502-505.
Trèves 2006, pp. 506-508.
Trèves 2006, pp. 505.

Bibliography

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Dubinsky, Ed (1979). The Structure of Nuclear Fréchet Spaces. Lecture Notes in Mathematics. 720. Berlin New York: Springer-Verlag. ISBN 978-3-540-09504-0. OCLC 5126156.
Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. OCLC 316549583.
Hogbe-Nlend, Henri; Moscatelli, V. B. (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. 52. Amsterdam New York New York: North Holland. ISBN 978-0-08-087163-9. OCLC 316564345.
Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-05644-9. OCLC 539541.
Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
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Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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External links

Nuclear space at ncatlab

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[FOOTNOTESchaeferWolff1999168-1] Schaefer & Wolff 1999, p. 168.

[FOOTNOTETrèves2006500-502-2] Trèves 2006, pp. 500-502.

[FOOTNOTESchaeferWolff1999169-3] Schaefer & Wolff 1999, p. 169.

[FOOTNOTETrèves2006502-505-4] Trèves 2006, pp. 502-505.

[FOOTNOTETrèves2006506-508-5] Trèves 2006, pp. 506-508.

[FOOTNOTETrèves2006505-6] Trèves 2006, pp. 505.

Topological tensor products and nuclear spaces
Basic concepts	Auxiliary normed spaces Nuclear space Tensor product (of Hilbert spaces)
Topologies	Inductive tensor product Injective tensor product Projective tensor product
Operators/Maps	Hypocontinuity Integral Nuclear (between Banach spaces) Trace class