Integral linear operator

An integral bilinear form is a bilinear functional that belongs to the continuous dual space of ${\displaystyle X{\widehat {\otimes }}_{\epsilon }Y}$, 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

See also: Injective tensor product and Projective tensor product

Let X and Y be locally convex TVSs, let ${\displaystyle X\otimes _{\pi }Y}$ denote the projective tensor product, ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ denote its completion, let ${\displaystyle X\otimes _{\epsilon }Y}$ denote the injective tensor product, and ${\displaystyle X{\widehat {\otimes }}_{\epsilon }Y}$ denote its completion. Suppose that ${\displaystyle \operatorname {In} :X\otimes _{\epsilon }Y\to X{\widehat {\otimes }}_{\epsilon }Y}$ denotes the TVS-embedding of ${\displaystyle X\otimes _{\epsilon }Y}$ into its completion and let ${\displaystyle {}^{t}\operatorname {In}$ :\left(X{\widehat {\otimes }}_{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }} be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of ${\displaystyle X\otimes _{\epsilon }Y}$ as being identical to the continuous dual space of ${\displaystyle X{\widehat {\otimes }}_{\epsilon }Y}$.

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

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

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

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

Integral linear maps

A continuous linear map ${\displaystyle \kappa :X\to Y^{\prime }}$ is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by ${\displaystyle (x,y)\in X\times Y\mapsto (\kappa x)(y)}$.[3] It follows that an integral map ${\displaystyle \kappa :X\to Y^{\prime }}$ is of the form:[3]

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

for suitable weakly closed and equicontinuous subsets S and T of ${\displaystyle X^{\prime }}$ and ${\displaystyle Y^{\prime }}$, respectively, and some positive Radon measure ${\displaystyle \mu }$ of total mass ≤ 1. The above integral is the weak integral, so the equality holds if and only if for every ${\displaystyle y\in Y}$, ${\displaystyle \left\langle \kappa (x),y\right\rangle =\int _{S\times T}\left\langle x^{\prime },x\right\rangle \left\langle y^{\prime },y\right\rangle \operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)}$.

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

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

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

Relation to Hilbert spaces

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

Proposition:[5] Suppose that ${\displaystyle u:X\to Y}$ is an integral map between locally convex TVS with Y Hausdorff and complete. There exists a Hilbert space H and two continuous linear mappings ${\displaystyle \alpha :X\to H}$ and ${\displaystyle \beta :H\to Y}$ such that ${\displaystyle u=\beta \circ \alpha }$.

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

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

• Every nuclear map is integral.[4]
  • An important partial converse is that every integral operator between two Hilbert spaces is nuclear.[5]
• If ${\displaystyle u:X\to Y}$ is integral then so is its transpose ${\displaystyle {}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }}$.[4]
  • Suppose that the transpose ${\displaystyle {}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }}$ of the continuous linear map ${\displaystyle u:X\to Y}$ is integral. Then ${\displaystyle u:X\to Y}$ is integral if the canonical injections ${\displaystyle \operatorname {In} _{X}:X\to X^{\prime \prime }}$ (defined by ${\displaystyle x\mapsto }$ value at x) and ${\displaystyle \operatorname {In} _{Y}:Y\to Y^{\prime \prime }}$ are TVS-embeddings (which happens if, for instance, X and ${\displaystyle Y_{b}^{\prime }}$ are barreled or metrizable).[4]
• If X and Y are normable spaces then ${\displaystyle u:X\to Y}$ is integral if and only if ${\displaystyle {}^{t}u:Y^{\prime }\to X^{\prime }}$ is integral.[6]
• Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that ${\displaystyle \alpha :A\to B}$, ${\displaystyle \beta :B\to C}$, and ${\displaystyle \gamma :C\to D}$ are all continuous linear operators. If ${\displaystyle \beta :B\to C}$ is an integral operator then so is the composition ${\displaystyle \gamma \circ \beta \circ \alpha :A\to D}$.[5]

Properties

• Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete. If ${\displaystyle \alpha :A\to B}$, ${\displaystyle \beta :B\to C}$, and ${\displaystyle \gamma :C\to D}$ are all integral linear maps then their composition ${\displaystyle \gamma \circ \beta \circ \alpha :A\to D}$ is nuclear.[5]
  • Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection ${\displaystyle u:X\to X}$ 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

References

1. Schaefer 1999, p. 168. sfn error: no target: CITEREFSchaefer1999 (help)
2. Treves 2006, pp. 500-502. sfn error: no target: CITEREFTreves2006 (help)
3. Schaefer 1999, p. 169. sfn error: no target: CITEREFSchaefer1999 (help)
4. Treves 2006, pp. 502-505. sfn error: no target: CITEREFTreves2006 (help)
5. Treves 2006, pp. 506-508. sfn error: no target: CITEREFTreves2006 (help)
6. Treves 2006, pp. 505. sfn error: no target: CITEREFTreves2006 (help)

• Diestel, Joe (2008). The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society. ISBN 0-8218-4440-7. OCLC 185095773.CS1 maint: ref=harv (link)
• Dubinsky, Ed (1979). The structure of nuclear Fréchet spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09504-7. OCLC 5126156.CS1 maint: ref=harv (link)
• Grothendieck, Alexander (1966). Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society. ISBN 0-8218-1216-5. OCLC 1315788.CS1 maint: ref=harv (link)
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
• Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Nlend, H (1977). Bornologies and functional analysis : introductory course on the theory of duality topology-bornology and its use in functional analysis. Amsterdam New York New York: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier-North Holland. ISBN 0-7204-0712-5. OCLC 2798822.CS1 maint: ref=harv (link)
• Nlend, H (1981). Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality. Amsterdam New York New York, N.Y: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland. ISBN 0-444-86207-2. OCLC 7553061.CS1 maint: ref=harv (link)
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin,New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.CS1 maint: ref=harv (link)
• Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.CS1 maint: ref=harv (link)
• Rudin, Walter (January 1, 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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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.CS1 maint: ref=harv (link)
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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External links

• Nuclear space at ncatlab