Integral linear operator

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An integral bilinear form is a bilinear functional that belongs to the continuous dual space of , 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

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Let X and Y be locally convex TVSs, let   denote the projective tensor product,   denote its completion, let   denote the injective tensor product, and   denote its completion. Suppose that   denotes the TVS-embedding of   into its completion and let   be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of   as being identical to the continuous dual space of  .

Let   denote the identity map and   denote its transpose, which is a continuous injection. Recall that   is canonically identified with  , the space of continuous bilinear maps on  . In this way, the continuous dual space of   can be canonically identified as a vector subspace of  , denoted by  . The elements of   are called integral (bilinear) forms on  . The following theorem justifies the word integral.

Theorem[1][2] — The dual J(X, Y) of   consists of exactly of the continuous bilinear forms u on   of the form

 

where S and T are respectively some weakly closed and equicontinuous (hence weakly compact) subsets of the duals   and  , and   is a (necessarily bounded) positive Radon measure on the (compact) set  .

There is also a closely related formulation [3] of the theorem above that can also be used to explain the terminology integral bilinear form: a continuous bilinear form   on the product   of locally convex spaces is integral if and only if there is a compact topological space   equipped with a (necessarily bounded) positive Radon measure   and continuous linear maps   and   from   and   to the Banach space   such that

 ,

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

Integral linear maps

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A continuous linear map   is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by  .[4] It follows that an integral map   is of the form:[4]

 

for suitable weakly closed and equicontinuous subsets S and T of   and  , respectively, and some positive Radon measure   of total mass ≤ 1. The above integral is the weak integral, so the equality holds if and only if for every  ,  .

Given a linear map  , one can define a canonical bilinear form  , called the associated bilinear form on  , by  . A continuous map   is called integral if its associated bilinear form is an integral bilinear form.[5] An integral map   is of the form, for every   and  :

 

for suitable weakly closed and equicontinuous aubsets   and   of   and  , respectively, and some positive Radon measure   of total mass  .

Relation to Hilbert spaces

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

Proposition:[6] Suppose that   is an integral map between locally convex TVS with Y Hausdorff and complete. There exists a Hilbert space H and two continuous linear mappings   and   such that  .

Furthermore, every integral operator between two Hilbert spaces is nuclear.[6] Thus a continuous linear operator between two Hilbert spaces is nuclear if and only if it is integral.

Sufficient conditions

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Every nuclear map is integral.[5] An important partial converse is that every integral operator between two Hilbert spaces is nuclear.[6]

Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that  ,  , and   are all continuous linear operators. If   is an integral operator then so is the composition  .[6]

If   is a continuous linear operator between two normed space then   is integral if and only if   is integral.[7]

Suppose that   is a continuous linear map between locally convex TVSs. If   is integral then so is its transpose  .[5] Now suppose that the transpose   of the continuous linear map   is integral. Then   is integral if the canonical injections   (defined by   value at x) and   are TVS-embeddings (which happens if, for instance,   and   are barreled or metrizable).[5]

Properties

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Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete. If  ,  , and   are all integral linear maps then their composition   is nuclear.[6] Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection   cannot be an integral operator.

See also

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References

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  1. ^ Schaefer & Wolff 1999, p. 168.
  2. ^ Trèves 2006, pp. 500–502.
  3. ^ Grothendieck 1955, pp. 124–126.
  4. ^ a b Schaefer & Wolff 1999, p. 169.
  5. ^ a b c d Trèves 2006, pp. 502–505.
  6. ^ a b c d e Trèves 2006, pp. 506–508.
  7. ^ Trèves 2006, pp. 505.

Bibliography

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  • Diestel, Joe (2008). The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. Vol. 16. Providence, R.I.: American Mathematical Society. ISBN 9781470424831. OCLC 185095773.
  • Dubinsky, Ed (1979). The Structure of Nuclear Fréchet Spaces. Lecture Notes in Mathematics. Vol. 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). 16. Providence: American Mathematical Society. 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. Vol. 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. Vol. 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. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. 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. Vol. 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. Vol. 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. Vol. 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. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  • Ryan, Raymond A. (2002). Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. London New York: Springer. ISBN 978-1-85233-437-6. OCLC 48092184.
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