Integral linear operator

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

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 those continuous bilinear forms c on that can be represented in the form of a map

where S and T are some closed, equicontinuous subsets of and , respectively, and is a positive Radon measure on the compact set with total mass Furthermore, if A is an equicontinuous subset of J(X, Y) then the elements can be represented with fixed and running through a norm bounded subset of the space of Radon measures on

Integral linear maps

A continuous linear map is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by .[3] It follows that an integral map is of the form:[3]

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.[4] 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

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

Proposition:[5] 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.[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 , , and are all continuous linear operators. If is an integral operator then so is the composition .[5]

If is a continuous linear operator between two normed space then is integral if and only if is integral.[6]

Suppose that is a continuous linear map between locally convex TVSs. If is integral then so is its transpose .[4] 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).[4]

Properties

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

See also

References

  1. Schaefer & Wolff 1999, p. 168.
  2. Trèves 2006, pp. 500-502.
  3. Schaefer & Wolff 1999, p. 169.
  4. Trèves 2006, pp. 502-505.
  5. Trèves 2006, pp. 506-508.
  6. Trèves 2006, pp. 505.

Bibliography

  • Diestel, Joe (2008). The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. 16. Providence, R.I.: American Mathematical Society. ISBN 9781470424831. OCLC 185095773.
  • 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.
  • 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.
  • 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.
  • Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.
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