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
[edit]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
[edit]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
[edit]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
[edit]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
[edit]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
[edit]- Auxiliary normed spaces
- Final topology
- Injective tensor product
- Nuclear operators
- Nuclear spaces
- Projective tensor product
- Topological tensor product
References
[edit]- ^ Schaefer & Wolff 1999, p. 168.
- ^ Trèves 2006, pp. 500–502.
- ^ Grothendieck 1955, pp. 124–126.
- ^ a b Schaefer & Wolff 1999, p. 169.
- ^ a b c d Trèves 2006, pp. 502–505.
- ^ a b c d e Trèves 2006, pp. 506–508.
- ^ Trèves 2006, pp. 505.
Bibliography
[edit]- 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.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 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. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.