Proceedings of the International Geometry Center

ISSN-print: 2072-9812
ISSN-online: 2409-8906
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On tensor products of nuclear operators in Banach spaces

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Published Dec 22, 2021
DOI https://doi.org/10.15673/tmgc.v14i3.2083

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Oleg Reinov
Saint Petersburg State University

Abstract

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

Keywords:
factorization, Lorentz-Schatten class, nuclear operator

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How to Cite
Reinov, O. (2021). On tensor products of nuclear operators in Banach spaces. Proceedings of the International Geometry Center, 14(3), 187-205. https://doi.org/10.15673/tmgc.v14i3.2083
Issue
Vol 14 No 3 (2021)
Section
Papers

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References

1. M. Sh. Birman, M. Z. Solomjak. Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchev and V. Peller.
2. Torsten Carleman. Uber die Fourierkoeffizienten einer stetigen Funktion. Acta Math., 41(1):377-384, 1916, doi: http://dx.doi.org/10.1007/BF02422951. Aus einem Brief an Herrn A. Wiman.
3. Karel de Leeuw, Yitzhak Katznelson, Jean-Pierre Kahane. Sur les coefficients de Fourier des fonctions continues. C. R. Acad. Sci. Paris Ser. A-B, 285(16):A1001-A1003, 1977.
4. J. Diestel, J. J. Uhl, Jr. Vector measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis.
5. Joe Diestel, Hans Jarchow, Andrew Tonge. Absolutely summing operators, volume 43 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995, doi: http://dx.doi.org/10.1017/CBO9780511526138.
6. Alexandre Grothendieck. Produits tensoriels topologiques et espaces nucleaires, volume 16. 1955.
7. Alexandre Grothendieck. La theorie de Fredholm. pages Exp. No. 91, 377-384, 1995.
8. Aicke Hinrichs, Albrecht Pietsch. p-nuclear operators in the sense of Grothendieck. Math. Nachr., 283(2):232-261, 2010, doi: http://dx.doi.org/10.1002/mana.200910128.
9. Alfred Horn. On the singular values of a product of completely continuous operators. Proc. Nat. Acad. Sci. U.S.A., 36:374-375, 1950, doi: http://dx.doi.org/10.1073/pnas.36.7.374.
10. R. J. Kaiser, J. R. Retherford. Preassigning eigenvalues and zeros of nuclear operators. Studia Math., 81(2):127-133, 1985, doi: http://dx.doi.org/10.4064/sm-81-2-127-133.
11. Hermann Konig. Weyl-type inequalities for operators in Banach spaces. In Functional analysis: surveys and recent results, II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979), volume 68 of Notas Mat., pages 297-317. North-Holland, Amsterdam-New York, 1980.
12. Hermann Konig. Eigenvalue distribution of compact operators, volume 16 of Operator Theory: Advances and Applications. Birkhauser Verlag, Basel, 1986, doi: http://dx.doi.org/10.1007/978-3-0348-6278-3.
13. Richard O'Neil. Integral transforms and tensor products on Orlicz spaces and L(p,,q) spaces. J. Analyse Math., 21:1-276, 1968, doi: http://dx.doi.org/10.1007/bf02787670.
14. Albrecht Pietsch. Weyl numbers and eigenvalues of operators in Banach spaces. Math. Ann., 247(2):149-168, 1980, doi: http://dx.doi.org/10.1007/BF01364141.
15. Albrecht Pietsch. Eigenvalues and s-numbers, volume 13 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1987.
16. Albrecht Pietsch. History of Banach spaces and linear operators. Birkhauser Boston, Inc., Boston, MA, 2007.
17. G. Pisier. Factorization of Linear Operators and Geometry of Banach Spaces. Amer. Math. Soc., Providence, Rhode Island, 1985.
18. O.I. Reinov. On products of nuclear operators. Funct. Anal. Its Appl., 51:316-317, 2017.
19. W. Rudin, editor. Fourier analysis on groups. Interscience, New York, 1962.
20. P. Saab. Convolution operators that factor through a hilbert space. Quaestiones Mathematicae, 31(1):79-87, 2008.
21. Robert Schatten. A Theory of Cross-Spaces. Annals of Mathematics Studies, No. 26. Princeton University Press, Princeton, N. J., 1950.
22. Hans Triebel. Uber die Verteilung der Approximationszahlen kompakter Operatoren in Sobolev-Besov-Raumen. Invent. Math., 4:275-293, 1967, doi: http://dx.doi.org/10.1007/BF01425385.