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On a Schur-Type Product for Matrices with Operator Entries

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In this paper, we will introduce a new Schur-type product for matrices with operator entries, and explore some of its properties. We shall see a connection between this product and the classical Schur product that will allow us to prove that this set of matrices endowed with such new product defines a Banach algebra. Also, a way to compute the operator and multiplier norms of matrices with operator entries in terms of norms of scalar matrices will be provided. As applications, we present a way to obtain multipliers for one of the products from a multiplier for the other product and show a method to construct a countable amount of elements belonging to different vector measure spaces, from a single element of \(L^\infty (\mathbb {T})\).

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  1. 1.

    Badea, C., Paulsen, V.: Schur multipliers and operator-valued Foguel–Hankel operators. Indiana Univ. Math. J 50, 1509–1522 (2001)

  2. 2.

    Banica, T.: Complex Hadamard matrices with noncommutative entries. Ann. Funct. Anal. 9(3), 354–368 (2018)

  3. 3.

    Bennett, G.: Schur multipliers. Duke Math. J. 44, 603–639 (1977)

  4. 4.

    Blasco, O., García-Bayona, I.: A class of Schur multipliers of matrices with operator entries. Mediterr. J. Math. 16, 82 (2019)

  5. 5.

    Blasco, O., García-Bayona, I.: New spaces of matrices with operator-valued entries. Quaest, Math (2019)

  6. 6.

    Blasco, O., García-Bayona, I.: Schur product with operator-valued entries. Taiwan J. Math. 23(5), 1175–1199 (2019)

  7. 7.

    Choi, D.: Inequalities related to partial trace and block Hadamard product. Linear Multilinear A. 66(8), 1619–1625 (2017)

  8. 8.

    Christensen, E.: On the complete boundedness of the Schur block product. Proc. Am. Math. Soc. 147(2), 523–532 (2019)

  9. 9.

    Das, P.K., Vashisht, L.K.: Traces of Hadamard and Kronecker products of matrices. Math. Appl. 6, 143–150 (2017)

  10. 10.

    Diestel, J., Uhl, Jun, J.J.: Vector measures. Integration, American Mathematical Society (AMS). XIII, 322 p. 35–60 (1977)

  11. 11.

    García-Bayona, I.: Traces of Schur and Kronecker products for block matrices. Khayyam J. Math. 5(2), 40–50 (2019)

  12. 12.

    Gohberg, I., Kaashoek, M.A., Goldberg, S.: Classes of Linear Operators, vol. II. Birkhauser Verlag, Basel (1993)

  13. 13.

    Günter, M., Klotz, L.: Schur’s theorem for a block Hadamard product. Linear Algebra Appl. 437, 948–956 (2012)

  14. 14.

    Halmos, P.R.: Measure Theory. Springer, New York (1974)

  15. 15.

    Horn, R.A.: The Hadamard product. Proc. Symp. Appl. Math. 40, 87–169 (1990)

  16. 16.

    Horn, R.A., Mathias, R., Nakamura, Y.: Inequalities for unitarily invariant norms and bilinear matrix products. Linear Multilinear A. 30(4), 303–314 (1991)

  17. 17.

    Kluvánek, I., Knowles, G.: Vector Measures and Control Systems, vol. 20. North-Holland Mathematics Studies, Amsterdam (1976)

  18. 18.

    Kwapien, S., Pelczyinski, A.: The main triangle projection in matrix spaces and its applications. Stud. Math. 34, 43–68 (1970)

  19. 19.

    Livshits, L.: Block-matrix generalizations of infinite-dimensional Schur products and Schur multipliers. Linear Multilinear A. 38(1–2), 59–78 (1994)

  20. 20.

    Magnus, J. R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics. 2nd ed., Wiley, Chichester (1999). J. Operator Theory1 17–56 (1995)

  21. 21.

    Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge studies in advanced mathematics, vol. 78. Cambridge University Press, Cambridge (2002)

  22. 22.

    Persson, L.E., Popa, N.: Matrix Spaces and Schur Multipliers: Matriceal Harmonic Analysis. World Scientific, Hackensack (2014)

  23. 23.

    Pommerenke, C.: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen (1975)

  24. 24.

    Schur, J.: Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Verandlichen. J. Reine Angew. Math. 140, 1–28 (1911)

  25. 25.

    Shapiro, H.S., Shields, A.L.: On some interpolations problems for analytic functions. Am. J. Math. 83, 513–532 (1961)

  26. 26.

    Sothanaphan, N.: Determinants of block matrices with noncommuting blocks. Linear Algebra Appl. 512, 202–218 (2017)

  27. 27.

    Stout, Q.F.: Schur products of operators and the essential numerical range. Trans. Am. Math. Soc. 264(1), 39–47 (1981)

  28. 28.

    Styan, G.P.H.: Hadamard products and multivariate statistical analysis. Linear Algebra Appl. 6, 217–240 (1973)

  29. 29.

    Toeplitz, O.: Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veranderlichen. Math. Ann. 70, 351–376 (1911)

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The author is thankful to the anonymous reviewers for their comments and suggestions that allowed to improve the paper, and also acknowledges the support provided by MINECO (Spain) under the Project MTM2014-53009-P and by MCIU (Spain) under the Grant FPU14/01032.

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Correspondence to Ismael García-Bayona.

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Partially supported by MTM2014-53009-P (MINECO Spain) and FPU14/01032 (MCIU Spain)

Communicated by Ali Armandnejad.

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García-Bayona, I. On a Schur-Type Product for Matrices with Operator Entries. Bull. Iran. Math. Soc. (2020).

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  • Schur product
  • Schur multipliers
  • Block matrices
  • Toeplitz matrices
  • Vector-valued measures

Mathematics Subject Classification

  • 47L10
  • 47A56
  • 15B05
  • 46G10