A Class of Schur Multipliers of Matrices with Operator Entries

  • Oscar BlascoEmail author
  • Ismael García-Bayona


In this paper, we will consider matrices with entries in the space of operators \(\mathcal {B}(H)\), where H is a separable Hilbert space, and consider the class of (left or right) Schur multipliers that can be approached in the multiplier norm by matrices with a finite number of diagonals. We will concentrate on the case of Toeplitz matrices and of upper triangular matrices to get some connections with spaces of vector-valued functions.


Schur product Toeplitz matrix Schur multiplier vector-valued measure vector-valued function 

Mathematics Subject Classification

Primary 47L10 46E40 Secondary 47A56 15B05 46G10 



  1. 1.
    Alexandrov, A.B., Peller, V.V.: Hankel and Toeplitz–Schur multipliers. Math. Ann. 324, 277–327 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barza, S., Persson, L.-E., Popa, N.: A matriceal analogue of Fejer’s theory. Math. Nachr. 260, 14–20 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bennett, G.: Schur multipliers. Duke Math. J. 44, 603–639 (1977)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Blasco, O., García-Bayona, I.: Schur product with operator-valued entries. Taiwan. J. Math. (2018). CrossRefzbMATHGoogle Scholar
  5. 5.
    Blasco, O., García-Bayona, I.: New spaces of matrices with operator-valued entries. Quaest. Math. (2019). CrossRefzbMATHGoogle Scholar
  6. 6.
    Blasco, O.: Fourier Analysis on vector measures on locally compact abelian groups. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 110(2), 519–539 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Böttcher, A., Grudsky, S.: Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis. Hindustan Book Agency, New Delhi (2000) and Birkhäuser, Basel (2000)zbMATHGoogle Scholar
  8. 8.
    Bukhvalov, A.V., Danilevich, A.A.: Boundary properties of analytic and harmonic functions with values in Banach spaces. Mat. Zametki 31, 203–214 (1982). English translation: Math. Notes 31 (1982), 104–110MathSciNetzbMATHGoogle Scholar
  9. 9.
    Diestel, J., Fourie, J., Swart, J.: The Metric Theory of Tensor Products. Grothendieck’s Résumé Revisited. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  10. 10.
    Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)CrossRefGoogle Scholar
  11. 11.
    Duren, P.L.: Theory of \(H^p\) Spaces. Academic Press, New York (1970)zbMATHGoogle Scholar
  12. 12.
    Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981)zbMATHGoogle Scholar
  13. 13.
    Hytonen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Spaces. Vol. I. Martingales and Littlewood-Paley Theory. Series of Modern Surveys in Mathematics, vol. 63. Springer, Cham (2016)CrossRefGoogle Scholar
  14. 14.
    Persson, L.-E., Popa, N.: Matrix Spaces and Schur Multipliers: Matriceal Harmonic Analysis. World Scientific, Hackensack (2014)CrossRefGoogle Scholar
  15. 15.
    Schur, J.: Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Verandlichen. J. Reine Angew. Math. 140, 1–28 (1911)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Toeplitz, O.: Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veranderlichen. Math. Annalen 70, 351–376 (1911)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de ValenciaBurjassotSpain

Personalised recommendations