Abstract
In this paper we study block matrices and multipliers with respect to a Schur-type product operation on them. A subclass of the block Toeplitz matrices is explored, obtaining generalizations of Toeplitz’s and Bennett’s classical theorems that show identifications with complex functions and measures on the bitorus. We also investigate the classes of matrices that can be approximated in the operator and multiplier norms by a subclass of the polynomial matrices and describe them using matrices generated by two-dimensional summability kernels, paying attention to the case of Toeplitz matrices and finding the corresponding connections with spaces of functions.
Similar content being viewed by others
References
Alexandrov, A.B., Peller, V.V.: Hankel and Toeplitz-Schur multipliers. Math. Ann. 324, 277–327 (2002)
Badea, C., Paulsen, V.: Schur multipliers and operator-valued Foguel-Hankel operators. Indiana Univ. Math. J. 50, 1509–1522 (2001)
Banica, T.: Complex Hadamard matrices with noncommutative entries. Ann. Funct. Anal. 9(3), 354–368 (2018)
Barza, S., Lie, V., Popa, N.: Approximation of infinite matrices by matricial Haar polynomials. Arkiv för Matematik 43(2), 251–269 (2005)
Barza, S., Persson, L.-E., Popa, N.: A matricial analogue of Fèjer theory. Math. Nachr. 260, 14–20 (2003)
Bennett, G.: Schur multipliers. Duke Math. J. 44, 603–639 (1977)
Blasco, O., García-Bayona, I.: A class of Schur multipliers of matrices with operator entries. Mediterr. J. Math. 16, 82 (2019)
Blasco, O., García-Bayona, I.: New spaces of matrices with operator entries. Quaest. Math. 43(5–6), 651–674 (2019)
Blasco, O., García-Bayona, I.: Schur product with operator-valued entries. Taiwanese J. Math. 23(5), 1175–1199 (2019)
Böttcher, A., Grudsky, S.: Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis. Hindustan Book Agency, New Delhi, 2000 and Birkhäuser Verlag, Basel, Boston, Berlin, (2000)
Choi, D.: Inequalities related to partial trace and block Hadamard product. Linear Multilinear A. 66(8), 1619–1625 (2017)
Christensen, E.: On the complete boundedness of the Schur block product. Proc. Amer. Math. Soc. 147(2), 523–532 (2019)
Coine, C.: Schur multipliers on \({\cal{B}}(L^p, L^q)\). J. Operator Theory 79(2), 301–326 (2018)
Davidson, K.R., Donsig, A.P.: Norms of Schur multipliers. Illinois J. Math. 51(3), 743–766 (2007)
Diestel, J., Jun, J.J.: Uhl, Vector measures, Integration, American Mathematical Society (AMS). XIII, 322 p. 35.60 (1977)
García-Bayona, I.: Traces of Schur and Kronecker products for block matrices. Khayyam J. Math. 5(2), 40–50 (2019)
García-Bayona, I.: On a Schur-type product for matrices with operator entries, Bull. Iran. Math. Soc. 46(6), 1775–1789 (2020)
Gohberg, I., Kaashoek, M.A., Goldberg, S.: Classes of Linear Operators, vol. II. Birkhauser, Basel (1993)
Günter, M., Klotz, L.: Schur’s theorem for a block Hadamard product. Linear Algebra Appl. 437, 948–956 (2012)
Halmos, P.R.: Measure Theory. Springer, New-York (1974)
Horn, R.A.: The Hadamard product. Proc. Symp. Appl. Math. 40, 87–169 (1990)
Jocić, D.R., Krtinić, D.: Schur-Laurent multipliers for block matrices and geometric characterization of continuous matrices. Linear and Multilinear A. 58(4), 523–534 (2009)
Kwapien, S., PeŁczyiński, A.: The main triangle projection in matrix spaces and its applications. Studia Math. 34, 43–68 (1970)
Livshits, L.: Block-matrix generalizations of infinite- dimensional Schur products and Schur multipliers. J. Lin. Mult. Alg. 38(1–2), 59–78 (1994)
Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd ed., Wiley, Chichester, UK, (1999). J. Operator Theory 1, 17–56 (1995)
Marcoci, A.N., Marcoci, L.G.: A new class of linear operators on and Schur multipliers for them. J. Funct. Spaces Appl. 5(2), 151–165 (2007)
Paulsen, V.: Completely bounded maps and operator algebras, Cambridge studies in advanced mathematics, vol 78. Cambridge University Press, Cambridge (2002)
PeŁczyiński, A., Sukochev, F.: Some remarks on Toeplitz multipliers and Hankel matrices. Stud. Math. 175(2), 175–204 (2006)
Persson, L.E., Popa, N.: Matrix Spaces and Schur Multipliers? Matriceal Harmonic Analysis. World Scientific, Hackensack (2014)
Pommerenke, C.: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen (1975)
Schur, J.: Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. J. Reine Angew. Math. 140, 1–28 (1911)
Shapiro, V.L.: Fourier series in several variables. Bull. Amer. Math. Soc. 70(1), 48–93 (1964)
Shapiro, H.S., Shields, A.L.: On some interpolations problems for analytic functions. Amer. J. Math. 83, 513–532 (1961)
Sothanaphan, N.: Determinants of block matrices with noncommuting blocks. Linear Algebra Appl. 512, 202–218 (2017)
Stout, Q.F.: Schur products of operators and the essential numerical range. Trans. Amer. Math. Soc. 264(1), 39–47 (1981)
Styan, G.P.H.: Hadamard products and multivariate statistical analysis. Linear Algebra Appl. 6, 217–240 (1973)
Toeplitz, O.: Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veränderlichen. Math. Ann. 70(3), 351–376 (1911)
Zygmund, A.: Trigonometric Series, vol. I, II, 3rd edn. Cambridge University Press (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
García-Bayona, I. Matrix Multipliers for a Block Schur Product. Results Math 76, 83 (2021). https://doi.org/10.1007/s00025-021-01399-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-021-01399-1
Keywords
- Schur product
- block matrices
- infinite matrices
- toeplitz matrices
- schur multipliers
- complex functions
- complex measures