Toeplitz and Hankel matrices as Hadamard–Schur multipliers

Nikolskaya, L.; Farforovskaya, Yu.

doi:10.1090/S1061-0022-04-00838-6

Toeplitz and Hankel matrices as Hadamard–Schur multipliers
HTML articles powered by AMS MathViewer

by L. N. Nikolskaya and Yu. B. Farforovskaya
Translated by: the authors

St. Petersburg Math. J. 15 (2004), 915-928

DOI: https://doi.org/10.1090/S1061-0022-04-00838-6

Published electronically: November 15, 2004

PDF | Request permission

Abstract:

The Hadamard product of two matrices $M= (m_{ij})$ and $A= (a_{ij})$ is defined by $M\circ A= (m_{ij}a_{ij})$. A matrix $M$ is a Hadamard–Schur multiplier (in short, HSM) if $\Vert M\Vert _{{\mathcal H}}= \sup \{\Vert M\circ A\Vert \mid A:l^{2}\longrightarrow l^{2}, \Vert A\Vert \le 1\}< \infty$. Let $\mu$ be a complex measure on the circle. An exact formula is found for the multiplier norm of the Toeplitz matrix $T_{\mu }=$ $(\hat \mu (i-j))_{i,j\ge 0}$: $\Vert T_{\mu }\Vert _{{\mathcal H}}=$ $\Vert \mu \Vert _{{\mathfrak M}}$. For the Hankel matrices $\Gamma _{\mu }= (\hat \mu (i+j))_{i,j\ge 0}$, we have $\Vert \Gamma _{\mu }\Vert _{{\mathcal H}}\le \Vert \mu \Vert _{{\mathfrak M}/H^{1}_{-}}$, and for more general “skew diagonal” matrices we have $\Vert (\hat \mu (im+jl))_{i,j\ge 0}\Vert _{{\mathcal H}}\le \Vert \mu \Vert _{{\mathfrak M}}$, where $l,m\in \Bbb {Z}$. Analogs of these results for matrix-valued measures and the corresponding block HSMs are established. Next, a necessary condition of Peller’s type for $\Vert \Gamma \Vert _{{\mathcal H}} < \infty$ is given. It is also shown that, for $\Lambda \subset \Bbb {Z}_{+}$, the norm $\Vert \Gamma \Vert _{{\mathcal H}}$ is equivalent to $\sup _{k\ge 1}\vert \gamma _{k}\vert$ on the set of Hankel matrices $\Gamma = (\gamma _{i+j})$ with $\gamma _{k}= 0$ for $k\in \Bbb {Z}_{+}{\backslash }\Lambda$ if and only if $\Lambda$ is a finite union of lacunary sequences.

References

Mihály Bakonyi and Dan Timotin, On an extension problem for polynomials, Bull. London Math. Soc. 33 (2001), no. 5, 599–605. MR 1844558, DOI 10.1112/S0024609301008268
G. Bennett, Schur multipliers, Duke Math. J. 44 (1977), no. 3, 603–639. MR 493490
M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals. III, Problems of mathematical physics, No. 6 (Russian), Izdat. Leningrad. Univ., Leningrad, 1973, pp. 27–53 (Russian). MR 0348494
Aline Bonami and Joaquim Bruna, On truncations of Hankel and Toeplitz operators, Publ. Mat. 43 (1999), no. 1, 235–250. MR 1697523, DOI 10.5565/PUBLMAT_{4}3199_{1}0
Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, Springer-Verlag, Berlin, 1990. MR 1071374, DOI 10.1007/978-3-662-02652-6
Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
R. E. Edwards and G. I. Gaudry, Littlewood-Paley and multiplier theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90, Springer-Verlag, Berlin-New York, 1977. MR 0618663
Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek, Classes of linear operators. Vol. II, Operator Theory: Advances and Applications, vol. 63, Birkhäuser Verlag, Basel, 1993. MR 1246332, DOI 10.1007/978-3-0348-8558-4_{1}
A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Resenhas 2 (1996), no. 4, 401–480 (French). Reprint of Bol. Soc. Mat. São Paulo 8 (1953), 1–79 [ MR0094682 (20 #1194)]. MR 1466414
Henry Helson, Harmonic analysis, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. MR 729682
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. XXXI, American Mathematical Society, Providence, R.I., 1974. Third printing of the revised edition of 1957. MR 0423094
Roger A. Horn, The Hadamard product, Matrix theory and applications (Phoenix, AZ, 1989) Proc. Sympos. Appl. Math., vol. 40, Amer. Math. Soc., Providence, RI, 1990, pp. 87–169. MR 1059485, DOI 10.1090/psapm/040/1059485
V. I. Macaev, A class of completely continuous operators, Dokl. Akad. Nauk SSSR 139 (1961), 548–551 (Russian). MR 0131769
Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
V. Olevskii and M. Solomyak, An estimate for Schur multipliers in $S_p$-classes, Linear Algebra Appl. 208/209 (1994), 57–64. MR 1287339, DOI 10.1016/0024-3795(94)90430-8
Sing-Cheong Ong, On the Schur multiplier norm of matrices, Linear Algebra Appl. 56 (1984), 45–55. MR 724547, DOI 10.1016/0024-3795(84)90112-5
Vladimir V. Peller, Estimates of functions of power bounded operators on Hilbert spaces, J. Operator Theory 7 (1982), no. 2, 341–372. MR 658618
V. V. Peller, Hankel Schur multipliers and multipliers of the space $H^{1}$, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984), 113–119 (Russian, with English summary). Investigations on linear operators and the theory of functions, XIII. MR 741701
V. V. Peller, Hankel operators in the theory of perturbations of unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen. 19 (1985), no. 2, 37–51, 96 (Russian). MR 800919
Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047, DOI 10.1007/b55674
N. Th. Varopoulos, Tensor algebras over discrete spaces, J. Functional Analysis 3 (1969), 321–335. MR 0250087, DOI 10.1016/0022-1236(69)90046-9