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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Toeplitz and Hankel matrices as Hadamard-Schur multipliers

Authors: L. N. Nikolskaya and Yu. B. Farforovskaya
Translated by: the authors
Original publication: Algebra i Analiz, tom 15 (2003), nomer 6.
Journal: St. Petersburg Math. J. 15 (2004), 915-928
MSC (2000): Primary 47B35
Published electronically: November 15, 2004
MathSciNet review: 2044634
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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.

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Additional Information

L. N. Nikolskaya
Affiliation: Laboratoire de Mathématiques Pures, UFR Maths et Info, Université de Bordeaux I, 33405 TALENCE Cedex France

Yu. B. Farforovskaya
Affiliation: Mathematics Department, St. Petersburg University of Electric Engineering, St. Petersburg, Russia

Keywords: Toeplitz matrix, Hankel matrix, Hadamard--Schur multiplier
Received by editor(s): June 3, 2003
Published electronically: November 15, 2004
Additional Notes: Supported by the European Network “Analysis, Operators, Applications” (Bordeaux team).
Article copyright: © Copyright 2004 American Mathematical Society