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Zbl 1030.47019
Aleksandrov, A.B.; Peller, V.V.
Hankel and Toeplitz-Schur multipliers.
(English)
[J] Math. Ann. 324, No.2, 277-327 (2002). ISSN 0025-5831; ISSN 1432-1807/e

The Schur product of matrices $A=\{a_{jk}\}_{j,k\ge 0}$ and $B=\{b_{jk}\}_{j,k\ge 0}$ is the matrix $A\star B=\{a_{jk}b_{jk}\}_{j,k\ge 0}$. Given a class $\cal{X}$ of bounded linear operators on the space $l^2$, a matrix $A$ is called a Schur multiplier of $\cal{X}$ if $A\star B\in\cal{X}$ for every $B\in\cal{X}$. For $0<p<\infty$, let $\frak{M}_p$ be the space of Schur multipliers of the Schatten-von Neumann class ${\bold S}_p$ and let $$\|A\|_{\frak{M}_p}=\sup\left\{\|A\star B\|_{{\bold S}_p}:\|B\|_{{\bold S}_p}\le 1\right\}.$$ For $p\ge 1$, $\|\cdot\|_{\frak{M}_p}$ is a norm on $\frak{M}_p$ and $$\|A_1+A_2\|^p_{\frak{M}_p}\le\|A_1\|^p_{\frak{M}_p}+ \|A_2\|^p_{\frak{M}_p}\quad\text{ if}\quad 0<p\le 1.$$ \par The present paper is devoted to studying the Hankel-Schur multipliers of ${\bold S}_p$, i.e., matrices of the form $\{\gamma_{j+k}\}_{j,k\ge 0}$ in $\frak{M}_p$, and the Toeplitz-Schur multipliers of ${\bold S}_p$, i.e., matrices of the form $\{t_{j-k}\}_{j,k\ge 0}$ in $\frak{M}_p$, for $0<p<1$. Some general results concerning Schur multipliers of ${\bold S}_p$ are established. Several sharp necessary conditions and sufficient conditions for Hankel matrices to be Schur multipliers of ${\bold S}_p$ are found. A characterization of the Hankel-Schur multipliers of ${\bold S}_p$ whose symbols have lacunary power series is obtained. Applying the results on Hankel-Schur multipliers of ${\bold S}_p$, the authors give the following characterization of the Toeplitz-Schur multipliers of ${\bold S}_p$. A Toeplitz matrix $\{t_{j-k}\}_{j,k\ge 0}$ is a Schur multiplier of ${\bold S}_p$ in the case $0<p<1$ if and only if there exists a discrete measure $\mu$ on $\Bbb{T}$ of the form $$\mu=\sum_{j\in\Bbb{Z}}\alpha_j\delta_{\tau_j},\quad \alpha_j\in\Bbb{C},\quad \tau_j\in\Bbb{T},$$ with $$\|\mu\|_{\cal{M}_p}:=\biggl(\sum_j |\alpha_j|^p\biggr)^{1/p}<\infty,$$ such that $t_j=\widehat{\mu}(j)$, $j\in\Bbb{Z}$. Moreover, in that case $$\big\|\{t_{j-k}\}_{j,k\ge 0}\}\big\|_{\frak{M}_p}= \|\mu\|_{\cal{M}_p}.$$ Finally, the Hankel-Schur multipliers of ${\bold S}_p$ whose symbols are complex measures on $\Bbb{T}$ are studied.
MSC 2000:
*47B35 Toeplitz operators, etc.
47B10 Operators defined by summability properties

Keywords: Hankel-Schur and Toeplitz-Schur multipliers; Schatten-von Neumann class; Besov space; lacunary power series; complex measure

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