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Hankel and Toeplitz-Schur multipliers. (English) Zbl 1030.47019

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

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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