## 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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