Toeplitz and Hankel matrices as Hadamard--Schur multipliers

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.