Hyponormality of SlantWeighted Toeplitz Operatos on the Torus

Abstract

Here we consider a sequence of positive numbers $\beta=\{ \beta_k \}_{k \in \mathbb{Z}^n}$ with $\beta_0=1$, and assume that there exists $0< r \leq 1 $ such that for each $i=1, 2, \dots , n$ and $k=(k_1,\dots, k_n)\in \mathbb{Z}^n$, we have $ r \leq \frac{\beta_k}{\beta_{k+\epsilon_i}} \leq 1$ if $ k_i \geq 0 $, and $ r \leq \frac{\beta_{k+\epsilon_i}}{\beta_k} \leq 1$ if $k_i < 0$. For such a weight sequence $\beta$, we define the weighted sequence space $L^2(\mathbb{T}^n,\beta)$ to be the set of all $f(z)=\sum_{k \in \mathbb{Z}^n} a_kz^k$ for which $\sum_{k \in \mathbb{Z}^n}|a_k|^2\beta_k^2<\infty$. Here $\mathbb{T}$ is the unit circle in the complex plane, and for $n\geq 1,\,\mathbb{T}^n$ denotes the n-Torus which is the cartesian product of $n$ copies of $\mathbb{T}$. For $\varphi\in L^\infty(\mathbb{T}^n,\beta)$, we define the slant weighted Toeplitz operator $A_\varphi$ on $L^2(\mathbb{T}^n,\beta)$ and give the necessary and sufficient conditions for a bounded linear operator on $L^2(\mathbb{T}^n,\beta)$ to be a slant weighted Toeplitz operator. We also prove that for a trigonometric polynomial $\varphi\in L^\infty(\mathbb{T}^n,\beta),\,\,A_\varphi$ cannot be hyponormal unless $\varphi\equiv 0$.