Generalization of Toeplitz + Hankel operators in the polydisc

Abstract

In this paper, we obtain a complete classification of Toeplitz + Hankel operators on the vector-valued Hardy space $H_{\mathcal{E}}^2({\mathbb{D}}^n)$ over the polydisc ${\mathbb{D}}^n$ in ${\mathbb{C}}^n$ for $n\ge 1$. We also characterize the paired operators on $L^2({\mathbb{T}}^n)$. Furthermore, we give a complete characterization for the class of essentially Toeplitz + essentially Hankel operators on the vector-valued Hardy space $H_{\mathcal{E}}^2(\mathbb{D})$ for finite-dimensional Hilbert space $\mathcal{E}$.

arXiv commentPages 23. Title revised. Sections 2 and 3 revised to introduce a new notion of Hankel-type operators on the polydisc