Compact Toeplitz operators products on Hardy–Sobolev spaces over the unit polydisk

Abstract

We study the compactness of finite sums of products of two Toeplitz operators on Hardy–Sobolev spaces over the unit polydisk $H_{\beta }^2\left({\mathbb{𝔻}}^n\right)$. We calculate the essential norm of these operators and answer the question of when a Toeplitz operator on $H_{\beta }^2\left({\mathbb{𝔻}}^n\right)$ is Fredholm.