Characterization of paired and Toeplitz + Hankel operators on the polydisc

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 $L^2({\mathbb{T}}^n)$. For an inner function $\theta \in H^{\mathrm{\infty }}({\mathbb{D}}^n)$, we further characterize $\theta$-paired operators on $H^2({\mathbb{D}}^n)$ with respect to the Beurling submodule $\theta H^2({\mathbb{D}}^n)$.