Toeplitz and Asymptotic Toeplitz operators on $H^2(\mathbb{D}^n)$

We initiate a study of asymptotic Toeplitz operators on the Hardy space $H^2(\mathbb{D}^n)$ (over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$). We also study the Toeplitz operators in the polydisc setting. Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let $T_{z_i}$ denote the multiplication operator on $H^2(\mathbb{D}^n)$ by the $i^{th}$ coordinate function $z_i$, $i =1, \ldots, n$, and let $T$ be a bounded linear operator on $H^2(\mathbb{D}^n)$. Then the following hold: (i) $T$ is a Toeplitz operator (that is, $T = P_{H^2(\mathbb{D}^n)} M_{\varphi}|_{H^2(\mathbb{D}^n)}$, where $M_{\varphi}$ is the Laurent operator on $L^{2}(\mathbb{T}^n)$ for some $\varphi \in L^\infty(\mathbb{T}^n)$) if and only if $T_{z_i}^* T T_{z_i} = T$ for all $i = 1, \ldots, n$. (ii) $T$ is an asymptotic Toeplitz operator if and only if $T = \mbox{~Toeplitz} + \mbox{~compact}$. The case $n = 1$ is the well known results of Brown and Halmos, and Feintuch, respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.