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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 $H^2(\\mathbb{D}^n)$. 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