Toeplitz and asymptotic Toeplitz operators on H2(Dn)

We initiate a study of Toeplitz operators and asymptotic Toeplitz operators on the Hardy space H2(Dn) (over the unit polydisc Dn in Cn). Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let Tzi denote the multiplication operator on H2(Dn) by the i-th coordinate function zi, i=1,…,n, and let T be a bounded linear operator on H2(Dn). Then the following hold: