Toeplitz and asymptotic Toeplitz operators on [formula omitted

Abstract

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:(i)T is a Toeplitz operator (that is, T=PH2(Dn)Mφ|H2(Dn), where Mφ is the Laurent operator on L2(Tn) for some φ∈L∞(Tn)) if and only if Tzi⁎TTzi=T for all i=1,…,n.(ii)T is an asymptotic Toeplitz operator if and only if T=Toeplitz+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.