Toeplitz operators on the Hardy space over the infinite-dimensional polydisc

We study infinite multiplicative Toeplitz matrices and Toeplitz operators on the Hardy space H-2(T-infinity) over the infinite-dimensional polydisc. This pair is a companion to the pair of Toeplitz matrices and Toeplitz operators on the Hardy space over the unit disc. We obtain a Brown- Halmos type theorem and the spectral inclusion theorem. The conditions for the semi-commutator TfTg-T-f (g) of Tocplitz operators T-f and T-g to be of finite rank are obtained if one of f and g depends only on finitely many variables. It is also shown that the Toeplitz algebra T(infinity)(F)generated by Toeplitz operators with bounded symbols depending only on finitely many variables on H-2(T-infinity) doesn't contain any nonzero compact operator. In particular, the Toeplitz algebra generated by Toeplitz operators with continuous symbols on H-2(T-infinity), as a C+-subalgebra of T(infinity)(F)contains no nonzero compact operators, and this is in sharp contrast to the case of finite-dimensional polydiscs. Moreover, the symbolic calculus for the Toeplitz algebra generated by all bounded Toeplitz operators is established.