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

Abstract

We study infinite multiplicative Toeplitz matrices and Toeplitz operators on the Hardy space $$H^2(\mathbb{T}^{\infty})$$ 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 $$ T_{f}T_{g}-T_{fg} $$ of Toeplitz operators $$ T_{f} $$ and $$T_{g} $$ to be of finite rank are obtained if one of ƒ and g depends only on finitely many variables. It is also shown that the Toeplitz algebra $$\mathcal{T}^{F}_{\infty} $$ generated by Toeplitz operators with bounded symbols depending only on finitely many variables on $$H^2(\mathbb{T}^{\infty}) $$ doesn’t contain any nonzero compact operator. In particular, the Toeplitz algebra generated by Toeplitz operators with continuous symbols on $$H^2(\mathbb{T}^{\infty}) $$ , as a C*-subalgebra of $$\mathcal{T}^{F}_{\infty} $$ , 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.