On products and partial isometry of Toeplitz operators with operator-valued symbols

Abstract

We solve the following problems associated with Toeplitz operators $T_{\hat{I}}¦$ on Hilbert space-valued Hardy spaces $H_{\mathcal{E}}^2({\mathbb{D}}^n)$ over the unit polydisc ${\mathbb{D}}^n$. $(I)$ Given operator-valued bounded analytic functions $\hat{I}“,\hat{I}¨$ on ${\mathbb{D}}^n$, we completely characterize when the product $M_{\hat{I}}“M_{\hat{I}}{¨}^{\ast }$ becomes a Toeplitz operator by identifying tractable conditions on the functions. Furthermore, these conditions can be used to explicitly write the product into a sum of simple Toeplitz operators. $(II)$ We prove that partially isometric Toeplitz operators admit the following factorization: \[ T_Φ = M_Γ M_Ψ^*, \] where, $\hat{I}“,\hat{I}¨$ are operator-valued inner functions on ${\mathbb{D}}^n$. A few of the immediate consequences are: $(a)$ every partially isometric Toeplitz operator has a partially isometric symbol almost everywhere on ${\mathbb{T}}^n$ (distinguished boundary of ${\mathbb{D}}^n$), $(b)$ any partially isometric analytic Toeplitz operator is of the form $M_{\hat{I}“V^{\ast }}$, where $\hat{I}“$ is an operator-valued inner function and $V$ is an constant isometry. In connection with the result $(ii)$, we establish and use a crucial phenomenon: the range of partially isometric Toeplitz operators is always a Beurling-type invariant subspace of $H_{\mathcal{E}}^2({\mathbb{D}}^n)$. Our results are new even in the case of Hardy spaces over the unit disc and extend the work of Brown--Douglas, Deepak--Pradhan--Sarkar on scalar-valued spaces.

Significant revision with new results added in Sections 3 and 5. A new section has been added on abstract Toeplitz operators