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

Abstract

We solve the following problems associated with Toeplitz operators $T_{\Phi}$ 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 $\Gamma, \Psi$ on $\mathbb{D}^n$, we completely characterize when the product $M_{\Gamma}M_{\Psi}^*$ 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: \[ T_{\Phi} = M_{\Gamma} M_{\Psi}^*, \] where, $\Gamma, \Psi$ 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 $M_{\Gamma V^*}$, where $\Gamma$ 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.