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

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