Partially isometric Toeplitz operators with operator-valued symbols

In this paper, we study partially isometric Toeplitz operators TΦ on Hilbert space-valued Hardy spaces HE²(Dⁿ) over the unit polydisc. We establish the following crucial phenomenon: the range of partially isometric Toeplitz operators is always a Beurling-type invariant subspace of HE²(Dⁿ). Using this result, we prove that partially isometric Toeplitz operators always admit the following factorization: \[ TΦ = MΓ MΨ^*, \] where Γ(z), Ψ(z) are operator-valued inner functions on Dⁿ, governed by certain conditions that force MΓ MΨ^* to become a Toeplitz operator. 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 Hardy spaces.