Unitary parts of Toeplitz operators with operator-valued symbols

Motivated by the canonical decomposition of contractions on Hilbert spaces, we investigate when contractive Toeplitz operators on vector-valued Hardy spaces on the unit disc admit a non-zero reducing subspace on which its restriction is unitary. We show that for a Hilbert space $\mathcal{E}$ and operator-valued symbol $\mathrm{\Phi }\in L_{\mathcal{B}(\mathcal{E})}^{\mathrm{\infty }}(\mathbb{T})$, the Toeplitz operator $T_{\mathrm{\Phi }}$ on $H_{\mathcal{E}}^2(\mathbb{D})$ has such a unitary subspace if and only if there exists a Hilbert space $\mathcal{F}$, an inner function $\mathrm{\Theta }(z)\in H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\mathrm{\infty }}(\mathbb{D})$, and a unitary $U:\mathcal{F}\to \mathcal{F}$ such that $$\mathrm{\Phi }(e^{it})\mathrm{\Theta }(e^{it})=\mathrm{\Theta }(e^{it})U\text{and}\mathrm{\Phi }(e^{it}{)}^{\ast }\mathrm{\Theta }(e^{it})=\mathrm{\Theta }(e^{it})U^{\ast }(\text{ a.e. on }\mathbb{T})\mathrm{.}$$ This result can be seen as a generalization of the corresponding result for Toeplitz operators on $H^2(\mathbb{D})$ by Goor in [13]. We provide finer characterizations for analytic Toeplitz operators by finding the correspondence between the unitary parts of $T_{\mathrm{\Phi }}$ on $H_{\mathcal{E}}^2(\mathbb{D})$ and $\mathrm{\Phi }(0)$ on $\mathcal{E}$.