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 $\Phi \in L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T})$, the Toeplitz operator $T_{\Phi}$ 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 $\Theta(z) \in H_{\mathcal{B}(\mathcal{F}, \mathcal{E})}^{\infty}(\mathbb{D})$, and a unitary $U:\mathcal{F} \rightarrow \mathcal{F}$ such that $$\Phi(e^{it}) \Theta(e^{it}) = \Theta(e^{it}) U \quad \text{and} \quad \Phi(e^{it})^* \Theta(e^{it}) = \Theta(e^{it}) U^* \quad (\text{ a.e. on }\mathbb{T}).$$ 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 $T_{\Phi}$ on $H_{\mathcal{E}}^2(\mathbb{D})$ and $\Phi(0)$ on $\mathcal{E}$.