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 $\mathcal{ E}$.
Categories: math.FA