Unitary parts of Toeplitz operators with operator-valued symbols
Volume 282 / 2025
Abstract
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 their restriction is unitary. We show that for a Hilbert space $\mathcal {E}$ and operator-valued function $\Phi (e^{it}) \in L_{\mathcal {B}(\mathcal {E})}^{\infty }(\mathbb {T})$, the Toeplitz operator $T_{\Phi }$ on $H_{\mathcal {E}}^2(\mathbb {D})$ with $\|T_{\Phi }\| \leq 1$, has such a non-trivial unitary subspace if and only if there exists a non-zero 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 \[ \Phi (e^{it}) \Theta (e^{it}) = \Theta (e^{it}) U,\ \quad \Phi (e^{it})^* \Theta (e^{it}) = \Theta (e^{it}) U^* \quad \ (\text {a.e. on}\ \mathbb {T}). \] In particular, we get the following intertwining relationships: \[ T_{\Phi } M_{\Theta } = M_{\Theta } U,\ \quad T_{\Phi }^* M_{\Theta } = M_{\Theta } U^*, \] where $M_{\Theta }:H_{\mathcal {F}}^2(\mathbb {D}) \rightarrow H_{\mathcal {E}}^2(\mathbb {D})$ denotes the multiplication operator corresponding to the symbol $\Theta (z)$. As an immediate consequence, we establish the corresponding result for Toeplitz operators on $H^2(\mathbb {D})$ by Goor (1972). Furthermore, we obtain the following applications:
(I) Finer characterizations for the non-trivial unitary parts of analytic Toeplitz operators $T_{\Phi }$ on $H_{\mathcal {E}}^2(\mathbb {D})$ by finding their correspondence with the non-trivial unitary parts of $\Phi (0)$ on $\mathcal {E}$.
(II) Tractable conditions for $T_{\Phi }$ to be completely non-unitary.
