Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity

Abstract

Let $\mathcal{E}$ be a separable Hilbert space, $B\in H_1^{\infty }(L(\mathcal{E}))$, and $\alpha >-1$. Associated with the Toeplitz operator $T_B$ on the vector-valued weighted Bergman space $A_{\alpha ,\mathcal{E}}^2$ are the sub-Bergman spaces ${\mathcal{A}}_{\alpha }(B)$ and ${\mathcal{A}}_{\alpha }(B^{\ast })$. In this paper, we prove that \[ \mathcal A_α(B)\supset A^2_{α-1,\mathcal H(B(0))} \quad\text{and}\quad \mathcal A_α(B^*)\supset A^2_{α-1,\mathcal H(B(0)^*)}. \] This confirms a conjecture of \cite{GLM26}. We also construct an infinite-dimensional two-sided inner function $B$ such that $\Vert B(0)\Vert <1$ and \[ \mathcal A_α(B)\approx \mathcal A_α(B^*)\approx A^2_{α-1,\mathcal E}, \] but $B$ is not a finite Blaschke-Potapov product. This example shows that the finite-dimensional rigidity theorem of \cite{GLM26} does not extend to infinite-dimensional coefficient spaces.