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

Let $\mathcal E$ be a separable Hilbert space, $B\in H^\infty_1(L(\mathcal E))$, and $α>-1$. Associated with the Toeplitz operator $T_B$ on the vector-valued weighted Bergman space $A^2_{α,\mathcal E}$ are the sub-Bergman spaces $\mathcal A_α(B)$ and $\mathcal A_α(B^*)$. In this paper, we prove that