The $2 \times 2$ block matrices associated with an annulus

A bounded Hilbert space operator $T$ for which the closure of the annulus $$\mathbb{A}_r=\{z \ : \ r<|z|<1\} \subseteq \mathbb{C}, \qquad (0<r<1)$$ is a spectral set is called an $\mathbb A_r$-contraction. A celebrated theorem due to Douglas, Muhly and Pearcy gives a necessary and sufficient condition such that a $2 \times 2$ block matrix of operators $\begin{bmatrix} T_1 & X 0 & T_2 \end{bmatrix}$ is a contraction. We seek an answer to the same question in the setting of annulus, i.e., under what conditions $\widetilde{T}_Y=\begin{bmatrix} T_1 & Y 0 & T_2 \end{bmatrix}$ becomes an $\mathbb A_r$-contraction. For a pair of $\mathbb A_r$-contractions $T_1,T_2$ and an operator $X$ that commutes with $T_1,T_2$, here we find a necessary and sufficient condition such that each of the block matrices $$T_X= \begin{bmatrix} T & X 0 & T \end{bmatrix} \,, \quad \widehat{T}_X=\begin{bmatrix} T_1 & X(T_1-T_2) 0 & T_2 \end{bmatrix}$$ becomes an $\mathbb A_r$-contraction. Thus, the general block matrix $\widetilde{T}_Y$ is an $\mathbb A_r$-contraction if $T_1-T_2$ (as in $\widehat{T}_X$) is invertible.