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

A bounded Hilbert space operator $T$ for which the closure of the annulus $${\mathbb{A}}_r=\{z:r<\mathrm{\mid }z\mathrm{\mid }<1\}\subseteq \mathbb{C},(0<r<1)$$ 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 $\left[\begin{array}{ccc}{T}_1& X0& T_2\end{array}\right]$ is a contraction. We seek an answer to the same question in the setting of annulus, i.e., under what conditions ${\tilde{T}}_Y=\left[\begin{array}{ccc}{T}_1& Y0& T_2\end{array}\right]$ 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 $$T_X=\left[\begin{array}{ccc}T& X0& T\end{array}\right],{\widehat{T}}_X=\left[\begin{array}{ccc}{T}_1& X(T_1-T_2)0& T_2\end{array}\right]$$ becomes an ${\mathbb{A}}_r$-contraction. Thus, the general block matrix ${\tilde{T}}_Y$ is an ${\mathbb{A}}_r$-contraction if $T_1-T_2$ (as in ${\widehat{T}}_X$) is invertible.