The \(2 \times 2\) block matrices associated with an annulus

A bounded Hilbert space operator T for which the closure of the annulus \(\begin{aligned} \mathbb {A}_r=\{z: \ r<|z|<1\} \subseteq \mathbb {C}, \qquad (0<r<1) \end{aligned}\) $$\begin{array}{c}\hfill {\mathbb{A}}_r=\{z:r<|z|<1\}\subseteq \mathbb{C},(0<r<1)\end{array}$$ is a spectral set is called an \(\mathbb {A}_r\) ${\mathbb{A}}_r$ -contraction. A celebrated theorem due to Douglas, Muhly, and Pearcy gives a necessary and sufficient condition such that a \(2 \times 2\) $2\times 2$ block matrix of operators \( \begin{bmatrix} T_1 & X \\ 0 & T_2 \end{bmatrix} \) $\left[\begin{array}{cc}{T}_1& X\\ 0& T_2\end{array}\right]$ is a contraction. We seek an answer to the same question in the setting of an annulus, i.e., under what conditions does \(\widetilde{T}_Y=\begin{bmatrix} T_1 & Y\\ 0 & T_2\\ \end{bmatrix} \) ${\tilde{T}}_Y=\left[\begin{array}{cc}{T}_1& Y\\ 0& T_2\\ \end{array}\right]$ become an \(\mathbb {A}_r\) ${\mathbb{A}}_r$ -contraction? For \(\mathbb {A}_r\) ${\mathbb{A}}_r$ -contractions \(T, T_1,T_2\) $T,T_1,T_2$ and an operator X that commutes with \(T, T_1,T_2\) $T,T_1,T_2$ , here we find a necessary and sufficient condition such that each of the block matrices \(\begin{aligned} T_X= \begin{bmatrix} T & X\\ 0 & T\\ \end{bmatrix} , \quad \widehat{T}_X=\begin{bmatrix} T_1 & X(T_1-T_2)\\ 0 & T_2\\ \end{bmatrix} \end{aligned}\) $$\begin{array}{c}\hfill T_X=\left[\begin{array}{cc}T& X\\ 0& T\\ \end{array}\right],{\widehat{T}}_X=\left[\begin{array}{cc}{T}_1& X(T_1-T_2)\\ 0& T_2\\ \end{array}\right]\end{array}$$ becomes an \(\mathbb {A}_r\) ${\mathbb{A}}_r$ -contraction.