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

A bounded Hilbert space operator T for which the closure of the annulus

$$\begin{aligned} \mathbb {A}_r=\{z: \ r<|z|<1\} \subseteq \mathbb {C}, \qquad (0<r<1) \end{aligned}$$

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 an annulus, i.e., under what conditions does \(\widetilde{T}_Y=\begin{bmatrix} T_1 & Y\\ 0 & T_2\\ \end{bmatrix} \) become an \(\mathbb {A}_r\)-contraction? For \(\mathbb {A}_r\)-contractions \(T, T_1,T_2\) and an operator X that commutes with \(T, T_1,T_2\), here we find a necessary and sufficient condition such that each of the block matrices

$$\begin{aligned} T_X= \begin{bmatrix} T & X\\ 0 & T\\ \end{bmatrix} , \quad \widehat{T}_X=\begin{bmatrix} T_1 & X(T_1-T_2)\\ 0 & T_2\\ \end{bmatrix} \end{aligned}$$

becomes an \(\mathbb {A}_r\)-contraction.