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

Abstract

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 $\left[\begin{array}{ccc}{T}_1& X0& T_2\end{array}\right]$ 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 \[ T_X= \begin{bmatrix} T & X 0 & T \end{bmatrix} \,, \quad \widehat{T}_X=\begin{bmatrix} T_1 & X(T_1-T_2) 0 & T_2 \end{bmatrix} \] becomes an ${\mathbb{A}}_r$-contraction. Thus, the general block matrix ${\tilde{T}}_Y$ is an ${\mathbb{A}}_r$-contraction if $T_1-T_2$ (as in ${\hat{T}}_X$) is invertible.

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