Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 277-294.

On doubly commuting operators in $C_{1, r}$ class and quantum annulus

Authors: Nitin Tomar
Author institution: Department of Mathematics, Indian Institute of Technology Bombay, Maharashtra, 400076, India

Summary: For $r1$ between $0$ and $1$, let $\mathbb{A}_r = \{ z \in \mathbb{C} : r < |z| < 1 \}$. We study the class $$ C_{1,r} = \{ T : T \text{ is invertible and } \|T\|, \|rT^{-1}\| \leqslant 1 \}.$$ The class $C_{1, r}$ is closely related to the \textit{quantum annulus} given by $QA_r = \{ T : T \text{ is invertible and } \|rT\|,$ $ \|rT^{-1}\| \leqslant 1 \}$. McCullough and Pascoe proved that $T \in QA_r$ dilates to an operator $S$ satisfying $(r^{-2} + r^2)I - S^*S - S^{-1}S^{-*} = 0$. We extend this dilation result to doubly commuting tuples of operators in quantum annulus as well as in $C_{1,r}$ class. We also provide characterizations and decomposition results for such tuples.

DOI: http://dx.doi.org/10.7900/jot.2025apr06.2514
Keywords: $C_{1,r}$ class, quantum annulus, decomposition, dilation

Contents Full-Text PDF