2025/03/31 by Tomar, Nitin
#47A15 #47A20 #47A25 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2503.23754
For 0 < r < 1 , let \mathbbAr = \ z ∈ ℂ : r < |z| < 1 \ be the annulus with boundary ∂ \mathbbAr = \mathbbT ∪ r\mathbbT , where \mathbbT is the unit circle in the complex plane \mathbb C. We study the class of operators C1,r = \ T : T is invertible and ‖T‖, ‖rT-1‖ ≤ 1 \, introduced by Bello and Yakubovich. Any operator T for which the closed annulus \mathbbAr is a spectral set is in C1,r. The class C1, r is closely related to the quantum annulus which is given by QAr = \ T : T is invertible and ‖rT‖, ‖rT-1‖ ≤ 1 \. McCullough and Pascoe proved that an operator in QAr admits a dilation to an operator S satisfying (r-2 + r2)I - S^*S - S-1S-* = 0. An analogous dilation result holds for operators in C1,r class. We extend these dilation results to doubly commuting tuples of operators in quantum annulus as well as in C1,r class. We also provide characterizations and decomposition results for such tuples.