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SPECTRAL CONSTANTS FOR THE QUANTUM ANNULUS
Journal article   Open access   Peer reviewed

SPECTRAL CONSTANTS FOR THE QUANTUM ANNULUS

Sourav Pal, James e. Pascoe and Nitin Tomar
Communications on pure and applied analysis, Forthcoming
02 Jun 2026
DOI: https://doi.org/10.3934/cpaa.2026084
url
https://doi.org/10.3934/cpaa.2026084View
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Abstract

Mathematics, Applied Science & Technology Mathematics Physical Sciences
For A(r)={z is an element of C:r(-1)<|z|< r}withr >1, we consider thecollection QA(r)={T:Tis an invertible operator and parallel to T parallel to,parallel to T-1 parallel to <= r},which is referred to as thequantum annulus. McCullough-Pascoe [7] proved adilation theorem for operators in QA(r). In this article, we refine this dilationtheorem and explicitly construct such a dilation. LetK(A(r)) be the smallestpositive constant for which Aris a K(A(r))-spectral set for operators in QA(r). Asignificant result due to Tsikalas established the lower boundK(A(r))>= 2, refin-ing earlier estimates. Recently, Pascoe proved that K(A(r))<= 2(1 +(2r2)(r4-1))and henceK(A(r))-> 2 as(r)->infinity. In this article, two alternative proofs of Pas-coe's upper bound are presented. The first one capitalizes a dilation theoremdue to McCullough and Pascoe, while the second involves a certain variety inthe Euclidean biball. In the multivariable setting, we show that the biannulus A(r)(2) is aK-spectral set for some K >0 for commuting pairs of operators in QA(r). Furthermore, we derive upper and lower bounds on the smallest spectralconstantKfor which certain classes of operator tuples in QA(r) have the closed polyannulus A(r)(n) as a K-spectral set. If we denote the smallest constant byK(A(r)(2)) for commuting pairs, andKdc(A(r)(n)) for doubly commutingn-tuples inQA(r), then the resulting bounds are given by 2(n)<= K-dc(A(r)(n))<= (( 3r2-1)(r2-1 ))(n ) and 2(2)<= K(A(r)(2))<="4 +((r2+ 1)(r2-1 ))(2)+ 4 ( (r2+ 1)(r2-1 )) (1/2)#, which further imply that 2(n)<= K-lim(r ->infinity )dc(A(r)(n))<= 3 (n) and 2(2)<= K-lim(r ->infinity)(A(r)(2))<= 3(2)

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