{"title":"Complete Nevanlinna–Pick kernels and the curvature invariant","authors":"Tirthankar Bhattacharyya, Abhay Jindal","doi":"10.1007/s10231-024-01524-1","DOIUrl":null,"url":null,"abstract":"<div><p>We consider a unitarily invariant complete Nevanlinna–Pick kernel denoted by <i>s</i> and a commuting <i>d</i>-tuple of bounded operators <span>\\(\\varvec{T}= (T_{1}, \\dots , T_{d})\\)</span> satisfying a natural contractivity condition with respect to <i>s</i>. We associate with <span>\\(\\varvec{T}\\)</span> its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of <span>\\(\\varvec{T}\\)</span>. The instrument that makes this possible is the characteristic function developed in Adv Math 426:109089, 2023, https://doi.org/10.1016/j.aim.2023.109089. We present an asymptotic formula for the curvature invariant. In the special case when <span>\\(\\varvec{T}\\)</span> is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of <span>\\(\\varvec{T}\\)</span> specifically when its characteristic function is a polynomial.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":"204 3","pages":"1183 - 1197"},"PeriodicalIF":1.0000,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-024-01524-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider a unitarily invariant complete Nevanlinna–Pick kernel denoted by s and a commuting d-tuple of bounded operators \(\varvec{T}= (T_{1}, \dots , T_{d})\) satisfying a natural contractivity condition with respect to s. We associate with \(\varvec{T}\) its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of \(\varvec{T}\). The instrument that makes this possible is the characteristic function developed in Adv Math 426:109089, 2023, https://doi.org/10.1016/j.aim.2023.109089. We present an asymptotic formula for the curvature invariant. In the special case when \(\varvec{T}\) is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of \(\varvec{T}\) specifically when its characteristic function is a polynomial.
期刊介绍:
This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it).
A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.