Complete Nevanlinna–Pick kernels and the curvature invariant

IF 1 3区 数学 Q1 MATHEMATICS
Tirthankar Bhattacharyya, Abhay Jindal
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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.

完成奈万林纳-匹克核和曲率不变量
我们考虑一个由s表示的一元不变完备Nevanlinna-Pick核和一个由有界算子\(\varvec{T}= (T_{1}, \dots , T_{d})\)组成的交换d元组,满足s的自然收缩条件。我们将其曲率不变量\(\varvec{T}\)联系起来,该曲率不变量是一个非负实数,其上由\(\varvec{T}\)缺陷空间的维数限定。使这成为可能的仪器是Adv Math 426:109089, 2023, https://doi.org/10.1016/j.aim.2023.109089中开发的特征函数。给出了曲率不变量的渐近公式。在\(\varvec{T}\)为纯的特殊情况下,我们提供了一个明显更简单的公式,揭示了在这种情况下,曲率不变量是一个整数。我们进一步研究了它与称为纤维维数的代数不变量的联系。此外,我们还得到了\(\varvec{T}\)在特征函数为多项式时曲率不变量的一个精炼简化的渐近公式。
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来源期刊
CiteScore
2.10
自引率
10.00%
发文量
99
审稿时长
>12 weeks
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.
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