Complete Nevanlinna-Pick Kernels and the Curvature Invariant

2024/01/28 by Bhattacharyya, Tirthankar, Jindal, Abhay
#46E22 #47A13 #47A15 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2401.15591

Abstract

We consider a unitarily invariant complete Nevanlinna-Pick kernel denoted by s and a commuting d-tuple of bounded operators T = (T₁, …, T_d) satisfying a natural contractivity condition with respect to s. We associate with T its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of \bfT. The instrument which makes this possible is the characteristic function developed in \citeBJ. \medskip We present an asymptotic formula for the curvature invariant. In the special case when \bfT 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 fibre dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of \bfT specifically when its characteristic function is a polynomial.

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