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Complete Nevanlinna-Pick Kernels and the Curvature Invariant

Tirthankar Bhattacharyya, Abhay Jindal

Abstract

We consider a unitarily invariant complete Nevanlinna-Pick kernel denoted by $s$ and a commuting $d$-tuple of bounded operators $T = (T_{1}, \dots, T_{d})$ 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 \cite{BJ}. \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.

Complete Nevanlinna-Pick Kernels and the Curvature Invariant

Abstract

We consider a unitarily invariant complete Nevanlinna-Pick kernel denoted by and a commuting -tuple of bounded operators satisfying a natural contractivity condition with respect to . We associate with its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of . The instrument which makes this possible is the characteristic function developed in \cite{BJ}. \medskip We present an asymptotic formula for the curvature invariant. In the special case when 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 specifically when its characteristic function is a polynomial.
Paper Structure (4 sections, 16 theorems, 88 equations)

This paper contains 4 sections, 16 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The curvature invariant
  3. The asymptotic formula for multipliers
  4. The asymptotic formula for the curvature invariant

Key Result

Lemma 3.1

The inclusion map $\delta: H_{s} \otimes {\rm Ran} \Delta_{\textit{T}} \to H^{2}(\partial \mathbb{B}_{d}) \otimes {\rm Ran} \Delta_{\textit{T}}$ has the following properties:

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 22 more