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Kernels with complete Nevanlinna-Pick factors and the characteristic function

Tirthankar Bhattacharyya, Abhay Jindal

Abstract

The Sz.-Nagy Foias characteristic function for a contraction has had a rejuvenation in recent times due to a number of authors. Such a classical object relates to an object of very contemporary interest, viz., the complete Nevanlinna-Pick kernels. Indeed, a unitarily invariant kernel on the unit ball {\em admits} a characteristic function if and only if it is a complete Nevanlinna-Pick kernel. However, what has captured our curiosity are the recent advancements in constructing characteristic functions for kernels that do not have complete Nevanlinna-Pick property. In such cases, the reproducing kernel Hilbert space which has served as the domain of the multiplication operator has always been the vector-valued Drury-Arveson space (thus the Hardy space in case of the unit disc). We present a unified framework for deriving characteristic functions for kernels that allow a complete Nevanlinna-Pick factor. Notably, our approach not only encapsulates all previously documented cases but also achieves a remarkable level of generalization, thereby expanding the concept of the characteristic function substantially. We also provide an explanation for the prominence of the Drury-Arveson kernel in all previously established results by showing that the Drury-Arveson kernel was the natural choice inherently suitable for those situations.

Kernels with complete Nevanlinna-Pick factors and the characteristic function

Abstract

The Sz.-Nagy Foias characteristic function for a contraction has had a rejuvenation in recent times due to a number of authors. Such a classical object relates to an object of very contemporary interest, viz., the complete Nevanlinna-Pick kernels. Indeed, a unitarily invariant kernel on the unit ball {\em admits} a characteristic function if and only if it is a complete Nevanlinna-Pick kernel. However, what has captured our curiosity are the recent advancements in constructing characteristic functions for kernels that do not have complete Nevanlinna-Pick property. In such cases, the reproducing kernel Hilbert space which has served as the domain of the multiplication operator has always been the vector-valued Drury-Arveson space (thus the Hardy space in case of the unit disc). We present a unified framework for deriving characteristic functions for kernels that allow a complete Nevanlinna-Pick factor. Notably, our approach not only encapsulates all previously documented cases but also achieves a remarkable level of generalization, thereby expanding the concept of the characteristic function substantially. We also provide an explanation for the prominence of the Drury-Arveson kernel in all previously established results by showing that the Drury-Arveson kernel was the natural choice inherently suitable for those situations.
Paper Structure (9 sections, 15 theorems, 83 equations)

This paper contains 9 sections, 15 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Generalization of the characteristic function
  3. The operators
  4. The kernels
  5. The Characteristic Function
  6. The Construction
  7. Kernels with a CNP factor
  8. The Taylor joint spectrum and functional calculus
  9. Construction of the characteristic function

Key Result

Theorem 2.3

Suppose we have a pure $1/k$-contraction $\textit{T} = (T_{1}, \ldots,T_{d})$ acting on a Hilbert space ${\mathcal{H}}$. Then the linear map $V_{\textit{T}}:{\mathcal{H}} \to H_{k} \otimes \overline{\rm Ran} \Delta_{\textit{T}}$ given by is an isometry. Moreover it satisfies the relation for all $i=1,\dots,d.$

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 26 more