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Cyclic Operators, Linear Functionals and RKHS

Dexie Lin, Yi Wang

TL;DR

The paper develops a kernel- and distribution-driven framework to study near-subnormality through Agler’s linear functional $\Lambda_{\mathbf{T},h}$. It introduces the two-variable kernel $F_{\mathbf{T},h}$ and shows that the off-diagonal growth of $F_{\mathbf{T},h}$ precisely characterizes when $\Lambda_{\mathbf{T},h}$ is a (compactly supported) distribution, via Paley–Wiener–Schwartz theory. A function-model is constructed on the reproducing kernel Hilbert space $\mathcal{H}(F_{\mathbf{T},h})$, yielding a unitary equivalence $T_i^*=U^*\partial_i U$ and enabling a framework to study joint spectra and eigenvalues through holomorphic function theory. In the matrix case, the results specialize to a Jordan-decomposition criterion, and the paper develops a convolution/norm theory for RKHS kernels, showing how product kernels correspond to joined operator tuples with norm bounds. A rich suite of examples, including P^2(μ), the Drury–Arveson space, and Jordan/subjordan models, illustrates the approach and highlights potential for new operator-model constructions beyond classical subnormality.

Abstract

Given a commuting $n$-tuple of bounded linear operators on a Hilbert space, together with a distinguished cyclic vector, Jim Agler defined a linear functional $Λ_{\mathbf{T},h}$ on the polynomial ring $\mathbb{C}[\mathbf{z},\bar{\mathbf{z}}]$. ``Near subnormality properties'' of an operator $T$ are translated into positivity properties of $Λ_{T,h}$. In this paper, we approach ``near subnormality properties'' in a different way by answering the following question: when is $Λ_{\mathbf{T},h}$ given by a compactly supported distribution? The answer is in terms of the off-diagonal growth condition of a two-variable kernel function $F_{\mathbf{T},h}$ on $\mathbb{C}^n$. Using the reproducing kernel Hilbert spaces (RKHS) defined by the kernel function $F_{\mathbf{T},h}$, we give a function model for all cyclic commuting $n$-tuples. This potentially gives a different approach to operator models. The reproducing kernels of the Fock space are used in the construction of $F_{\mathbf{T},h}$, but one may also replace the Fock space by other RKHS. We give many examples in the last section.

Cyclic Operators, Linear Functionals and RKHS

TL;DR

The paper develops a kernel- and distribution-driven framework to study near-subnormality through Agler’s linear functional . It introduces the two-variable kernel and shows that the off-diagonal growth of precisely characterizes when is a (compactly supported) distribution, via Paley–Wiener–Schwartz theory. A function-model is constructed on the reproducing kernel Hilbert space , yielding a unitary equivalence and enabling a framework to study joint spectra and eigenvalues through holomorphic function theory. In the matrix case, the results specialize to a Jordan-decomposition criterion, and the paper develops a convolution/norm theory for RKHS kernels, showing how product kernels correspond to joined operator tuples with norm bounds. A rich suite of examples, including P^2(μ), the Drury–Arveson space, and Jordan/subjordan models, illustrates the approach and highlights potential for new operator-model constructions beyond classical subnormality.

Abstract

Given a commuting -tuple of bounded linear operators on a Hilbert space, together with a distinguished cyclic vector, Jim Agler defined a linear functional on the polynomial ring . ``Near subnormality properties'' of an operator are translated into positivity properties of . In this paper, we approach ``near subnormality properties'' in a different way by answering the following question: when is given by a compactly supported distribution? The answer is in terms of the off-diagonal growth condition of a two-variable kernel function on . Using the reproducing kernel Hilbert spaces (RKHS) defined by the kernel function , we give a function model for all cyclic commuting -tuples. This potentially gives a different approach to operator models. The reproducing kernels of the Fock space are used in the construction of , but one may also replace the Fock space by other RKHS. We give many examples in the last section.

Paper Structure

This paper contains 6 sections, 30 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm: intro ans to Question 1']}
  3. Proof of Theorem \ref{['thm: intro H(F) and T']}
  4. Proof of Theorem \ref{['thm: intro matrix case']}
  5. Proof of Theorem \ref{['thm: intro convolution norm']}
  6. Examples

Key Result

Theorem 1.1

For a cyclic commuting $n$-tuple $(\mathbf{T}, h)$ on a Hilbert space $\mathcal{H}$ and a compact convex set $K\subset\mathbb{C}^n$, the following hold.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2: HormanderPDEI2003 Definition 4.3.1
  • Theorem 2.3: Paley-Wiener-Schwartz HormanderPDEI2003 Theorem 7.3.1
  • Definition 2.4
  • proof : Proof of Theorem \ref{['thm: intro ans to Question 1']}
  • Lemma 3.1
  • ...and 51 more