Cyclic Operators, Linear Functionals and RKHS

TL;DR

This paper explores when a linear functional associated with a cyclic commuting operator tuple can be represented by a compactly supported distribution, using RKHS and kernel functions to model such operators.

Contribution

It introduces a new characterization of near subnormality via off-diagonal growth conditions of kernel functions and constructs a function model for cyclic commuting operator tuples.

Findings

Characterization of when the linear functional is given by a compactly supported distribution.

Development of a function model for cyclic commuting n-tuples using RKHS.

Application of Fock space kernels and alternative RKHS in the construction of operator models.

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 $\Lambda_{\mathbf{T},h}$ on the polynomial ring $\mathbb{C}[\mathbf{z},\bar{\mathbf{z}}]$. ``Near subnormality properties'' of an operator $T$ are translated into positivity properties of $\Lambda_{T,h}$. In this paper, we approach ``near subnormality properties'' in a different way by answering the following question: when is $\Lambda_{\mathbf{T},h}$ 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.