An operator model in the annulus

TL;DR

This paper develops a functional model for invertible operators on Hilbert spaces satisfying a specific positivity condition related to an annulus, establishing the annulus as a complete spectral set and connecting to classical operator theories.

Contribution

It introduces a concrete unitarily equivalent model for operators with a positivity condition tied to the annulus, extending the understanding of spectral sets and operator classes.

Findings

The annulus is a complete K-spectral set for the operator.

A new functional model is constructed for operators satisfying the positivity condition.

Connections are made with Sz.-Nagy--Foias model and observability gramian.

Abstract

For an invertible linear operator $T$ on a Hilbert space $H$, put \[ \alpha(T^*,T) := -T^{*2}T^2 + (1+r^2) T^* T - r^2 I, \] where $I$ stands for the identity operator on $H$ and $r\in (0,1)$; this expression comes from applying Agler's hereditary functional calculus to the polynomial $\alpha(t)=(1-t) (t-r^2)$. We give a concrete unitarily equivalent functional model for operators satisfying $\alpha(T^*,T)\ge0$. In particular, we prove that the closed annulus $r\le |z|\le 1$ is a complete $K$-spectral set for $T$. We explain the relation of the model with the Sz.-Nagy--Foias one and with the observability gramian and discuss the relationship of this class with other operator classes related to the annulus.