Journal of Operator Theory

Volume 90, Issue 1, Summer 2023 pp. 25-40.

An operator model in the annulus

Authors: Glenier Bello (1) and Dmitry V. Yakubovich (2)
Author institution: (1) Departamento de Matematicas, Universidad Autonoma de Madrid, 28049, Spain
(2) Departamento de Matematicas, Universidad Autonoma de Madrid, 28049, Spain

Summary: For an invertible linear operator $T$ on a Hilbert space $\mathcal{H}$, put \begin{equation*} \alpha(T^*,T) := -T^{*2}T^2 + (1+r^2) T^* T - r^2 I, \end{equation*} where $I$ stands for the identity operator on $\mathcal{H}$ 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 $\alpha(T^*,T)\geqslant 0$. In particular, we prove that the closed annulus $r\leqslant |z|\leqslant 1$ is a complete $\sqrt{2}$-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.

DOI: http://dx.doi.org/10.7900/jot.2021sep05.2346
Keywords: Dilation, functional model, operator inequality, annulus.

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