# 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.

Contents
Full-Text PDF