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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 H, put α(T,T):=T2T2+(1+r2)TTr2I, where I stands for the identity operator on H and r(0,1); this expression comes from applying Agler's hereditary functional calculus to the polynomial α(t)=(1t)(tr2). We give a concrete unitarily equivalent functional model for operators satisfying α(T,T). 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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