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):=−T∗2T2+(1+r2)T∗T−r2I,
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)=(1−t)(t−r2).
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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