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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1325–1360 | Cite as

Operator Inequalities Implying Similarity to a Contraction

  • Glenier Bello-Burguet
  • Dmitry YakubovichEmail author
Article
  • 40 Downloads

Abstract

Let T be a bounded linear operator on a Hilbert space H such that
$$\begin{aligned} \alpha \left[ T^*,T\right] :=\sum _{n=0}^\infty \alpha _n T^{*n}T^n\ge 0, \end{aligned}$$
where \(\alpha (t)=\sum _{n=0}^\infty \alpha _n t^n\) is a suitable analytic function in the unit disc \({\mathbb {D}}\) with real coefficients. We prove that if \(\alpha (t) = (1-t) \tilde{\alpha } (t)\), where \(\tilde{\alpha }\) has no zeros in [0, 1], then T is similar to a contraction. Operators of this type have been investigated by Agler, Müller, Olofsson, Pott and others, however, we treat cases where their techniques do not apply. We write down an explicit Nagy–Foias type model of an operator in this class and discuss its usual consequences (completeness of eigenfunctions, similarity to a normal operator, etc.). We also show that the limits of \(\Vert T^nh\Vert \) as \(n\rightarrow \infty \), \(h\in H\), do not exist in general, but do exist if an additional assumption on \(\alpha \) is imposed. Our approach is based on a factorization lemma for certain weighted Wiener algebras.

Keywords

Similarity to a contraction Dilation Functional model Operator inequality The Wiener algebra 

Notes

Acknowledgements

The first author acknowledges the Grant Severo-Ochoa La Caixa for undergraduate studies. Both authors are partially supported by Plan Nacional I+D Grant No. MTM2015-66157-C2-1-P. The authors also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). The authors thank Maria Gamal’ and Mihai Putinar for useful suggestions and the unknown referee for suggestions that improved the exposition.

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Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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