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Level sets of \((p,e-p)\) outer generalized pseudo spectrum

Proceedings: ICMAA 2016
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Abstract

Let \(\mathscr {A}\) be a complex Banach algebra with unit e. Let p be a non trivial idempotent element in \(\mathscr {A}\) and \(\varepsilon >0.\) For \(a \in \mathscr {A},\) it is proved that the interior of the level set of \((p,e-p)-\varepsilon \) pseudo spectrum of a is empty in the unbounded component of \((p,e-p)\) resolvent set of a. An example is constructed to show that the condition ‘unbounded component’ can not be dropped. Further, it is proved this ‘unbounded component’ can be dropped in the case when \(\mathscr {A}\) is B(X) where X is a complex uniformly convex Banach space. That is, if \(T \in B(X)\) then interior of the level set of \((p,I-p)-\varepsilon \) pseudo spectrum is empty in \((p,I-p)\) resolvent set of T.

Keywords

Analytic vector valued map \((p, q)-\varepsilon \) pseudo spectrum Complex uniformly convex Banach space 

Mathematics Subject Classification

Primary 46H05; Secondary 47A10 15A09 

Notes

Acknowledgements

The authors are thankful to the anonymous referee for the valuable suggestions towards the improvement of this paper. Research of the first author was supported by the Department of Science and Technology (DST), India (No: SB/FTP/MS-015/2013). Second author thanks the University Grants Commission (UGC), India for the financial support provided as a form of Research Fellowship to carry out this research work at IIT Hyderabad.

Compliance with ethical standards

Conflict of interest

The authors have equally contributed and give their consent for publication.The authors declare that they have no conflict of interest.

Research involving human participants and/or animals

This paper does not contain any studies involving with human participants/ animals.

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Copyright information

© Forum D'Analystes, Chennai 2017

Authors and Affiliations

  1. 1.Department of MathematicsIIT HyderabadHyderabadIndia

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