Annals of Functional Analysis

Level sets of the condition spectrum

D. Sukumar and S. Veeramani

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For 0<ϵ1 and an element a of a complex unital Banach algebra A, we prove the following two topological properties about the level sets of the condition spectrum. (1) If ϵ=1, then the 1-level set of the condition spectrum of a has an empty interior unless a is a scalar multiple of the unity. (2) If 0<ϵ<1, then the ϵ-level set of the condition spectrum of a has an empty interior in the unbounded component of the resolvent set of a. Further, we show that, if the Banach space X is complex uniformly convex or if X is complex uniformly convex, then, for any operator T acting on X, the level set of the ϵ-condition spectrum of T has an empty interior.

Article information

Ann. Funct. Anal., Volume 8, Number 3 (2017), 314-328.

Received: 24 August 2016
Accepted: 5 October 2016
First available in Project Euclid: 4 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46H05: General theory of topological algebras
Secondary: 47A10: Spectrum, resolvent

condition spectrum vector-valued analytic functions complex uniformly convex Banach space


Sukumar, D.; Veeramani, S. Level sets of the condition spectrum. Ann. Funct. Anal. 8 (2017), no. 3, 314--328. doi:10.1215/20088752-0000016X.

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