Annals of Functional Analysis

Level sets of the condition spectrum

D. Sukumar and S. Veeramani

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.afa/1491280440

Digital Object Identifier
doi:10.1215/20088752-0000016X

Mathematical Reviews number (MathSciNet)
MR3689995

Zentralblatt MATH identifier
1378.46034

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

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

Citation

Sukumar, D.; Veeramani, S. Level sets of the condition spectrum. Ann. Funct. Anal. 8 (2017), no. 3, 314--328. doi:10.1215/20088752-0000016X. https://projecteuclid.org/euclid.afa/1491280440


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