## Annals of Functional Analysis

### Level sets of the condition spectrum

#### Abstract

For $0\lt \epsilon\leq1$ and an element $a$ of a complex unital Banach algebra $\mathcal{A}$, we prove the following two topological properties about the level sets of the condition spectrum. (1) If $\epsilon=1$, 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 $0\lt \epsilon\lt 1$, then the $\epsilon$-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 $\epsilon$-condition spectrum of $T$ has an empty interior.

#### Article information

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

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

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

#### 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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