## Annals of Functional Analysis

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

### Level sets of the condition spectrum

#### Abstract

For $0<\u03f5\le 1$ 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 $\u03f5=1$, 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 $0<\u03f5<1$, then the $\u03f5$-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 ${X}^{\ast}$ is complex uniformly convex, then, for any operator $T$ acting on $X$, the level set of the $\u03f5$-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