Level sets of the condition spectrum

Abstract

For $0<ϵ\le 1$ 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 $ϵ=1$, 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 $X^{\ast }$ 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.