Level sets of condition spectrum

Abstract

For $0\u3c\epsilon\leq 1$ and an element $a$ of a complex Banach algebra $\mathcal{A}$ with unit $e$, the level set of $\epsilon$- condition spectrum is defined as $$L_{\epsilon}(a)\coloneqq\left\{\lambda\in \mathbb{C} : \|(a-\lambda.e)\|\left\|(a-\lambda.e)^{-1}\right\|=\frac{1}{\epsilon}\right\}.$$ We prove the following topological properties about $L_{\epsilon}(a)$ \begin{enumerate} \item If $\epsilon=1$ then $L_{1}(a)$ has an empty interior unless $a$ is a scalar multiple of the unit. %$L_{1}(a)$ has non empty interior for $a = \lambda$ where $\lambda\in \mathbb{C}$ and \item If $0\u3c\epsilon\u3c1$ then $L_{\epsilon}(a)$ has an empty interior %for $a = \lambda$ where $\lambda\in \mathbb{C}$ and also for any $a$ which is not a scalar multiple of the unit, $L_{\epsilon}(a)$ has 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 $X^{*}$ is complex uniformly convex, then for any operator $T\in B(X)$, $L_{\epsilon}(T)$ has an empty interior. \end{enumerate