The level sets of the resolvent norm and convexity properties of Banach spaces (Q1039941)

\textit{J.\,Globevnik} [Ill.\ J.\ Math.\ 20, 503--506 (1976; Zbl 0322.30040)] posed the question whether or not the norm of the resolvent of an element of a unital Banach algebra can be constant on an open subset of the resolvent set. The same question was posed as an open problem by A.\,Böttcher in 1994 at a Banach Semester in Warsaw. In the 1990s, A.\,Böttcher highlighted the importance of the question of Globevnik for computational spectral analysis. \textit{E.\,Sharogodsky} constructed in [Bull.\ Lond.\ Math.\ Soc.\ 40, No.\,3, 493--504 (2008; Zbl 1147.47007)] an invertible bounded linear operator on a Banach space and a closely densely defined operator on a Hilbert space with resolvent norms that are constant in a neighbourhood of zero, and then gives an affirmative answer to the Globevnik-Böttcher question. The main theorem of this article is that there exist a separable, reflexive and strictly convex Banach space \(X\) and an operator \(T\in L(X)\) such that \(\|(T-\lambda I)^{-1}\|\) is constant in a neighborhood of \(\lambda =0\). Some useful remarks are also made.