A characterization of alternatively convex or smooth Banach spaces (Q1637586)

A Banach space \(X\) is called alternatively convex or smooth if the following holds: Whenever \(x,y\in X\) are of norm one such that the midpoint of \(x\) and \(y\) lies on the unit sphere and \(f\) is a norm-one functional on \(X\) such that \(f(x)=1\), then one also has \(f(y)=1\). This notion was introduced by \textit{V. M. Kadets} in [Quaest. Math. 19, No. 1--2, 225--235 (1996; Zbl 0855.47021)] in the study of the Daugavet equation. An operator \(T\) on \(X\) satisfies the Daugavet equation if \(\|I+T\|=1+\|T\|\), where \(I\) is the identity operator on \(X\). For instance, this is true if \(\|T\|\in \sigma(T)\), where \(\sigma(T)\) is the spectrum of \(T\). If \(M\) is a set of operators on \(X\), then \(X\) is said to have the anti-Daugavet property with respect to \(M\) if every \(T\in M\) which satisfies the Daugavet equation also satisties \(\|T\|\in \sigma(T)\). It was proved by Kadets in [loc. cit.] that a finite-dimensional space \(X\) has the anti-Daugavet property with respect to all operators on \(X\) if and only if \(X\) is alternatively convex or smooth. The authors of the present paper introduce the following notion: \(X\) has the anti-alternative Daugavet property with respect to \(M\) if the equivalence \[ \max_{0\leq \theta\leq 2\pi}\|I+e^{i\theta}T\|=1+\|T\| \;\;\Leftrightarrow \;\;\rho(T)=\|T\| \] holds for every \(T\in M\), where \(\rho(T)\) is the spectral radius of \(T\). The numerical radius of an operator \(T\) on \(X\) is defined as \[ r(T):=\sup\{|f(Tx)|:x\in X, \, f\in X^*, \,f(x)=1=\|x\|=\|f\|\}. \] \(T\) is said to attain its numerical radius if there are \(x\in X\) and \(f\in X^*\) with \(f(x)=1=\|x\|=\|f\|\) such that \(|f(Tx)|=r(T)\). Denote by \(\mathrm{NRA}(X)\) the set of all operators on \(X\) which attain their numerical radius. The main result of the paper is the following: \(X\) is alternatively convex or smooth if and only if \(X\) has the anti-alternative Daugavet property with respect to \(\mathrm{NRA}(X)\) if and only if \(X\) has the anti-alternative Daugavet property with respect to the class of all rank-one operators in \(\mathrm{NRA}(X)\). A similar characterization for the anti-alternative Daugavet property with respect to the class of compact operators is also proved.