Selfadjoint operators on real or complex Banach spaces
A map \(J_0:X\to X^*\) is said to be a selector if \(\|J_0(x)\|=\|x\|\) and \(J_0(x)(x)=\|x\|^2\) for all \(x\in X\). If \(J_0(x)\) is the only element in \(X^*\) satisfying \(\|J_0(x)\|=\|x\|\) and \(J_0(x)(x)=\|x\|^2\), then \(J_0\) is called dual map. It is well known that a smooth Banach space is reflexive if and only if \(J_0\) is surjective, and is strictly convex if and only if \(J_0\) is injective. In [\textit{F. J. García-Pacheco} et al., Math. Proc. R. Ir. Acad. 116A, No. 2, 105--111 (2016; Zbl 1395.46010)] it was proved that a dual map \(J_0\) is additive if and only if \(X\) is a Hilbert space. An operator \(T:X\to X\) is then called \(J_0\)-selfadjoint if \(T^*\circ J_0 = J_0\circ T\). Note that, if \(X\) is a smooth Banach space, then there exists only one selector. In this interesting paper, the author provides examples of selfadjoint operators on general Banach spaces and extends the classical theory of eigenvalues and supporting vectors of selfadjoint positive operators to the setting of reflexive, strictly convex and smooth Banach spaces.
scientific article; zbMATH DE number 7187801