Selfadjoint operators on real or complex Banach spaces
type: Item
instance of: scholarly article
title: Selfadjoint operators on real or complex Banach spaces
zbMATH Open document ID: 1443.47021
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2020-04-07T00:00:00Z
review text: 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.
reviewed by: Daniele Puglisi
Mathematics Subject Classification ID: 47B01
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 7187801
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: smooth space
zbMATH Keywords: duality mapping
zbMATH Keywords: point spectrum
author: Francisco García-Pacheco
MaRDI profile type: Publication
full work available at URL: j.na.2019.111696
OpenAlex ID: W2989751538
cites work: 𝖫^{2}-summand vectors in Banach spaces
cites work: Q5608364
cites work: Hermitian operators on Lipschitz function spaces
cites work: Supporting vectors of continuous linear operators
cites work: Geometry of Banach spaces. Selected topics
cites work: Q4821622
cites work: Selectors of the duality mapping
cites work: Isometries of Minkowski geometries
cites work: Spear operators between Banach spaces
cites work: Some orthogonality equation with two functions
cites work: Semi-Inner-Product Spaces
cites work: The theory of self-adjoint operators in Banach spaces with a hermitian form
cites work: Self-adjoint operators on real Banach spaces
cites work: Fredholm equation in smooth Banach spaces and its applications
DOI: 10.1016/J.NA.2019.111696
description: scientific article; zbMATH DE number 7187801