Selfadjoint operators on real or complex Banach spaces. (March 2020)

• Record Type: Journal Article
• Authors: García-Pacheco, Francisco Javier
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: 46B20 -- 46B03 -- 46B07; Selfadjoint operator -- Smooth space -- Duality mapping -- Point spectrum; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111696
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Selfadjoint operators on Hilbert spaces have been extended to more general scopes such as certain real Banach spaces and complex Banach spaces endowed with a continuous Hermitian bilinear form. Here we propose a definition of selfadjoint operator that works for both real and complex Banach spaces and that naturally extends the classical concept of selfadjointness for operators on Hilbert spaces. We strongly rely on selectors of the duality mapping, obtaining a very natural extension in the case of smooth Banach spaces. We provide nontrivial examples of selfadjoint operators on nonHilbert smooth Banach spaces and on nonHilbert general Banach spaces. Finally, we extend the classical theory on eigenvalues and supporting vectors of selfadjoint positive operators on Hilbert spaces to the scope of reflexive, strictly convex and smooth Banach spaces.