Orthogonalities, linear operators, and characterization of inner product spaces
The paper under review explores the reversion of Birkhoff orthogonality under a bounded linear operator between real Banach spaces. Among other things, the authors obtain an interesting characterization of inner product spaces of dimension greater than or equal to \(3\), in terms of the existence of a bounded linear operator reversing Birkhoff orthogonality. The paper also relates isosceles orthogonality and Birkhoff orthogonality via a bounded linear operator, and the authors obtain some further characterizations of inner product spaces of dimension greater than or equal to \(3\). Furthermore, the connection between superior (inferior) semi-inner-products and Birkhoff orthogonality is explored to obtain a sufficient condition for a two-dimensional smooth Banach space to be a Radon plane. In the view of this reviewer, this is an interesting mathematical work that appeals to anyone interested in the study of various notions of orthogonality, or, in a broader sense, understanding the geometry of Banach spaces through orthogonality and bounded linear operators.
characterization of inner product spaces
scientific article; zbMATH DE number 6798721