On some geometric properties of operator spaces

TL;DR

This paper explores geometric properties like parallelism, orthogonality, and semi-rotundity in operator spaces, providing characterizations and introducing new concepts such as semi-rotund points.

Contribution

It offers complete characterizations of operator parallelism, investigates Birkhoff-James orthogonality, and introduces semi-rotundity in the context of operator spaces.

Findings

Characterization of parallelism for compact operators on reflexive spaces

Analysis of approximate parallelism in Hilbert space operators

Introduction and study of semi-rotund operators and spaces

Abstract

In this paper we study some geometric properties like parallelism, orthogonality and semi-rotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces $\mathbb{X} $ and $\mathbb{Y}$, assuming $\mathbb{X}$ to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces $\mathbb{X} $ and $\mathbb{Y}.$ We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff-James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semi-rotund points (semi-rotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semi-rotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semi-rotund space which is not strictly convex, if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.