Orthogonality and parallelism of operators on various Banach spaces

TL;DR

This paper explores the properties of orthogonality and parallelism of operators on Banach spaces, focusing on Schatten ideals and convex spaces, with characterizations and applications of norm parallelism.

Contribution

It provides new characterizations of norm parallelism for operators in Schatten ideals and convex spaces, including conditions when operators are parallel to the identity.

Findings

Characterization of orthogonality and support disjointness in Schatten ideals

Conditions for norm parallelism of operators in Banach spaces

Results on norm parallelism of compact operators

Abstract

We present some properties of orthogonality and relate them with support disjoint and norm inequalities in p Schatten ideals. In addition, we investigate the problem of characterization of norm parallelism for bounded linear operators. We consider the characterization of norm parallelism problem in p Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm parallel to the identity operator. Finally, we give some equivalence assertions about the norm parallelism of compact operators. Some applications and generalizations are discussed for certain operators.