Orthogonality of bounded linear operators on complex Banach spaces

TL;DR

This paper characterizes Birkhoff-James orthogonality of bounded linear operators on complex Banach spaces, extending real case results and exploring orthogonality and symmetry properties in complex settings.

Contribution

It provides a complete characterization of Birkhoff-James orthogonality for operators on complex Banach spaces, introducing new definitions and distinguishing complex from real cases.

Findings

Complete characterization of orthogonality in complex Banach spaces

Equivalent conditions for orthogonality of compact operators

Zero operator is the only left symmetric operator on complex 2D l_p spaces

Abstract

We study Birkhoff-James orthogonality of bounded linear operators on complex Banach spaces and obtain a complete characterization of the same. By means of introducing new definitions, we illustrate that it is possible in the complex case, to develop a study of orthogonality of bounded (compact) linear operators, analogous to the real case. Furthermore, earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case, can be obtained as simple corollaries to our present study. In fact, we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case, in order to distinguish the complex case from the real case. We also study the left symmetric linear operators on complex two-dimensional $l_p$ spaces. We prove that $ T $ is a left symmetric linear operator on $ \ell_p^2{(\mathbb{C})}$ if and only if $ T $ is the zero operator.