Characterizations of $w_{\rho}$-Birkhoff--James orthogonality and $w_{\rho}$-parallelism

TL;DR

This paper characterizes Birkhoff--James orthogonality and parallelism in Hilbert space operators using the operator radius norm $w_{\rho}$, extending classical results and providing new insights into operator geometry.

Contribution

It provides complete characterizations of $w_{\rho}$-based orthogonality and parallelism, generalizing known results for classical Birkhoff--James orthogonality.

Findings

Complete characterization of $w_{\rho}$-orthogonality.

Complete characterization of $w_{\rho}$-parallelism.

Extension of Bhatia and Šemrl's classical orthogonality result.

Abstract

We study the concepts of Birkhoff--James orthogonality and parallelism in Hilbert space operators, induced by the operator radius norm $w_{\rho}(\cdot)$. In particular, we completely characterize Birkhoff--James orthogonality and parallelism with respect to $w_{\rho}(\cdot)$. As an application of the results presented, we obtain a well-known characterization due to R.~Bhatia and P.~\v{S}emrl for the classical Birkhoff--James orthogonality of Hilbert space operators. Some other related results are also discussed.