Characterizations of $w_ρ$-Birkhoff--James orthogonality and $w_ρ$-parallelism

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.