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

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.~Šemrl for the classical Birkhoff–James orthogonality of Hilbert space operators. Some other related results are also discussed.