We study the concepts of Birkhoff--James orthogonality and parallelism in Hilbert space operators, induced by the operator radius norm wρ(·). In particular, we completely characterize Birkhoff--James orthogonality and parallelism with respect to wρ(·). As an application of the results presented, we obtain a well-known characterization due to R.~Bhatia and P.~{S}emrl for the classical Birkhoff--James orthogonality of Hilbert space operators. Some other related results are also discussed.
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Kıttaneh et al. (2024) studied this question.
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