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We explore Birkhoff-James orthogonality of two elements in a complex Banachspace by using the directional approach. Our investigation illustrates thegeometric distinctions between a smooth point and a non-smooth point in acomplex Banach space. As a concrete outcome of our study, we obtain a new proofof the Bhatia-\v{S}emrl Theorem on orthogonality of linear operators.

Birkhoff-James orthogonality in complex Banach spaces and Bhatia-\v{S}emrl Theorem revisited