Mappings preserving different types of orthogonality on B(H)

Let B(H) be the algebra of all bounded linear operators acting on a complex Hilbert space H. Linear preserver problems on B(H) concern the characterizations of all linear mappings on B(H) that leave certain functions, subsets, relations, etc. invariant We characterize the class of surjective (conjugate) linear mappings on B(H) that preserve the (strong) Birkhoff-James orthogonality in both directions. We also describe mappings on B(H) preserving the orthogonality of ranges of Hilbert space operators in terms of the Roberts orthogonality.