In this talk we discuss three concepts of orthogonality in Hilbert C*-modules: the Birkhoff-James orthogonality, the strong Birkhoff-James orthogonality, and the orthogonality with respect to the inner product. We shall consider relations between them and characterize classes of Hilbert C*-modules in which two of them coincide. Also, we shall characterize linear surjective mappings on B(H) which preserve the strong Birkhoff-James orthogonality in both directions.
This is a joint work with Rajna Rajić.