Inner product C∗-modules generalize inner product spaces by allowing inner prod-ucts to take values in an arbitrary C∗-algebra A instead of the C∗-algebra of complex numbers C. The classical Birkhoff–James orthogonality says that if x, y are elements of a complex normed linear space (X,  · ), then x is orthogonal to y in the Birkhoff–James sense, in short x ⊥B y, if x + λy ≥ x (λ ∈ C). As a natural generalization of this notion, the con-cept of strong Birkhoff–James orthogonality, which involves modular structure of a Hilbert C∗-module, states that if x and y are elements of a Hilbert A-module X, x is orthogonal to y in the strong Birkhoff–James sense, in short x ⊥sB y, if x + ya ≥ x (a ∈ A),. In this talk, we present some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert C∗-modules and certain elements of B(H). We also discuss some types of approximate orthogonality.

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