Utilizing the Birkhoff–James orthogonality, we present some characterizations of the norm-parallelism for elements of B(ℋ)B(ℋ) defined on a finite dimensional Hilbert space, elements of a Hilbert C∗C∗-module over the C∗C∗-algebra of compact operators and elements of an arbitrary C∗C∗-algebra. We also consider the characterization of norm parallelism problem for operators on a finite dimensional Hilbert space when the operator norm is replaced by the Schatten pp-norm. Some applications and generalizations are discussed for certain elements of a Hilbert C∗C∗-module.

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