Norm-parallelism in the geometry of Hilbert $C^{\ast }$-modules

Abstract

Utilizing the Birkhoff--James orthogonality, we present some characterizations of the norm-parallelism for elements of $\mathbb{B}(\mathcal{H})$ defined on a finite dimensional Hilbert space, elements of a Hilbert $C^{\ast }$-module over the $C^{\ast }$-algebra of compact operators and elements of an arbitrary $C^{\ast }$-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 $p$-norm. Some applications and generalizations are discussed for certain elements of a Hilbert $C^{\ast }$-module.

17 pages, to appear in Indag. Math

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