Norm-parallelism in the geometry of Hilbert $C^*$-modules

TL;DR

This paper characterizes norm-parallelism in Hilbert $C^*$-modules and operators using Birkhoff--James orthogonality, extending to Schatten $p$-norms and providing applications and generalizations.

Contribution

It offers new characterizations of norm-parallelism in Hilbert $C^*$-modules and operators, including Schatten $p$-norms, with applications and broader generalizations.

Findings

Characterizations of norm-parallelism in finite-dimensional Hilbert spaces.

Extension of norm-parallelism to Schatten $p$-norms.

Applications to elements of Hilbert $C^*$-modules.

Abstract

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