Parallelism in Hilbert $K(\mathcal{H})$-modules

Abstract

Let $(\mathcal{H},[\cdot ,\cdot ])$ be a Hilbert space and $K(\mathcal{H})$ be the $C^{\ast }$-algebra of compact operators on $\mathcal{H}$. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert $K(\mathcal{H})$-module $\mathcal{E}$ by employing the minimal projections on $\mathcal{H}$. Let $T,S\in \mathcal{L}(\mathcal{E})$. We show that $T\Vert S$ if and only if there exists a sequence of basic vectors $\{x_n{\}}^{{\xi }_n}$ in $\mathcal{E}$ such that ${\lim }_n[⟨T{x}_n,S{x}_n⟩{\xi }_n,{\xi }_n]=\lambda \Vert T\Vert \Vert S\Vert$ for some $\lambda \in \mathbb{T}$. In addition, we give some equivalence assertions about the norm-parallelism of "compact" operators on a Hilbert $C^{\ast }$-module.