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

Abstract

Let $(\mathcal{H}, [\cdot, \cdot ])$ be a Hilbert space and $K(\mathcal{H})$ be the $C^*$-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 $T \| S$ if and only if there exists a sequence of basic vectors $\{x_n\}^{\xi_n}$ in $\mathcal{E}$ such that $\lim_n [\langle Tx_n, Sx_n \rangle \xi_n, \xi_n ] = \lambda\| T\| \| S\|$ for some $\lambda \in \mathbb{T}$. In addition, we give some equivalence assertions about the norm-parallelism of "compact" operators on a Hilbert $C^*$-module.