Separated Pairs of Submodules in Hilbert $C^*$-modules

We introduce the notion of the separated pair of closed submodules in the setting of Hilbert $C^*$-modules. We demonstrate that even in the case of Hilbert spaces this concept has several nice characterizations enriching the theory of separated pairs of subspaces in Hilbert spaces. Let $\mathscr H$ and $\mathscr K$ be orthogonally complemented closed submodules of a Hilbert $C^*$-module $\mathscr E$. We establish that $(\mathscr H,\mathscr K)$ is a separated pair in $\mathscr{E}$ if and only if there are idempotents $\Pi_1$ and $\Pi_2$ such that $\Pi_1\Pi_2=\Pi_2\Pi_1=0$ and $\mathscr R(\Pi_1)=\mathscr H$ and $\mathscr R(\Pi_2)=\mathscr K$. We show that $\mathscr R(\Pi_1+\lambda\Pi_2)$ is closed for each $\lambda\in \mathbb{C}$ if and only if $\mathscr R(\Pi_1+\Pi_2)$ is closed. We use the localization of Hilbert $C^*$-modules to define the angle between closed submodules. We prove that if $(\mathscr H^\perp,\mathscr K^\perp)$ is concordant, then $(\mathscr H^{\perp\perp},\mathscr K^{\perp\perp})$ is a separated pair if the cosine of this angle is less than one. We also present some surprising examples to illustrate our results.