Separated Pairs of Submodules in Hilbert $C^{\ast }$-modules

Abstract

We introduce the notion of the separated pair of closed submodules in the setting of Hilbert $C^{\ast }$-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 $\mathcal{H}$ and $\mathcal{K}$ be orthogonally complemented closed submodules of a Hilbert $C^{\ast }$-module $\mathcal{E}$. We establish that $(\mathcal{H},\mathcal{K})$ is a separated pair in $\mathcal{E}$ if and only if there are idempotents ${\hat{I}}_1$ and ${\hat{I}}_2$ such that ${\hat{I}}_1{\hat{I}}_2={\hat{I}}_2{\hat{I}}_1=0$ and $\mathcal{R}({\hat{I}}_1)=\mathcal{H}$ and $\mathcal{R}({\hat{I}}_2)=\mathcal{K}$. We show that $\mathcal{R}({\hat{I}}_1+\hat{I}»{\hat{I}}_2)$ is closed for each $\hat{I}»\in \mathbb{C}$ if and only if $\mathcal{R}({\hat{I}}_1+{\hat{I}}_2)$ is closed. We use the localization of Hilbert $C^{\ast }$-modules to define the angle between closed submodules. We prove that if $({\mathcal{H}}^{\perp },{\mathcal{K}}^{\perp })$ is concordant, then $({\mathcal{H}}^{\perp \perp },{\mathcal{K}}^{\perp \perp })$ is a separated pair if the cosine of this angle is less than one. We also present some surprising examples to illustrate our results.

20 pages