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

TL;DR

This paper introduces the concept of separated pairs of submodules in Hilbert $C^*$-modules, providing characterizations, conditions for closed sums, and exploring angles between submodules, enriching the theory of submodule interactions.

Contribution

The paper extends the notion of separated pairs to Hilbert $C^*$-modules, characterizes them via idempotents, and introduces a new angle concept using localization, with conditions for closed sums.

Findings

Separated pairs characterized by orthogonal idempotents.

Closedness of sums linked to sum of projections.

Angle between submodules influences separation properties.

Abstract

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.