# Halmos' two projections theorem for Hilbert $C^*$-module operators and the Friedrichs

## TL;DR

This paper extends Halmos' two projections theorem to Hilbert $C^*$-modules using the concept of harmonious projections, providing new characterizations of submodules and a norm equation related to the Friedrichs angle.

## Contribution

It introduces harmonious projections in Hilbert $C^*$-modules and generalizes Halmos' theorem to this setting, with applications to Friedrichs angle characterization.

## Key findings

- Extended Halmos' theorem to harmonious projections on Hilbert $C^*$-modules.
- Provided new characterizations of closed submodules and projections.
- Proved a norm equation related to Friedrichs angle in this framework.

## Abstract

Halmos' two projections theorem for Hilbert space operators is one of the fundamental results in operator theory. In this paper, we introduce the term of two harmonious projections in the context of adjointable operators on Hilbert $C^*$-modules, extend Halmos' two projections theorem to the case of two harmonious projections. We also give some new characterizations of the closed submodules and their associated projections. As an application, a norm equation associated to a characterization of the Friedrichs angle is proved to be true in the framework of Hilbert $C^*$-modules.