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

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.