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

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