On the orthogonal complementarity of closed submodules of Hilbert $C^*$-modules

It is known that a closed submodule of a Hilbert $C^*$-module may fail to be orthogonally complemented. Due to this weakness, some new phenomena may happen compared with that in Hilbert spaces. In the framework of adjointable operators on Hilbert $C^*$-modules, I will report some recent progress on the generalized Douglas theorem, the generalized parallel sum, the generalized Halmos' two projections theorem. Some applications will also be presented.