Compatibility and Schur complements of operators on Hilbert C*-module

Let E be a Hilbert C*-module, and S be an orthogonally complemented closed submodule of E. The authors generalize the definitions of S-complementability and S-compatibility for general (adjointable) operators from Hilbert space to Hilbert C*-module, and discuss the relationship between each other. Several equivalent statements about S-complementability and S-compatibility, and several representations of Schur complements of S-complementable operators (especially, of S-compatible operators and of positive S-compatible operators) on a Hilbert C*-module are obtained. In addition, the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of S-complementable operators and S*-complementable operators on a Hilbert C*-module.