Compatibility and Schur complements of operators on Hilbert \(C^*\)-module (Q2430340)

The authors generalize the definitions of \(S\)-complementability and \(S\)-compatibility for operators from Hilbert spaces to Hilbert \(C^*\)-modules as follows. Let \(E\) be a Hilbert \(C^*\)-module and \(S\) be an orthogonally complemented closed submodule of \(E\). An adjointable operator \(T\) on \(E\) is called \(S\)-complementable if there exist adjointable operators \(M_l, M_r\) on \(E\) such that \(P_SM_r = M_r, M_lP_S = M_l\), \(P_STM_r = P_ST\), \(M_lTP_S = TP_S\), where \(P_S\) is the projection onto \(S\). The pair \((A, S)\), where \(A\) is an adjointable operator on \(E\), is called \(S\)-compatible if there exists an idempotent \(Q\) with range \(S\) such that \(\langle Qx, y\rangle_A= \langle x, Qy\rangle_A\), where \(\langle x, y\rangle_A:=\langle Ax, y\rangle\). The authors present several equivalent statements about \(S\)-complementability and \(S\)-compatibility, and several representations of Schur complements of \(S\)-complementable operators on a Hilbert \(C^*\)-module. They also investigate the quotient property for Schur complements of \(S\)-complementable operators on a Hilbert \(C^*\)-module.