RANGE OF IDEMPOTENT ADJOINTABLE OPERATORS ON HILBERT C^*-MODULES

RANGE OF IDEMPOTENT ADJOINTABLE OPERATORS ON HILBERT C MODULES

Let T and S be idempotent adjointable operators on the Hilbert C -module H over a C -algebra A. We establish that if there exist constants 1; 2 0 such that for all x 2 R(T) and y 2 R(S) jx + yj 1jxj and jx + yj 2jyj; then R(T) R(S) = f0g and R(T) + R(S) is orthogonality comple- mented submodule of H. We also show that if 1; 2 are idempotents in L(E) such that R( 1) R( 2) = f0g and R( 1) + R( 2) is an or- thogonally complemented submodule of E, Then R( 1 + 2) is closed in E if and only if R( 1 ???? 2) is closed in E. Acknowledgment. This is a joint work with Professors W. Luo, M.S. Moslehian, Q. Xu, and H. Zhang.