range of idempotent adjointable operators on hilbert c^*-modules

Let t and s be idempotent adjointable operators on thehilbert c-module h over a c-algebra a. we establish that if thereexist 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 idempotentsin 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 closedin 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.