Norm Inequalities Associated with Two Projections

Suppose that p and q are projections in a unitalC *-algebra U such that parallel to p(1-q) parallel to < 1. It is shown that there exists a unitary u in U which is homotopic to the unit of U, and satisfies pup = pu* p, u( pqp)u * = qpq and parallel to 1 - u parallel to <= root 2 parallel to(qp)(dagger)parallel to / 1+ parallel to (qp)(dagger) parallel to center dot parallel to p(1 - q) parallel to, where (qp)(dagger) denotes the Moore-Penrose inverse of qp. Under the same restriction of parallel to p(1- q) parallel to < 1, it is proved that parallel to p- q parallel to < 1 if and only if there exists a unitary u in U such that pup is normal and q = upu*. An example is constructed to show that there exist certain Hilbert space H and projections p and q on H such that parallel to p - q parallel to = 1 and q = upu* for some unitary operator u on H.