  As a continuation of the work on linear maps between operator algebras which
preserve certain subsets of operators with finite rank, or corank, here we
consider the problem inbetween, that is, we treat the question of preserving
operators with infinite rank and infinite corank. Since, as it turns out, in
this generality our preservers cannot be written in a nice form what we have
got used to when dealing with linear preserver problems, hence we restrict our
attention to certain important classes of operators like idempotents, or
projections, or partial isometries. We conclude the paper with a result on the
form of linear maps which preserve the left ideals in B(H).
