Some linear preserver problems on B(H) concerning rank and corank

As a continuation of the work on linear maps between operator algebras whic h preserve certain subsets of operators with finite rank, or finite corank, here we consider the problem in between, 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 fo rm what we have got used to when dealing with linear preserver problems, he nce we restrict our attention to certain important classes of operators lik e 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). (C) 1999 Elsevier Science Inc. All rights reserved.