Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space

Let and be infinite-dimensional separable Hilbert spaces and the lattice of all closed subspaces oh . We describe the general form of pairs of bijective maps having the property that for every pair we have . Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.