Preservers of unitary group or equivalence by unitaries

Abstract

We give the complete description of bijective real-linear transformations on B(H), the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H, where every unitary U∈B(H) is mapped to a unitary ϕ(U). In turn, bijective real-linear maps on B(H) preserving equivalence by unitaries will be determined.