Maps preserving the truncation of products of operators

Let H be a complex Hilbert space with dimH >= 2 and B(H) the algebra of all bounded linear operators on H. For A, B is an element of B(H), A is called a truncation of B if A = PABPA*, where P-A and P-A* denote projections onto the closures of the range of A and A*, respectively. In this paper, we determine the structures of all bijections on B(H) preserving the truncation of products of operators in both directions. Moreover, we also consider all maps preserving the truncation of triple Jordan products of operators in both directions.