Self-homotopy equivalences of product spaces

The group of self-homotopy equivalences Aut(X x Y) is represented as a product of two subgroups under the assumption that the self equivalences of X x Y can be diagonalized. Moreover, an analogous result holds for the subgroup Aut(#)(X x Y) of self-equivalences, which induce identity automorphisms on homotopy groups. Other methods for the computation of Aut(X x Y) are studied, especially when the spaces involved have an H- or coH-structure, and several examples are considered, among others, some non-simply connected H-spaces of rank 2.