Multiplicative trace and spectrum preservers on stochastic matrices

We characterize maps ϕᵢ: S → S, i=1, …, m and m≥ 1, that have the multiplicative spectrum or trace preserving property: {eqnarray*} spec (ϕ_1(A_1)⋯ ϕ_m(A_m)) &=& spec (A_1⋯ A_m), tr (ϕ_1(A_1)⋯ ϕ_m(A_m)) &=& tr (A_1⋯ A_m), {eqnarray*} where S is the set of n× n doubly stochastic, row stochastic, or column stochastic matrices, or the space spanned by one of these sets. Linearity is assumed when $m=1$. We show that every stochastic matrix contains a real doubly stochastic component that carries the spectral information. In consequence, the multiplicative spectrum or trace preservers on these sets S are linked to the corresponding preservers on the space of doubly stochastic matrices. Moreover, when m≥ 3, multiplicative trace preservers always coincide with multiplicative spectrum preservers.