Multiplicative trace and spectrum preservers on stochastic matrices

We characterize maps $\phi_i: \mathcal{S} \to \mathcal{S}$, $i=1, \ldots, m$ and $m\ge 1$, that have the multiplicative spectrum or trace preserving property: \begin{eqnarray*} \textrm{spec} (\phi_1(A_1)\cdots \phi_m(A_m)) &=& \textrm{spec} (A_1\cdots A_m),\quad\text{or}\quad \textrm{tr} (\phi_1(A_1)\cdots \phi_m(A_m)) &=& \textrm{tr} (A_1\cdots A_m), \end{eqnarray*} where $\mathcal{S}$ is the set of $n\times n$ 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 $\mathcal{S}$ are linked to the corresponding preservers on the space of doubly stochastic matrices. Moreover, when $m\ge 3$, multiplicative trace preservers always coincide with multiplicative spectrum preservers.