On factorization of the shift semigroup

Let $\text{E}$ be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup ${\mathrm{\S }}^{\text{E}}=(S_t^{\text{E}}{)}_{t\ge 0}$ on $L^2({\mathbb{R}}_{+},\text{E})$ into the product of $n$ commuting contractive semigroups, i.e., characterizes all $n$-tuples of commuting semigroups $({\text{V}}_1,{\text{V}}_2,\mathrm{.}\mathrm{.}\mathrm{.},{\text{V}}_n)$ where ${\text{V}}_i=(V_{i,t}{)}_{t\ge 0}$ for $i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},n$ are semigroups of contractions satisfying $V_{i,t}{V}_{j,t}=V_{j,t}{V}_{i,t}$ for all $i$ and $j$ and $S_t^{\text{E}}=V_{1,t}{V}_{2,t}\cdots V_{n,t}$ for all $t\ge 0.$ The factorizations are characterized by tuples of self-adjoint operators $\underset{‾}{A}=(A_1,A_2,\mathrm{.}\mathrm{.}\mathrm{.},A_n)$ and tuples of positive contractions $\underset{‾}{B}=(B_1,B_2,\mathrm{.}\mathrm{.}\mathrm{.},B_n)$ on $\text{E}$ satisfying certain conditions which are stated in \cref{thm:psi12}. One of the tools of our analysis is a convexity argument using the extreme points of the {\em Herglotz } class of functions $$P:=\{f:\text{D}\to \text{C}\text{ is analytic},\mathrm{\Re }f>0\text{ and }f(0)=1\}\mathrm{.}$$