On factorization of the shift semigroup

Let $\E$ be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup $\S^\E=(S_t^\E)_{t\ge 0}$ on $L^2(\R_+,\E)$ into the product of $n$ commuting contractive semigroups, i.e., characterizes all $n$-tuples of commuting semigroups $(\V_1,\V_2,...,\V_n)$ where $\V_i=(V_{i,t})_{t\ge 0}$ for $i=1,2,...,n$ are semigroups of contractions satisfying $V_{i,t}V_{j,t}=V_{j,t}V_{i,t}$ for all $i$ and $j$ and $S_t^\E=V_{1,t}V_{2,t}\cdots V_{n,t}$ for all $t\ge 0.$ The factorizations are characterized by tuples of self-adjoint operators $\underline{A}=(A_1,A_2,...,A_n)$ and tuples of positive contractions $\underline{B}=(B_1,B_2,...,B_n)$ on $\E$ 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 $$P:=\{f:\D\to \C \text{ is analytic}, \Re{f}>0 \text{ and }f(0)=1 \}.$$