Let \(\mathcal {E}\) be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup \({\mathcal {S}}^\mathcal {E}=(S_t^\mathcal {E})_{t\ge 0}\) on \(L^2(\mathbb {R}_+,\mathcal {E})\) into the product of n commuting contractive semigroups, i.e., characterizes all n-tuples of commuting semigroups \(({\mathcal {V}}_1,{\mathcal {V}}_2,\ldots ,{\mathcal {V}}_n)\) where \({\mathcal {V}}_i=(V_{i,t})_{t\ge 0}\) for \(i=1,2,\ldots ,n\) are semigroups of contractions satisfying \(V_{i,t}V_{j,t}=V_{j,t}V_{i,t}\) for all i and j and \(S_t^\mathcal {E}=V_{1,t}V_{2,t}\cdots V_{n,t}\) for all \(t\ge 0.\) The factorizations are characterized by tuples of self-adjoint operators \(\underline{A}=(A_1,A_2,\ldots ,A_n)\) and tuples of positive contractions \(\underline{B}=(B_1,B_2,\ldots ,B_n)\) on \(\mathcal {E}\) satisfying certain conditions which are stated in Theorem 4.10. One of the tools of our analysis is a convexity argument using the extreme points of the Herglotz class of functions \(\begin{aligned} P:=\{f:{{\mathbb {D}}}\rightarrow \mathbb {C}\text { is analytic}, \text {Re}\{f\}>0 \text { and }f(0)=1 \}. \end{aligned}\)
