Generalized Evolution Semigroups and General Dichotomies

The paper’s Dichotomy Theorem (Theorem 4.4) proves exactly the four-way equivalence and the Green-operator inverse in the μ-time setting using the similarity T = F^{-1} S F and the classical evolution semigroup theory; it also establishes spectral mapping and vertical translation invariance of the generator. The candidate solution reproduces this route: it performs the θ = μ(s) change of variables, shows similarity of semigroups, invokes the classical dichotomy theorem, derives the spectral condition σ(G) ∩ iR = ∅ and invertibility, and reconstructs the inverse via the Green kernel with the μ′ weight. Apart from not explicitly restating the growth assumption ensuring T is a C0-semigroup (condition (29)), the steps and logic coincide with the paper’s proof. See the similarity and spectrum results (32)–(34) and Corollary 3.7, and Theorem 4.4 with formula (37) for the inverse.