Use of Bundles of Locally Convex Spaces in Problems of Convergence of Semigroups of Operators. I - Silvestri, Benedetto (Eurasian Mathematical Journal, 2016)
In this work we construct certain general bundles \( M, ρ, X \) and \( B, η, X \) of Hausdorff locally convex spaces associated with a given Banach bundle \( E, π, X \). Then we present conditions ensuring the existence of bounded sections \( U Γ_x_∞(ρ) \) and \( P Γ_x_∞(η) \) both continuous at a point \( x_∞ X \), such that \( U(x) \) is a \( C_0 \)-semigroup of contractions on \( E_x \) and \( P(x) \) is a spectral projector of the infinitesimal generator of the semigroup \( U(x) \), for every \( x X \).