Use of Bundles of Locally Convex Spaces in Problems of Convergence of Semigroups of Operators. I

Abstract

In this work we construct certain general bundles \( \langle \mathfrak{M}, \rho, X \rangle \) and \( \langle \mathfrak{B}, \eta, X \rangle \) of Hausdorff locally convex spaces associated with a given Banach bundle \( \langle \mathfrak{E}, \pi, X \rangle \). Then we present conditions ensuring the existence of bounded sections \( U \in \Gamma_{x_\infty}(\rho) \) and \( P \in \Gamma_{x_\infty}(\eta) \) both continuous at a point \( x_\infty \in X \), such that \( U(x) \) is a \( C_0 \)-semigroup of contractions on \( \mathfrak{E}_x \) and \( P(x) \) is a spectral projector of the infinitesimal generator of the semigroup \( U(x) \), for every \( x \in X \).