Use of bundles of locally convex spaces in problems of convergence of semigroups of operators defined on different Banach spaces. Applications to spectral stability problems

TL;DR

This paper develops a framework using bundles of locally convex spaces to analyze the convergence of operator semigroups across different Banach spaces, with applications to spectral stability.

Contribution

It introduces new bundles of locally convex spaces associated with Banach bundles and provides conditions for the existence of continuous sections representing semigroups and spectral projectors.

Findings

Constructed general bundles of locally convex spaces from Banach bundles.

Established conditions for the existence of continuous sections of semigroups and spectral projectors.

Applied the framework to spectral stability problems of operator semigroups.

Abstract

In this work we construct certain general bundles $<\mathfrak{M},\rho,X>$ and $<\mathfrak{B},\eta,X>$ of Hausdorff locally convex spaces associated with a given Banach bundle $<\mathfrak{E},\pi,X>$. Then we present conditions ensuring the existence of bounded sections $\mathcal{U}\in \Gamma^{x_{\infty}}(\rho)$ and $\mathcal{P}\in \Gamma^{x_{\infty}}(\eta)$ both continuous at a point $x_{\infty}\in X$, such that $\mathcal{U}(x)$ is a $C_{0}-$semigroup of contractions on $\mathfrak{E}_{x}$ and $\mathcal{P}(x)$ is a spectral projector of the infinitesimal generator of the semigroup $\mathcal{U}(x)$, for every $x\in X$.