Use of bundles of locally convex spaces in problems of convergence of semigroups of operators. I

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 $\mathcal{U}\in\Gamma^{x_\infty}(\rho)$ 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 $x\in X$.