On the generation of groups of bounded linear operators on Fréchet spaces

Abstract

In this paper we present a general method for generation of uniformly continuous groups on abstract Fréchet spaces (without appealing to spectral theory) and apply it to a such space of distributions, namely $\mathcal{F}{L}_{loc}^2({\mathbb{R}}^n)$, so that the linear evolution problem \begin{equation*} \left\{\begin{array}{l} u_{t} = a(D)u, t \in \mathbb{R} \\ u(0) = u_0 \end{array} \right. \end{equation*}always has a unique solution in such a space, for every pseudodifferential operator $a(D)$ with constant coefficients. We also provide necessary and sufficient conditions so that the spaces $L^2$ and ${\mathcal{E}}^{\prime }$ are left invariant by this group; and we conclude that the solution of the heat equation on $\mathcal{F}{L}_{loc}^2({\mathbb{R}}^n)$ for all $t\in \mathbb{R}$ extends the standard solution on Hilbert spaces for $t⩾0$.

References in corpus (1)

• Strongly continuous and locally equi-continuous semigroups on locally convex spaces