On the generation of groups of bounded linear operators on Fr\'{e}chet spaces

In this paper we present a general method for generation of uniformly continuous groups on abstract Fr\'{e}chet spaces (without appealing to spectral theory) and apply it to a such space of distributions, namely ${\mathscr F}L^{2}_{loc}(\mathbb{R}^{n})$, 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 ${\mathscr E}'$ are left invariant by this group; and we conclude that the solution of the heat equation on ${\mathscr F}L^{2}_{loc}(\mathbb{R}^{n})$ for all $t \in \mathbb{R}$ extends the standard solution on Hilbert spaces for $t \geqslant 0$.