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

الأصل · EN

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 FL²loc(Rⁿ), so that the linear evolution problem equation* {arrayl uₜ = a(D)u, t ∈ R u(0) = u₀ array. 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 L² and E' are left invariant by this group; and we conclude that the solution of the heat equation on FL²loc(Rⁿ) for all t ∈ R extends the standard solution on Hilbert spaces for t 0.