Strongly compatible generators of groups on Fréchet spaces

Abstract

We consider the linear Cauchy problem(1){ut=a(D)u,t∈Ru(0)=u0, where a(D):X→X is a continuous linear operator on a Fréchet space X. By imposing a condition (which is neither stronger nor weaker than the equicontinuity of the powers of a(D)), we present the necessary and sufficient conditions for the generation of a uniformly continuous group on X, which provides the unique solution of (1). In addition, for every pseudodifferential operator a(D) with constant coefficients defined on FLloc2, which is a Fréchet space of distributions, we also provide the necessary and sufficient conditions such that the restriction {eta(D)}t⩾0 is a well defined semigroup on L2 and E′. We conclude that the heat equation solution on FLloc2 for all t∈R extends the standard solution on Hilbert spaces for t⩾0.