Strongly compatible generators of groups on Frechet spaces

We consider the linear Cauchy problem {u(t) = a(D)u, t is an element of R u(0) = u(0), (1) where a(D) : X -> X is a continuous linear operator on a Frechet 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 F L-loc(2), which is a Frechet space of distributions, we also provide the necessary and sufficient conditions such that the restriction {e(t) (a(D))}(t >= 0) is a well defined semigroup on L-2 and E'. We conclude that the heat equation solution on F L-loc(2) for all t is an element of R extends the standard solution on Hilbert spaces for t >= 0. (C) 2019 Elsevier Inc. All rights reserved.