Semigroup Forum Vol. 45 (1992) 26-37 9 1992 Springer-Verlag New York Inc. RESEARCH ARTICLE Interpolation of Semigroups and Integrated Semigroups* Wolfgang Arendt, Frank Neubrander** and Ulf Schlotterbeck Communicated by R. Nagel O. Introduction In 1971, S. G. Krein, G. I. Laptev and G. A. Cvetkova [K-L-C] proved that any linear (unbounded) operator A on a Banach space E such that the resolvent set contains a half-line (w, cx~), generates a Co-semigroup on a certain (maximal) subspace Z of E (see also [Ka], [M-O-O], [Ne3]). This is a very general result, expressing the popular belief that linear dynamic systems are good-natured in some sense. The fact that no information on the size of Z is available in general, suggests a comparison of these operators according to the actual size of Z; with Z = E as the best possible case and with Z = {0} as the worst, but still possible one (see [Be]). On the good side of this scale, the cases D(A k) C Z (k e N) are of particular interest. In this paper we want to show that this situation is actually characteristic for generators of k-times integrated semigroups which were intro- duced in order to treat the abstract Cauchy problem u'(t) = Au(t), u(O) = x in cases where the resolvent of the operator A exists and has polynomial growth in a right half-plane. Such operators frequently occur if one studies differential oper- ators in LP(I~n), (1 < p < cx~), systems of linear partial differential equations or higher order Cauchy problems, to mention just a few instances (see [Arl], JAr2], [A-K], [dL], [K-HI, [Nel], [Ne2], [Ne3], IN-S], [Oh], [So], IT-M1], [T-M2], [Wh]). It turned out that integrated semigroups share many properties with Co-semi- groups. The purpose of this paper is to show that every Co-semigroup on a Banach space F induces integrated semigroups on continuously embedded sub- spaces and that, conversely, every integrated semigroup on F can be "sand- wiched" by Co-semigroups on extrapolation- and interpolation spaces. In order to make this more precise we introduce some notation. An op- erator A on a Banach space E is the generator of k-times integrated semigroup (where k E N0 ) if there exist w > 0 and S(.): [0, ~) --*/:(E) strongly continuous such that (w, ~) is contained in the resolvent set of A, and ~0 ~ (#I - A)-lx = #k e-~,tS(t)xdt (x E E,# > w). The function S(.) is called k-times integrated semigroup. If there exists M > 0 such that IS(~)l _< MeWt for all t > 0, then S(.) is called exponentially bounded of type w. Thus, if S(.) is of type w, it is also of type w ~ for all w ~ > w. * A preliminary version of this paper appeared in: Semesterbericht Funktionalanaly- sis 15, Tfibingen (1988/89). ** Research supported in part by NSF Grant DMS-8601983 and by DFG (Deutsche Forschungsgemeinschaft )