Math. Nachr. zyxwvutsr 169 (1994) 207-218 Spectral Mapping Theorems for C,-groups Satisfying Non-quasianalytic Growth Conditions l) By RAINER NACEL and SENZHONG HUANC~) of Tiibingen (Received February 22, 1993) (Revised Version October 29, 1993) 1. Introduction Let zyxwvut F zyxw = zyxwvut (T(t))tzo be a C,-semigroup of bounded linear operators on a Banach space zy E and denote its generator by A. For many applications it is a fundamental problem to decide whether the spectrum of each operator T(t) can be obtained from the spectrum o(A) of A. In particular, one hopes that the Weak Spectral Mapping Theorem holds, i.e., (WSMT) a(T(t)) = exp (t . a(A)) for all t zyxwv 2 0 Already from HILLE-PHILLIPS [S, p. 6651 (see also [15], [16] and [12, A-111, Ex. 1.41) one knows that even for C,-groups the WSMT may fail. But in [12, A-111 and B-1111 it is shown that spectral mapping theorems hold if the semigroup satisfies certain regularity or positivity conditions (see also [l]). In addition, G. GREINER proved for Co-groups (see [12, A-111, Theorem 7.41 that WSMT holds if the norm 1) T(t)/) grows at most polynomially, i.e., there exists some polynomial p such that IIT(t)II p(t) for all t E zyxwvut R. In this paper we consider non-quasianalytic weights instead of polynomials. These are measurable locally bounded functions zyxwv w : R + R such that (1) 1 I o(t) and o(t + s) w(t) w(s) for t, s E R and . m -m Typical examples of non-quasianalytic weight are functions w(t) := elt1" for t E R and 0 5 CI < 1. In the following we show that WSMT holds for groups F = (T(t)),,R satisfying The paper is part of a research project supported by the Deutsche Forschungsgemeinschaft (DFG). Support by the Deutscher Akademischer Autauschdienst (DAAD) is gratefully acknowledged.