Math. Nachr. 212 (2000), 101 – 116 Evolution Semigroups and Product Formulas for Nonautono- mous Cauchy Problems By Gregor Nickel of T¨ ubingen (Received July 22, 1997) (Revised Version June 8, 1998) Abstract. In this paper, we study nonautonomous Cauchy problems (NCP )  ˙ u(t)= A(t)u(t) u(s)= x ∈ X for a family of linear operators (A(t)) t∈I on some Banach space X by means of evolution semigroups. In particular, we characterize “stability” in the so called “hyperbolic case” on the level of evolution semigroups and derive a product formula for the solutions of (NCP ). Moreover, in Section 4 we connect the “hyperbolic” and the “parabolic” case by showing, that integrals  t s A(τ ) dτ always define generators. This yields another product formula. 1. Kato’s stability condition A necessary and sufficient condition for wellposedness of nonautonomous Cauchy problems in terms of the family (A(t)) t∈I is still lacking (see, e. g., [Go1]). However, at least in the hyperbolic case, all results are based on the classical 1970 paper of Kato [Ka2] and his stability condition. In preliminary results (see, e. g., [Ka1]) the oper- ators A(t) were assumed to generate contraction semigroups, thus Kato’s stability condition was automatically satisfied (see below). It was used later, e. g., [DaP-Gr], or [DaP-Si] in combination with more or less complicated regularity conditions to obtain wellposedness of (NCP ). In our paper, we characterize Kato’s stability in the perspective of evolution semi- groups and derive two different approximation formulas, see Sections 3 and 4. We start with Kato’s basic definition of stability for a family of generators (A(t)) t∈I (cf. [Pa], p. 131). Since, for the moment, we do not use evolution semigroups, we consider compact intervals I := [0,T ]. Definition 1.1. (Kato – stability.) A family (A(t),D(A(t))) t∈I of generators of C 0 – semigroups on a Banach space X is called Kato – stable, if there exist constants M ≥ 1 1991 Mathematics Subject Classification. Primary: 47D06; Secondary: 47A55. Keywords and phrases. Evolution semigroups, noautonomous Cauchy problems.