Hyperbolic Linear Skew-Product Semiflows

A spectral theory for evolution operators on Banach spaces has been developed in [14, 15] considering associated Co-semigroups on vector-valued function spaces. It is then quite natural to substitute the shift on R by an arbitrary flow a on a topological space X and to substitute the evolution operator by a cocycle 4 over a. This task was performed by Latushkin and Stepin (cf. [8, 9]) for hyperbolic linear skew-product flows assuming some norm continuity of this flow. In general only strong continuity can be obtained (cf. Sacker and Sell [18) and Example 2 below). Following a suggestion by Hale [7: p. 601 we consider strongly continuous linear skew-product flows in Banach spaces and characterize hyperbolicity through a spectral condition.