Hyperbolic Linear Skew-Product Semiflows

Abstract

A spectral theory for evolution operators on Banach spaces has been developed in [14, 15] considering associated $C_0$-semigroups on vector-valued function spaces. It is then quite natural to substitute the shift on $\mathbb{R}$ by an arbitrary flow $\sigma$ on a topological space $X$ and to substitute the evolution operator by a cocycle $\Phi$ over $\sigma$. 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.