A spectral theory for evolution operators on Banach spaces has been developed in [14, 15] considering associated -semigroups on vector-valued function spaces. It is then quite natural to substitute the shift on by an arbitrary flow on a topological space and to substitute the evolution operator by a cocycle over . 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.
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R.T. Rau, Hyperbolic Linear Skew-Product Semiflows. Z. Anal. Anwend. 15 (1996), no. 4, pp. 865–880DOI 10.4171/ZAA/734