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Zeitschrift für Analysis und ihre Anwendungen

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Volume 15, Issue 4, 1996, pp. 865–880
DOI: 10.4171/ZAA/734

Published online: 1996-12-31

Hyperbolic Linear Skew-Product Semiflows

R.T. Rau[1]

(1) Universität Tübingen, Germany

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.

Keywords: $C_0$-semi groups, cocycles, exponential dichotomy, skew-product flows

Rau R.T.: Hyperbolic Linear Skew-Product Semiflows. Z. Anal. Anwend. 15 (1996), 865-880. doi: 10.4171/ZAA/734