Abstract

We study the operator Lu(t):= u′(t) − A(t) u(t) on Lp (R; X) for sectorial operators A(t), with t ∈ R, on a Banach space X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreni conditions, and have the property of maximal Lp-regularity. We establish optimal regularity on the time interval R showing that L is closed on its minimal domain. We further give conditions for ensuring that L is a semi-Fredholm operator. The Fredholm property is shown to persist under A(t)-bounded perturbations, provided they are compact or have small A(t)-bounds. We apply our results to parabolic systems and to generalized Ornstein–Uhlenbeck operators. 2000 Mathematics Subject Classification 35K20, 35K90, 47A53.

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