On Fredholm properties of Lu = u′ − A(t)u for paths of sectorial operators
Published online by Cambridge University Press: 12 July 2007
Abstract
We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 135 , Issue 1 , February 2005 , pp. 39 - 60
- Copyright
- Copyright © Royal Society of Edinburgh 2005
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