Fredholm differential operators with unbounded coefficients |
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Authors: | Yuri Latushkin Yuri Tomilov |
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Institution: | a Department of Mathematics, University of Missouri-Columbia, Mathematical Science Building, MO 65211, USA b Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland |
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Abstract: | We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg. |
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Keywords: | 47D06 35P05 35F10 58J20 (primary) 58E99 47A53 (secondary) |
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