Dichotomy theorems for evolution equations

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] In the first part of this work, under minimal assumptions, we characterize the Fredholm property and compute the Fredholm index of abstract differential operators -d/dt + A([dot]) acting on spaces of functions f : [double-struck R] [arrow] X. Here A(t) are (in general) unbounded operators on the Banach space X, and our results are formulated in terms of exponential dichotomies on two halflines for the propagator solving the evolution equation u'(t) = A(t)u(t) in the mild sense. In the second part of this work, we prove that the operator G, the closure of the first-order differential operator -d/dt+D(t) on L[superscript 2](R,X), is Fredholm if and only if the not well-posed equation u'(t) = D(t)u(t), t [element of] [double-struck R], has exponential dichotomies on [double-struck R][subscript +] and [double-struck R][subscript -] and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t) = A + B(t), A is the generator of a bi-semigroup, B([interpunct]) is a bounded piecewise strongly continuous operator valued function. Also, we prove some perturbations results and consider various PDE examples of not well-posed problems.