Yuri Latushkin
Department of Mathematics, University of Missouri
Fredholm Differential Operators and Dichotomy
Abstract
We prove that a first order linear differential operator G=-d/dt+A(t)
with unbounded operator coefficients A(t) is Fredholm on spaces of
vector functions on real line if and only if the corresponding differential
equation has exponential dichotomies on both positive and negative semi-lines,
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.
This provides a complete infinite dimensional generalization of well-known
finite dimensional results by K. Palmer, and by A. Ben-Artzi and I. Gohberg,
used in the study of stability of traveling waves for applied partial differential
equations. It also gives new connections of dichotomy to infinite dimensional
Morse theory, and the celebrated Atiyah-Patodi-Singer Index Theorem.
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