Fredholm differential operators with unbounded coefficients
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- 作者：Latushkin ; Yuri ; Tomilov ; Yuri
- 关键词：47D06 ; 35P05 ; 35F10 ; 58J20 (primary) ; 58E99 ; 47A53 (secondary) ; Exponential dichotomy ; Fredholm operators ; Fredholm index ; Differential and difference operators ; Evolution semigroups ; Pairs of subspaces ; Travelling waves ; Spectral flow
- 刊名：Journal of Differential Equations
- 出版年：2005
- 出版时间：January 15, 2005
- 年：2005
- 卷：208
- 期：2
- 页码：388-429
- 全文大小：520 K
文摘
We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on R with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both R+ and R− 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 A(t),t
R, the operator G is a closure of the operator −+A(t). Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.
R, the operator G is a closure of the operator −+A(t). Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.