Fredholm differential operators with unbounded coefficients

Abstract

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 þ and R À 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 0 ðtÞ ¼ AðtÞuðtÞ with, generally, unbounded operators AðtÞ; tAR; the operator G is a closure of the operator À d dt þ AðtÞ: Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg. r

Stability of Traveling Waves for a Class of Reaction-Diffusion Systems That Arise in Chemical Reaction Models

“…However, the proof in [5] also works for any L p (R), 1 ≤ p < ∞. The reason is that Palmer's theorem (see, e.g., [15]), which relates the Fredholm properties of first-order linear differential operators of the form U → ∂ ξ U − A(ξ)U (ξ) to the spectra of the constant-coefficient operators U → ∂ ξ U − A(±∞)U (ξ), is true not only in the spaces used in [5] but also in any L p (R), 1 ≤ p < ∞; see [10]. Theorem 1.1 implies in particular that if the traveling wave Y * is spectrally stable in any of the spaces L 1 (R), L 2 (R), H 1 (R), or BU C(R), then it is linearly exponentially stable in that space.…”

Robustness of Fredholm properties of parabolic evolution equations under boundary perturbations

“…These facts lead to a Fredholm alternative for (1.1) stated in Theorem 4.5. In Section 4 we use the Fredholm theory developed in [13], [14] and [16]. For further information on this subject we refer to the references therein and [6], [7].…”

Evans Functions, Jost Functions, and Fredholm Determinants

“…Indeed, by a well-known Dichotomy Theorem (sometimes called Palmer's theorem), the operator G is Fredholm on L 2 (R) d if and only if (1.1) has exponential dichotomies Q − on R − and Q + on R + ; see [3], [48], [49], [56] or [57, Theorem 3.2], and also [41], [53] for more recent versions of the dichotomy theorem. ✸ Next, we will discuss the Bohl exponents and exponential splittings for the perturbed equation (1.2).…”