Robustness of exponential dichotomies in infinite-dimensional dynamical systems

In this paper we examine the issue of the robustness, or stability, of an exponential dichotomy, or an exponential trichotomy, in a dynamical system on an Banach space W. These two hyperbolic structures describe long-time dynamical properties of the associated time-varying linearized equation ∂₁v + Av = B(t) v₁ where the linear operator B(t) is the evaluation of a suitable Fréchet derivative along a given solution in the set K in W. Our main objective is to show, under reasonable conditions, that if B(t) = B(λ, t) depends continuously on a parameter λ∈Λ and there is an exponential dichotomy, or exponential trichotomy, at a value λ₀∈Λ, then there is an exponential dichotomy, or exponential trichotomy, for all λ near λ₀. We present several illustrations indicating the significance of this robustness property.