Evolution Semigroups, Translation Algebras, and Exponential Dichotomy of Cocycles

Abstract

We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space 3 acting on a Banach space X. Our main tool is the associated evolution semigroup on C 0 (3; X ). We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mappingÂannular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 (Z; X ). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, we extend to our situation the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded, continuous, mild solutions to the inhomogeneous equation. Academic Press

On the uniform exponential stability of linear skew‐product semiflows

“…Also, it is interesting to see that almost all interesting infinite dimensional situations, as for instance flows originating from partial differential equations and functional differential equations, only yield strongly continuous cocycles. In this context, there has been studied the dichotomy of linear skew-product semiflows defined on compact spaces (see [2,3,4,5]), and on a locally compact spaces, respectively (see [15]). The idea of associating an evolution semigroup in the expanded case of exponential stability or dichotomy of linear skew-product flow on locally compact metric space Θ , has its origins in the works of Latushkin and Stepin [13], respectively Latushkin, Montgomery-Smith and Schnaubelt [14].…”

On Spectral Characterization of Nonuniform Hyperbolicity

“…Subsequent related results consider the general case of linear cocycles (or the so-called linear skew product flows) acting on Banach spaces. We refer to [23][24][25][26][27][28][29] and references therein. We stress that all of those works consider only uniform hyperbolic behaviour.…”

A lower bound for the stability radius of time-varying systems

“…In the last few years, the theory of linear skew-product semiflows has proved to be a very useful tool in the study of evolution equations with unbounded coefficients (see [4]- [6], [10], [11], [14]- [20]). Significant questions concerning the asymptotic behaviour of linear skew-product semiflows have been answered.…”

“…Significant questions concerning the asymptotic behaviour of linear skew-product semiflows have been answered. In this context, the theorems of Perron type or so-called "input-output" conditions of characterization of the asymptotic properties, have been obtained for uniform exponential stability (see [19]), for uniform exponential expansiveness (see [16]) and for pointwise and uniform exponential dichotomy, respectively, of discrete and continuous linear skewproduct semiflows (see [4]- [6], [10], [11], [15], [17], [18], [20]). …”