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**Evolution semigroups in dynamical systems and differential equations.**
*(English)*
Zbl 0970.47027

Mathematical Surveys and Monographs. 70. Providence, RI: American Mathematical Society (AMS). x, 361 p. (1999).

The main objective of this book is to characterize the asymptotic properties like stability, hyperbolicity, or exponential dichotomy of linear differential equations on Banach spaces and infinite dimensional dynamical systems in terms of spectral properties of the associated evolution semigroup. The book includes a systematic treatment of recent results in this area as well as a comprehensive survey of the vast literature related to the subject, and it discusses various applications. Each chapter contains a section with bibliography and remarks.

After the introduction and historical remarks in Chapter 1, Chapter 2 is devoted to the study of evolution semigroups associated with autonomous differential equations \(\dot x = A x\) with unbounded infinitesimal generator \(A\). Here the delicate relationship between the spectrum of the semigroup and the spectrum of its generator is presented and hyperbolic semigroups are introduced, the properties of which are closely related to the dichotomy properties of the solutions of autonomous differential equations. Chapter 3 deals with Howland semigroups (on the line and on the half-line) and nonautonomous differential equations \(\dot x = A(t) x\). The main theorems here are the spectral mapping theorem for evolution semigroups and the dichotomy theorem which states that an evolution family has exponential dichotomy if and only if the corresponding semigroup on \({\mathbb R}\) is hyperbolic. In Chapter 4 four characterizations of the existence of exponential dichotomy for evolution families are given, one of the main themes being the hyperbolicity of the associated discrete operators. In Chapter 5 applications of evolution semigroups to linear control theory and to the problem of persistence of exponential dichotomy under perturbations are considered. Chapter 6 is dedicated to the study of linear skew-product flows, their associated Mather semigroups and their spectral theory (Sacker-Sell spectral theory), and in Chapter 7 dichotomies of linear skew-product flows are studied by means of an associated family of discrete evolution operators. The final Chapter 8 collects the results connected to the exact Osedelets-Lyapunov exponents and the multiplicative ergodic theorem of V. Osedelets.

After the introduction and historical remarks in Chapter 1, Chapter 2 is devoted to the study of evolution semigroups associated with autonomous differential equations \(\dot x = A x\) with unbounded infinitesimal generator \(A\). Here the delicate relationship between the spectrum of the semigroup and the spectrum of its generator is presented and hyperbolic semigroups are introduced, the properties of which are closely related to the dichotomy properties of the solutions of autonomous differential equations. Chapter 3 deals with Howland semigroups (on the line and on the half-line) and nonautonomous differential equations \(\dot x = A(t) x\). The main theorems here are the spectral mapping theorem for evolution semigroups and the dichotomy theorem which states that an evolution family has exponential dichotomy if and only if the corresponding semigroup on \({\mathbb R}\) is hyperbolic. In Chapter 4 four characterizations of the existence of exponential dichotomy for evolution families are given, one of the main themes being the hyperbolicity of the associated discrete operators. In Chapter 5 applications of evolution semigroups to linear control theory and to the problem of persistence of exponential dichotomy under perturbations are considered. Chapter 6 is dedicated to the study of linear skew-product flows, their associated Mather semigroups and their spectral theory (Sacker-Sell spectral theory), and in Chapter 7 dichotomies of linear skew-product flows are studied by means of an associated family of discrete evolution operators. The final Chapter 8 collects the results connected to the exact Osedelets-Lyapunov exponents and the multiplicative ergodic theorem of V. Osedelets.

Reviewer: Christiane Tretter (Leicester)

### MSC:

47D06 | One-parameter semigroups and linear evolution equations |

34G10 | Linear differential equations in abstract spaces |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47E05 | General theory of ordinary differential operators |