This book presents a systematic account of the theory of asymptotic behaviour of semigroups of linear operators acting in a Banach space. The focus is on the relationship between asymptotic behaviour of the semigroup and spectral properties of its infinitesimal generator. The most recent developments in the field are included, such as the Arendt-Batty-Lyubich-Vu theorem, the spectral mapp- ing theorem of Latushkin and Montgomery-Smith, Weis's theorem on stability of positive semigroup in Lp-spaces, the stability theorem for semigroups whose resolvent is bounded in a half-plane, and a systematic theory of individual stability. Addressed to researchers and graduate students with interest in the fields of operator semigroups and evolution equations, this book is self-contained and provides complete proofs.

Klappentext

Over the past ten years, the asymptotic theory of one-parameter semigroups of operators has witnessed an explosive development. A number oflong-standing open problems have recently been solved and the theory seems to have obtained a certain degree of maturity. These notes, based on a course delivered at the University of Tiibingen in the academic year 1994-1995, represent a first attempt to organize the available material, most of which exists only in the form of research papers. If A is a bounded linear operator on a complex Banach space X, then it is an easy consequence of the spectral mapping theorem exp(tO"(A)) = O"(exp(tA)), t E JR, and Gelfand's formula for the spectral radius that the uniform growth bound of the wt family {exp(tA)h~o, i. e. the infimum of all wE JR such that II exp(tA)II :::: Me for some constant M and all t 2: 0, is equal to the spectral bound s(A) = sup{Re A : A E O"(A)} of A. This fact is known as Lyapunov's theorem. Its importance resides in the fact that the solutions of the initial value problem du(t) =A () dt u t , u(O) = x, are given by u(t) = exp(tA)x. Thus, Lyapunov's theorem implies that the expo­ nential growth of the solutions of the initial value problem associated to a bounded operator A is determined by the location of the spectrum of A.

Inhalt

1. Spectral bound and growth bound.- 1.1. C0semigroups and the abstract Cauchy problem.- 1.2. The spectral bound and growth bound of a semigroup.- 1.3. The Laplace transform and its complex inversion.- 1.4. Positive semigroups.- Notes.- 2. Spectral mapping theorems.- 2.1. The spectral mapping theorem for the point spectrum.- 2.2. The spectral mapping theorems of Greiner and Gearhart.- 2.3. Eventually uniformly continuous semigroups.- 2.4. Groups of non-quasianalytic growth.- 2.5. Latushkin - Montgomery-Smith theory.- Notes.- 3. Uniform exponential stability.- 3.1. The theorem of Datko and Pazy.- 3.2. The theorem of Rolewicz.- 3.3. Characterization by convolutions.- 3.4. Characterization by almost periodic functions.- 3.5. Positive semigroups on Lp-spaces.- 3.6. The essential spectrum.- Notes Ill.- 4. Boundedness of the resolvent.- 4.1. The convexity theorem of Weis and Wrobel.- 4.2. Stability and boundedness of the resolvent.- 4.3. Individual stability in B-convex Banach spaces.- 4.4. Individual stability in spaces with the analytic RNP.- 4.5. Individual stability in arbitrary Banach spaces.- 4.6. Scalarly integrable semigroups.- Notes.- 5. Countability of the unitary spectrum.- 5.1. The stability theorem of Arendt, Batty, Lyubich, and V?.- 5.2. The Katznelson-Tzafriri theorem.- 5.3. The unbounded case.- 5.4. Sets of spectral synthesis.- 5.5. A quantitative stability theorem.- 5.6. A Tauberian theorem for the Laplace transform.- 5.7. The splitting theorem of Glicksberg and DeLeeuw.- Notes.- Append.- Al. Fractional powers.- A2. Interpolation theory.- A3. Banach lattices.- A4. Banach function spaces.- References.- Symbols.