The Asymptotic Behaviour of Semigroups of Linear Operators

ISBN-13: 9783764354558 / Angielski / Twarda / 1996 / 241 str.

Jan Van Neerven

Opis

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.

Spis treści

1. Spectral bound and growth bound.- 1.1. C0—semigroups 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.