The Asymptotic Behaviour of Semigroups of Linear Operators

Inhaltsverzeichnis

• 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.
• 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.
• 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.
• Append.
• Al. Fractional powers.
• A2. Interpolation theory.
• A3. Banach lattices.
• A4. Banach function spaces.
• References.
• Symbols.