Stability of Operators and Operator Semigroups

This book is about stability of linear dynamical systems, discrete and continuous. More precisely, we discuss convergence to zero of strongly continuous semigroups of operators and of powers of a bounded linear operator, both with respect to different topologies. The discrete and the continuous cases are treated in parallel, and we systematically employ a comparison of methods and results in either case. Apart from classical results, many recent crucial developments in the area are presented, such as the resolvent approach to stability. Special attention is payed to stability with respect to the weak operator topology. We also connect stability in operator theory to its analogues in ergodic theory and harmonic analysis. The book is addressed to all researchers and graduate students interested in this field.

Zusammenfassung

The asymptotic behaviour, in particular "stability" in some sense, is studied systematically for discrete and for continuous linear dynamical systems on Banach spaces. Of particular concern is convergence to an equilibrium with respect to various topologies. Parallels and differences between the discrete and the continuous situation are emphasised.

Inhalt
Introduction.- Chapter I. Functional analytic tools.- 1. Structure of compact semigroups.- 2. Mean ergodicity.- 3. Tools from semigroup theory.- Chapter II. Stability of linear operators.- 1. Power boundedness.- 2. Strong stability.- 3. Weak stability.- 4. Almost weak stability.- 5. Abstract examples.- 6. Stability via Lyapunov equation.- Chapter III. Stability of C0-semigroups.- 1. Boundedness.- 2. Uniform exponential stability.- 3. Strong stability.- 4. Weak stability.- 5. Almost weak stability.- 6. Abstract examples.- 7. Stability via Lyapunov equation.- Chapter IV. Discrete vs. continuous.- 1. Embedding operators into C0-semigroups.- 2. Cogenerators.- Bibliography.