Rainer Nagel

One-parameter semigroups of positive operators

Basic results on semigroups on banach spaces.- Characterization of semigroups on banach spaces.- Spectral theory.- Asymptotics of semigroups on banach spaces.- Basic results on spaces Co(X).- Characterization of positive semigroups on Co(X).- Spectral theory of positive semigroups on Co(X).- Asymptotics of positive semigroups on Co(X).- Basic results on banach lattices and positive operators.- Characterization of positive semigroups on banach lattices.- Spectral theory of positive semigroups on banach lattices.- Asymptotics of positive semigroups on banach lattices.- Basic results on semigroups and operator algebras.- Characterization of positive semigroups on w*-algebras.- Spectral theory of positive semigroups on w*-algebras and their preduals.- Asymptotics of positive semigroups on c*-and w*-algebras.

Functional analytic methods for evolution equations

Preface.- Giuseppe Da Prato: An Introduction to Markov Semigroups.- Peer C. Kunstmann and Lutz Weis: Maximal

The spectrum of unbounded operator matrices with non-diagonal domain

L_p -regularity for Parabolic Equations, Fourier Multiplier Theorems and

Asymptotic Behavior of One-Parameter Semigroups of Positive Operators

H^\infty

Vertex control of flows in networks

-functional Calculus.- Irena Lasiecka: Optimal Control Problems and Riccati Equations for Systems with Unbounded Controls and Partially Analytic Generators-Applications to Boundary and Point Control Problems.- Alessandra Lunardi: An Introduction to Parabolic Moving Boundary Problems.- Roland Schnaubelt: Asymptotic Behaviour of Parabolic Nonautonomous Evolution Equations.

Approximation theorems for the propagators of higher order abstract Cauchy problems

Abstract Systems of linear evolution equations involving non-diagonal boundary conditions yield operator matrices having non-diagonal domains in products of Banach spaces. We show how to compute the spectrum of these matrices by using “virtual” matrix elements and characteristic operator functions.

IDENTIFICATION OF EXTRAPOLATION SPACES FOR UNBOUNDED OPERATORS

In this paper we survey the Perron-Frobenius spectral theory for positive semigroups on Banach lattices and indicate its applications to stability theory of retarded differential equations and quasi-periodic flows.

Semigroups for Initial-Boundary Value Problems

We study a transport equation in a network and control it in a single vertex. We describe all possible reachable states and prove a criterion of Kalman type for those vertices in which the problem is maximally controllable. The results are then applied to concrete networks to show the complexity of the problem.

LINEAR NEUTRAL PARTIAL DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH

In this paper, we present two quite general approximation theorems for the propagators of higher order (in time) abstract Cauchy problems, which extend largely the classical Trotter-Kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions. Then, we apply the approximation theorems to deal with the second order dynamical boundary value problems.

Superstable operators on Banach spaces

Abstract Abstract extrapolation spaces for strongly continuous semigroups of linear operators on Banach spaces have been constructed by various methods (see, e.g., [Am (1988)], [DaP-Gr (1984)], [Na (1983)], [Ne (1992)], [Wa (1986)]). Usually they appear as “artefacts” used in some intermediate step in order to solve the Cauchy problem on the original space. Only in a few cases (see the papers by the Dutch school on X ⊙*, e.g., [Ne (1992)]), and in sharp contrast to the situation for interpolation spaces (see, e.g., [Gr (1969)], [DiB (1991)], [Lu (1985)], [Ac-Te (1987)]), the extrapolation spaces have been identified in a concrete way. It is our intention to fill this gap and subsequently to give an application of the extrapolation method to a perturbation problem.