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Über dieses Buch

This book gives a gentle but up-to-date introduction into the theory of operator semigroups (or linear dynamical systems), which can be used with great success to describe the dynamics of complicated phenomena arising in many applications. Positivity is a property which naturally appears in physical, chemical, biological or economic processes. It adds a beautiful and far reaching mathematical structure to the dynamical systems and operators describing these processes.
In the first part, the finite dimensional theory in a coordinate-free way is developed, which is difficult to find in literature. This is a good opportunity to present the main ideas of the Perron-Frobenius theory in a way which can be used in the infinite dimensional situation. Applications to graph matrices, age structured population models and economic models are discussed.
The infinite dimensional theory of positive operator semigroups with their spectral and asymptotic theory is developed in the second part. Recent applications illustrate the theory, like population equations, neutron transport theory, delay equations or flows in networks. Each chapter is accompanied by a large set of exercises. An up-to-date bibliography and a detailed subject index help the interested reader.
The book is intended primarily for graduate and master students. The finite dimensional part, however, can be followed by an advanced bachelor with a solid knowledge of linear algebra and calculus.



Finite Dimensions


Chapter 1. An Invitation to Positive Matrices

In this chapter we set the stage for our story. We fix our notation and summarize the linear algebraic background that will be needed for the first part of the book. We present some motivating examples of positive matrices at the very beginning and shall return to these examples later on.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 2. Functional Calculus

Our second chapter develops an abstract functional calculus for matrices. It seems of great advantage to have this abstract tool at hand even if we will not need the full power of this calculus later. This will take us to known statements of linear algebra using a coordinate-free approach, allowing to introduce some ideas which will be rather beneficial when we treat infinite-dimensional problems.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 3. Powers of Matrices

In the previous chapter we presented known facts from the spectral theory of matrices in a coordinate-free way. We are, however, interested not simply in linear algebra, but mainly in the asymptotic behavior of dynamical systems, a central theme in this text.
We apply the knowledge we gained on the structure of linear operators on finite-dimensional vector spaces to investigate what happens to the sequence consisting of the powers of a matrix. Topics we cover include boundedness, convergence to zero, convergence, mean convergence (or Cesàro convergence), periodicity, and hyperbolic decomposition.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 4. The Matrix Exponential Function

We continue our investigation of the asymptotic behavior of dynamical systems described by matrices, which was started in last chapter, now moving to the continuous time case. This means that we investigate the asymptotic properties of the matrix exponential function.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 5. Positive Matrices

We call a real matrix positive if its entries are greater or equal to zero. Positivity naturally occurs in many applications and it turns out to have deep consequences on the spectral properties of the matrix. In this chapter we discuss spectral properties of positive matrices, incorporating seminal work by Perron, Frobenius, and Wielandt.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 6. Applications of Positive Matrices

We have now accumulated enough material to pause for a while to discuss its consequences in concrete situations. We have revised linear algebra facts from a functional analytic perspective and obtained a construction to get functions of matrices in a coordinate-free manner, without the use of the Jordan normal form. This was useful when we considered positive matrices, and enabled us to see important and deep spectral consequences of positivity.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 7. Positive Matrix Semigroups and Applications

Now we investigate positive one-parameter matrix semigroups, or, using a more common name, positive matrix exponentials. As expected, positivity and irreducibility in this case also lead to remarkable spectral and asymptotic properties.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 8. Positive Linear Systems

We present one important large field of applications to the theory developed so far: control theory. More specifically, we present an elementary introduction to positive linear systems.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Infinite Dimensions


Chapter 9. A Crash Course on Operator Semigroups

After studying matrix exponential functions, it is natural to ask whether similar properties can be proved in infinite-dimensional spaces. Indeed, we will see shortly that if we have a semigroup which is continuous (in the usual operator norm), then it is the exponential function of a bounded linear operator.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 10. Banach Lattices and Positive Operators

In the remaining chapters we shall try to extend the theory of positive matrices to infinite-dimensional spaces. One of the first questions is how to generalize concepts like positivity of vectors, or positivity, irreducibility, and imprimitivity of matrices. We have tried to have an abstract look at the finite-dimensional case, to motivate infinite-dimensional concepts. Still, the transition from finite to infinite dimensions is not easy.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 11. Generation Properties

In this chapter we continue to study the connection between semigroups and their generators. We characterize generators by the properties of their resolvents. As an important byproduct, we derive a characterization of positive semigroups by the positivity of the resolvent. This is a fundamental result and will be frequently used in the sequel.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 12. Spectral Theory for Positive Semigroups

We have discovered in the finite-dimensional case that exponential functions enjoy some rather special spectral properties. Such properties are, for example, that the spectrum of a semigroup operator is determined by the spectrum of its generator, or, that the stability of a semigroup is guaranteed whenever the spectrum of its generator lies in the left half-plane.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 13. Unbounded Positive Perturbations

For two unbounded linear operators A and B on a Banach space X it is not always evident how to define in a reasonable way their sum A+B.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Advanced Topics and Applications


Chapter 14. Advanced Spectral Theory and Asymptotics

In this chapter we continue our investigation of spectral properties of positive C 0-semigroups on Banach lattices and show how the Perron–Frobenius theory can be generalized to the infinite-dimensional setting. We also list some important properties of irreducible semigroups. We will see that many results valid for positive matrix semigroups continue to hold also in infinite dimensions.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 15. Positivity and Delay Equations

We have seen that, in general, the growth bound ω 0(T) of a C 0-semigroup (T(t)) t≥0 and the spectral bound s(A) of its generator A do not coincide, even if positivity is assumed. It turns out that in Hilbert spaces a deeper analysis is possible using the boundedness of the resolvent. This has the consequence that for a positive semigroup T(t)) t≥0 on a Hilbert space the equality s(A) = ω 0(T) holds. This is the most important result of Section 15.2.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 16. Koopman Semigroups

We present here a class of positive operator semigroups that arise in studying dynamical systems. The main idea is to linearize a given (nonlinear) system by considering another state space. The linear operator which acts on this new space is called the Koopman operator. It is named after B. O. Koopman, who used this in the 1930s together with G. D. Birkhoff and J. von Neumann to prove the so-called ergodic theorems.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 17. Linear Boltzmann Transport Equations with Scattering

In this chapter we give an application of positive semigroup theory to linear transport equations. This is a wonderful piece of mathematics modeling neutron transport in a reactor which uses much of the theory we developed in this text.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 18. Transport Problems in Networks

Consider a closed network of pipes or wires in which some material (water, electrons, information packets, goods, etc.) is flowing at constant speed on each edge, with no friction or loss. In the nodes of the network the material is redistributed into the pipes according to Kirchhoff’s laws. Simplifying the physical laws and concentrating on the structure of the network, this situation can be described by a system of linear transport equations on the edges of a graph.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi

Chapter 19. Population Equations with Diffusion

Many applications of positive semigroups occur in mathematical biology or chemistry. In the finite-dimensional part of our text we have already discussed a very simple discrete-time population model, called the Leslie model (see Section 6.3). In this chapter we present a time-continuous age-structured population model with spatial diffusion. We present a rather advanced model in order to show the reader some generalizations and applications.
András Bátkai, Marjeta Kramar Fijavž, Abdelaziz Rhandi


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