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2022 | Buch

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Completeness Theorems and Characteristic Matrix Functions

Applications to Integral and Differential Operators

verfasst von: Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Verlag: Springer International Publishing

Buchreihe : Operator Theory: Advances and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph presents necessary and sufficient conditions for completeness of the linear span of eigenvectors and generalized eigenvectors of operators that admit a characteristic matrix function in a Banach space setting. Classical conditions for completeness based on the theory of entire functions are further developed for this specific class of operators. The classes of bounded operators that are investigated include trace class and Hilbert-Schmidt operators, finite rank perturbations of Volterra operators, infinite Leslie operators, discrete semi-separable operators, integral operators with semi-separable kernels, and period maps corresponding to delay differential equations. The classes of unbounded operators that are investigated appear in a natural way in the study of infinite dimensional dynamical systems such as mixed type functional differential equations, age-dependent population dynamics, and in the analysis of the Markov semigroup connected to the recently introduced zig-zag process.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract

This chapter has an introductory character. Basic elements of operator theory are reviewed, and the notion of completeness is defined for Banach space operators and specified further for Hilbert space operators. Elements of spectral theory are presented, and a few illustrative examples are given. The chapter consists of three sections of which the first consists of four subsections. Throughout the Banach and Hilbert spaces are assumed to be complex spaces.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 2. Completeness Theorems for Compact Hilbert Space Operators

Abstract

This chapter consists of five sections. In the first and second section the first three main completeness theorems are presented with the second and third theorem (Theorems 2.2.1 and 2.2.2) being further refinements of the first theorem (Theorem 2.1.2). In the third section we illustrate the first completeness theorem using the operator T _g appearing in Sect. 1.3.1, and a few additional examples are presented in Sect. 2.3.1. In Sect. 2.4 a number of classical completeness theorems presented in [32] are reviewed from the point of view of Theorems 2.2.1 and 2.2.2. In the final section we present some auxiliary results that will be used to verify the assumptions of the completeness theorems in concrete cases. Elements of the theory of entire functions as presented in Chap. 14 play an important role in the analysis in this chapter.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 3. Compact Hilbert Space Operators of Order One

Abstract

In this chapter we further specify Theorem 2.2.2 for compact Hilbert space operators of order one. Such operators are Hilbert-Schmidt operators and not necessarily trace class operators. We begin with some remarks about the latter class of operators. Throughout this chapter we shall use terminology and basic facts from the theory of entire functions which can be found in Chap. 14.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 4. Completeness for a Class of Banach Space Operators

Abstract

In this chapter we consider the completeness problem for a more general class of bounded linear operators then those considered in Chaps. 2 and 3. Moreover noncompact operators are included too, and we allow the underlying spaces to be complex Banach spaces.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 5. Characteristic Matrix Functions for a Class of Operators

Abstract

In this chapter we extend the notion of characteristic matrix function, as defined in [48] for unbounded operators, to bounded operators. Classes of Banach space operators are introduced for which the assumptions of Theorem 4.1.3 can easily be verified.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 6. Finite Rank Perturbations of Volterra Operators

Abstract

In this chapter we introduce an important class of operators that have a characteristic matrix function in the sense of Definition 5.2.1. The chapter consists of three sections. In the first section the characteristic matrix function is defined. The main theorem is a completeness theorem which is proved in the second section. In the final session we show that the results of the first two sessions remain true if the Volterra operator is replaced by a quasi-nilpotent operator.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 7. Finite Rank Perturbations of Operators of Integration

Abstract

In this chapter we specify further the results of the previous chapter for the case when the Volterra operator V is an operator of integration. Completeness results will be given for three different cases. The first section has a preliminary character.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 8. Discrete Case: Infinite Leslie Operators

Abstract

In this chapter we discuss a class of operators acting on the Hilbert space \(\ell _+^2({\mathbb C})\). The operators are infinite versions of Leslie matrices, and therefore they are called Leslie operators. These operators are again of the form finite rank perturbation of a Volterra operator. The chapter consists of four sections. The first two present the definition of a Leslie operator and describe the connection with boundary value systems. The third section has an auxiliary character. In the fourth section we present a class of Leslie operators T with special completeness properties, namely T ^∗ has a complete span of eigenvectors and generalised eigenvectors, while the latter is not true for the operator T. In the final section we consider completeness for a generalised Leslie operator which is a finite rank perturbation of a non-compact quasi-nilpotent operator, i.e., the role of Volterra operators is taken over by quasi-nilpotent operator, as in Sect. 6.3.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 9. Semi-Separable Operators and Completeness

Abstract

In this chapter we specify further the results of Chaps. 4, 5 and 6 for semi-separable operators. The chapter consists of three sections (and two subsections). The first section deals with completeness results for a class of discrete semi-separable operators studied in Section V.3 of [29]. Such operators are quite different from the Leslie operators considered in Chap. 8. The theory developed in Chap. 4 is useful in studying completeness for these discrete semi-separable operators. The second section deals with completeness results for semi-separable integral operators. The results in Chapter IX of [32], together with the results developed in Chaps. 5 and 6, are used to derive completeness results for the semi-separable integral operators. The final section has a preparatory character, not directly related to semi-separable operators. We collect some results about the fundamental solution of an ordinary differential equation, and we present an explicit resolvent formula for a class of integral operators and related Volterra operators which will play a role in the next chapter and in Chap. 11.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 10. Periodic Delay Equations

Abstract

In this chapter we introduce linear periodic functional differential equations as infinite dimensional dynamical systems. The first two sections have a preliminary character and concern time dependent delay equations and the associated fundamental solutions. In the third section we show that a time dependent delay equation has an associated two parameter family of solution operators. In the last section of this chapter we introduce the period maps for periodic delay equations, and in Theorem 10.4.5 we collect the spectral properties of the period maps that will be used in the next chapter when studying completeness problems for period maps.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 11. Completeness Theorems for Period Maps

Abstract

We consider completeness problems for period maps associated with periodic functional differential equations. These maps are bounded linear operators acting on Banach spaces of continuous functions. In the first section we show that these period maps are compact operators which can be written as the sum of a Volterra operator and a finite rank operator. The significance of completeness theorems for period maps is explained in the second and third section. Two completeness theorems for the period map of certain concrete scalar periodic delay equations are presented in the fourth and the fifth section, first for one-periodic equations and next for two-periodic equations.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 12. Completeness for Perturbations of Unbounded Operators

Abstract

The operators considered in this chapter will be unbounded Banach space operators. In the first section the theory of characteristic matrix functions, as introduced in Chap. 5, is extended to closed operators. Here we follow the procedure as developed in [48]. In the second section the characteristic matrix function is used to develop an analogue of Theorem 5.2.6 for unbounded Banach space operators.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 13. Applications to Dynamical Systems

Abstract

In this chapter we consider three different classes of unbounded operators that arise in applications. In the first section we consider exponential dichotomous operators that appear in the study of functional differential equations. In the second section, we consider the infinitesimal generator of the semigroup associated with age-dependent population models. In the third section, we consider unbounded operators that arise in the study of Markov processes. In each of the three sections the unbounded operators concerned are operators A of the kind appearing in (12.2) of the previous chapter. The results concerning completeness obtained in this chapter can be viewed as generalisations of Theorem 6.2.1.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Chapter 14. Results from the Theory of Entire Functions

Abstract

In this chapter, which consists of nine sections, we present elements from the theory of entire functions that are used throughout the book. The emphasis is to derive corollaries of the classical results of Phragmén-Lindelöf and Paley-Wiener that are used to derive completeness results for classes of operators. We fine-tune these results in various directions so that natural conditions in operator theory allow us to directly apply fundamental results from the theory of entire functions. In particular, we focus on the connection between the distribution of zeros and the growth properties of an entire function of completely regular growth. Such functions play an important role in the completeness results derived in this book. Using entire functions of the form

$$\displaystyle f(z) = p(z) + q(z) \int _{-a}^a e^{-zt} \varphi (t)\,dt,\qquad z \in {\mathbb C}, $$

where p and q ≠ 0 are polynomials, 0 < a < ∞, and φ is a non-zero square integrable function on the interval [−a, a], we show how to use the classical results from complex analysis to derive very detailed properties of these functions.

Marinus A. Kaashoek, Sjoerd M. Verduyn Lunel

Backmatter

Titel: Completeness Theorems and Characteristic Matrix Functions
verfasst von: Marinus A. Kaashoek
Sjoerd M. Verduyn Lunel
Copyright-Jahr: 2022
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-04508-0
Print ISBN: 978-3-031-04507-3
DOI: https://doi.org/10.1007/978-3-031-04508-0

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