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2000 | Buch

Traces and Determinants of Linear Operators

verfasst von: Israel Gohberg, Seymour Goldberg, Nahum Krupnik

Verlag: Birkhäuser Basel

Buchreihe : Operator Theory: Advances and Applications

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SUCHEN

Über dieses Buch

The authors initially planned to write an article describing the origins and devel­ opments of the theory of Fredholm operators and to present their recollections of this topic. We started to read again classical papers and we were sidetracked by the literature concerned with the theory and applications of traces and determi­ nants of infinite matrices and integral operators. We were especially impressed by the papers of Poincare, von Koch, Fredholm, Hilbert and Carleman, as well as F. Riesz's book on infinite systems of linear equations. Consequently our plans were changed and we decided to write a paper on the history of determinants of infi­ nite matrices and operators. During the preparation of our paper we realized that many mathematical questions had to be answered in order to gain a more com­ plete understanding of the subject. So, we changed our plans again and decided to present the subject in a more advanced form which would satisfy our new require­ ments. This whole process took between four and five years of challenging, but enjoyable work. This entailed the study of the appropriate relatively recent results of Grothendieck, Ruston, Pietsch, Hermann Konig and others. After the papers [GGK1] and [GGK2] were published, we saw that the written material could serve as the basis of a book.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
As in the finite dimensional case, determinants of operators acting on infinite dimensional Banach spaces provide important tools for solving linear equations by exhibiting explicit formulas for their solutions. These formulas are generalizations of the famous Cramer’s rule for solving finite systems of linear equations. The most outstanding example is Fredholm’s theory of integral equations. By means of determinants, Fredholm was able to exhibit explicit solutions of the equations and to determine additional properties of the integral operators such as the existence of eigenvalues and the evaluation of their multiplicities [Fr].
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter I. Finite Rank Operators
Abstract
This chapter is of a preliminary character. Here we accumulate results about the traces of finite rank operators as well as the determinants of operators of the form I + F, where F is an operator of finite rank Also, various formulas of trace and determinant are presented for operators of the form mentioned above. A variant of an axiomatic definition of the trace and determinant is proposed. This material is basic for the remaining chapters of the book where the determinant and trace are defined for more general classes of operators via limits of finite rank operators. The chapter also contains the trace and determinant of operator pencils and applications of the determinant to inversions of operators of the form I + F, where F has finite rank.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter II. Continuous Extension of Trace and Determinant
Abstract
This is one of the main chapters in this book. It contains the extension theory of trace and determinant of finite rank operators to wider algebras of linear operators. Since the functionals trace and determinant are not continuous with respect to the operator norm on the finite rank operators acting in infinite dimensional spaces, we study extensions of these functions to algebras of operators with other norms. In this chapter we also investigate the main properties of the extended determinant. In particular the connection between zeros of the determinant and eigenvalues of the corresponding operators is given.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter III. First Examples
Abstract
The theory of determinants and traces in several matrix algebras are reviewed in this chapter. The results are based on the results of the previous chapter.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter IV. Trace Class and Hilbert-Schmidt Operators in Hilbert Space
Abstract
The study of traces and determinants of trace class and Hilbert-Schmidt operators on Hilbert space requires a familiarity with these classes of operators. This chapter discusses all the related concepts and theorems needed in this book. Included are properties of singular numbers of compact operators and Lidskii’s trace theorem. We then proceed to introduce the regularized determinants of Hilbert-Schmidt operators. The formulas for the traces of integral operators with continuous kernel and with smooth kernel are presented. The determinant of operator pencils also appears. The last section treats the S p spaces which are generalizations of the algebra of Hilbert-Schmidt operators.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter V. Nuclear Operators in Banach Spaces
Abstract
This chapter contains a brief exposition of nuclear operators in Banach space and their corresponding trace and determinant. It also contains generalizations to Banach space of some of the results in the preceding chapter. One of the main theorems is Theorem 3.1 concerning the trace and determinant of nuclear operators due to Grothendieck. Some asymptotic behavior of eigenvalues of nuclear operators in Banach spaces is presented.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter VI. The Fredholm Determinant
Abstract
In Section 1 we treat the trace and determinant for Fredholm integral operators. For the case of a continuous kernel, this theory was first introduced by Fredholm in the famous paper [Fr]. Some modifications of the Fredholm determinant for integral operators with discontinuous kernels are proposed in Sections 2 and 3. In contrast with the regularized determinant, which are usually used for discontinuous kernels, the modified determinants considered here are multiplicative functionals and can be included in the general theory constructed in Chapter II. Additional developments of the Fredholm determinant are given which are used to further analyze Hill’s method. Section 6 contains formulas for the determinant of systems of integral operators.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter VII. Possible Values of Traces and Determinants. Perelson Algebras
Abstract
It is shown how the trace and the determinant of an operator can vary with the algebra which contains the operator. The main conclusion is that if the operator is not of trace class, then generally speaking, the values of the trace constitute the whole complex plane and the values of the determinant are either a singleton {0} or the set C\{0}.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter VIII. Inversion Formulas
Abstract
Cramer’s rule and formulas of the resolvents for operators which belong to algebras considered in previous sections appear in this chapter. The results are applied to integral operators with continuous kernels and with kernels which have a jump discontinuity on the diagonal. The last section deals with Fredholm’s method of solutions of homogeneous integral equations.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter IX. Regularized Determinants
Abstract
A general theory of regularized determinants in normed algebras of operators acting in Banach spaces is proposed. In this approach regularized determinants are defined as continuous extensions of the corresponding regularized determinants of operators of finite rank. We characterize the algebras for which such extensions exist and describe the properties of the extended determinants.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter X. Hilbert-Carleman Determinants
Abstract
Chapter X presents examples of algebras on which the regularized determinants are computed. We show that in these algebras the regularized determinant coincides with the Hilbert-Carleman determinant. The first section, which is of a preliminary nature, contains basic formulas for the Hilbert-Carleman determinant for integral operators with degenerate kernel. These formulas are then used in Section 2 to introduce the Hilbert-Carleman determinant for integral operators in Banach spaces. The Hilbert-Carleman determinant on the algebra of Hilbert-Schmidt operators appears in Section 3. In Section 4 the Hilbert-Carleman determinant is defined on the Mikhlin-Itskovich algebra. This algebra is a natural generalization of the algebra of Hilbert-Schmidt operators. The discrete analogue of these results appears in Section 7. Section 5 contains an analysis of a specific algebra of integral operators. The so-called diagonally modified Hilbert-Carleman determinant is introduced and studied in Section 6.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
12. Regularized Determinants of Higher Order
Abstract
In Chapter II we introduced the determinant as an extension of the determinant on the algebra of operators of finite rank; the present chapter enlarges these investigations to regularized determinants of higher order. Here these higher order determinants are studied as extensions of the corresponding regularized determinants of operators of finite rank The main extension theorem is proved in Section1, analyticity of the determinant in Section 2. In Sections 3, 4 we prove a theorem (stated in Section IV.10) which is a generalization of the Lidskii trace theorem to polynomial operator pencils.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter XII. Inversion Formulas via Generalized Determinants
Abstract
In this chapter we obtain Cramer’s rule and the formulas for the resolvent which are expressed via the extended traces tr(A k ) of iterations and regularized determinants. This chapter may be viewed as a generalization of the results in Chapter VIII. In the main formulas in that chapter the determinants are replaced here by the corresponding regularized determinant.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter XIII. Determinants of Integral Operators with Semi-separable Kernels
Abstract
Chapter XIII is devoted to integral operators with semi-separable kernels. The main aim is to compute the Fredholm determinant and the Hilbert-Carleman determinant for those operators. The theory of systems with boundary conditions plays a crucial role here. Formulas for the resolvents of the above integral operators are also provided.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Chapter XIV. Algebras without the Approximation Property
Abstract
An example of the determinant of an algebra without the approximation property is discussed in this chapter. It is shown that, in this case, the determinant is not uniquely defined; i.e., there are many extensions of the determinant on the algebra of operators of finite rank.
Israel Gohberg, Seymour Goldberg, Nahum Krupnik
Backmatter
Metadaten
Titel
Traces and Determinants of Linear Operators
verfasst von
Israel Gohberg
Seymour Goldberg
Nahum Krupnik
Copyright-Jahr
2000
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8401-3
Print ISBN
978-3-0348-9551-4
DOI
https://doi.org/10.1007/978-3-0348-8401-3