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Traces and determinants of linear operators. (English) Zbl 0946.47013

Operator Theory: Advances and Applications. 116. Basel: Birkhäuser. viii, 258 p. (2000).
The book under review consists of 14 chapters. Contents: Preface. Introduction. I. Finite Rank Operators. II. Continuous Extension of Trace and Determinant. III. First Examples. IV. Trace Class and Hilbert-Schmidt Operators in Hilbert Space. V. Nuclear operators in Banach spaces. VI. The Fredholm Determinant. VII. Possible Values of Traces and Determinants. Perelson Algebras. VIII. Inversion Formulas. IX. Regularized determinants. X. Hilbert-Carleman Determinants. XI. Regularized Determinants of Higher Order. XII. Inversion Formulas via Generalized Determinants. XIII. Determinants of Integral Operators with Semi-separable Kernels. XIV. Algebras without the Approximation Property. Bibliography (64 items). Index. List of Symbols.
Each section is closed with historical Comments.
The authors’ motivation is well expressed in their preface. Namely, they write:
“The authors initially planned to write an article describing the origins and developments of the theory of Fredholm operators and to present their recollection 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 determinants of infinite matrices and integral operators. We were especially impressed by the papers of Poincaré, 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 infinite 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 complete 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 requirements. The 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 König and others. After the papers …were published, we saw that the written material could serve as the basis of a book. We also realized that the results of those papers still did not give a full picture of the theory and applications of traces and determinants of linear operators. There were numerous gaps to be filled and examples to be added. This was the real motivation for writing the book which we now present…”
Let \({\mathcal F}(B)\) be the algebra of all finite operators contained in the algebra \({\mathcal L}(B)\) of all linear bounded operators acting in a complex Banach space \(B\). The trace of an \(F \in {\mathcal F}(B)\) and the determinant \(\det (I+F)\) are uniquely determined and \[ \text{tr }F = \sum_j \lambda_j(F);\quad \det (I +F) = \prod_j (1 + \lambda_j(F)), \] where \(\lambda_j\) are non-zero eigenvalues of \(F\). Clearly, \(I + F\) is invertible in \({\mathcal L}(B)\) if and only if \(\det (I + F) \not= 0\). In order to define the trace and the determinant of an operator in a more general case, the authors considered subalgebras \({\mathcal D}\) of \({\mathcal L}(B)\) continuously embedded in \({\mathcal L}(B)\) such that the norm induced in \({\mathcal D}\) by the original one is submultiplicative and the set \({\mathcal F}_D\) = \({\mathcal F}(D)\cap {\mathcal D}\) is dense in \({\mathcal D}\) (i.e. \({\mathcal D}\) has the approximation property). If \(A \in {\mathcal D}\) satisfies one of the conditions which guarantee continuous extensions of the trace and determinant in \({\mathcal D}\) and \(\|K_n - A\|_{\mathcal D} \rightarrow 0\) for \(\{K_n\} \subset {\mathcal F}_{\mathcal D}\), then in \({\mathcal D}\) \[ \text{det}_{\mathcal D}(I + A) = \lim_{n \to \infty} \det (I + K_n)\quad and\quad \text{tr}_{\mathcal D}A = \lim_{n \to \infty} \text{tr} K_n. \] Note that this definition is independent of the choice of the sequence \(\{K_n\}\). Main applications of the developed theory are for integral operators with continuous and Hilbert-Schmidt kernels and to the Floquet theory for the Hill’s equation. Note that in the case of continuous kernels the determinants obtained here do coincide with the original Fredholm determinants. An infinite dimensional version of the Cramer theorem for systems of linear equations is also presented.
The book is well written and organized. However, the pleasure in reading it is slightly damaged by several inconsequences in denotations (may be just misprints). Typical errors are, for instance, the set of complexes is denoted by either \(\mathbb C\) or \(C\); the letters \(A\), \(B\), \(C\) denote incidentally operators and spaces; the set of all linear bounded operators acting in a Banach space incidentally is denoted by \(L(B)\), \({\mathcal L}(B)\), \({\mathcal L}({\mathcal B})\dots\) and so on. No matter, if the reader is more or less close to the subject. But for a freshman this could be quite misleading. Similarly, with French names: “Roushé” instead of “Rouché”: “La Theory de Fredholm” instead of “La Théorie de Fredholm”… Also the spelling of Russian names is not unified. For instance, “Liusternik” and “Ljusternik”. Through the book linear bounded operators acting in Banach spaces with index zero are considered. These restrictions seem to be not always necessary. For instance, A. Buraczewski [Stud. Math. 22, 265-307 (1963; Zbl 0121.33503)] constructed determinants for operators with finite nullity and deficiency in linear spaces (without any topology) and in Banach spaces. The algebraic part of his theory has been presented in the book of the reviewer and S. Rolewicz, “Equations in Linear Spaces”, Warszawa (1968; Zbl 0181.40501). It seems also that a part of results can be applied in order to consider Wronskians induced by linear operators [cf. Z. Dudek, Demonstratio Math. 11, 1115-1130 (1978; Zbl 0402.47007); 13, 987-993 (1980; Zbl 0463.47001); also the reviewer, “Algebraic analysis”, Dortrecht (1988; Zbl 0696.47002); Wronski theorems in algebras with logarithms, cf. the reviewer, “Logarithms and antilogarithms. An algebraic analysis approach”. Dordrecht (1998; Zbl 0898.46002)].

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A99 General theory of linear operators
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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