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Fredholm and local spectral theory. II: With application to Weyl-type theorems. (English) Zbl 1448.47002

Lecture Notes in Mathematics 2235. Cham: Springer (ISBN 978-3-030-02265-5/pbk; 978-3-030-02266-2/ebook). xi, 546 p. (2018).
This book is a sequel to [Fredholm and local spectral theory, with applications to multipliers. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1077.47001)], which by now has received more than 300 citations, showing great interest to several specialists for this subject. The monograph under review is oriented to Weyl-type and Browder-type theorems. Historically, it was H. Weyl [Rend. Circ. Mat. Palermo 27, 373–392, 402 (1909; JFM 40.0395.01)] who first noticed that the structure of the spectrum of selfadjoint operators has some special properties. A similar structure has been later observed for several Banach space operators, which are said to satisfy Weyl’s theorem. Noticing that there is no systematic approach to this subject, the author decided to write this book, whose purpose “is to provide the first general treatment of operators for which Weyl-type or Browder-type theorems hold, taking into account the more recent developments,” as the author puts it.
The aim of the first chapter is to introduce some classes of Banach space operators, related to the classical Fredholm theory. After some elementary notions, concerning the reduced minimum modulus and the spectrum of a Banach space operator, other spectra such as the approximate point spectrum and the surjectivity spectrum are introduced. Starting from the kernel and the range of an operator, the useful concepts of ascent and descent are then defined and their properties are investigated. The inclusion of the kernel in all ranges of the iterates of an operator, whose range is supposed to be closed, gives the definition of a semi-regular operator, originating from works by {T. Kato}.
Going back to a classical perturbation result by Kato, the concept of generalized Kato decomposition, firstly appearing in works by {M. Mbekhta}, is recalled. In this context, the class of Browder-type operator is discussed, including the larger classes called upper semi-Browder and lower semi-Browder operators. After dealing with operators having topological uniform descent, the author presents the class of quasi-Fredholm operators, introduced by J.-P. Labrousse [Rend. Circ. Mat. Palermo (2) 29, 161–258 (1980; Zbl 0474.47008)], which is stable under finite rank compact perturbations, and whose members have topological uniform descent. The next two sections of this chapter deal with an approach to Drazin invertible operators, which are, in fact, direct sums of nilpotents and invertible operators. A short section of comments, including mainly references, ends this chapter.
The second chapter contains a systematic study of local spectral theory, developed around the global and local single-valued extension property. The single-valued extension property (SVEP) originates in works by {N. Dunford} and was intensively used by I. Colojoară and C. Foiaş [Theory of generalized spectral operators. New York-London-Paris: Gordon and Breach Science Publishers (1968; Zbl 0189.44201)], while the global and local single valued property appears in works by Finch, Laursen, Neumann, and the author of this book as well, to cite only a few. The reviewer may be permitted to mention that the study of the local single-valued extension property and local spectra was initiated in his work [F. H. Vasilescu, Tohoku Math. J. (2) 21, 509–522 (1969; Zbl 0193.10001)], which contains Definition 2.1 and some properties as well, presented in Section 2.1 of this book. This chapter also includes information about the quasi-nilpotent part of an operator, the localized single-valued extension property in connection with the topological uniform descent, its stability under quasi-nilpotent equivalence, spectral mapping theorems for local spectra, and connections with Drazin invertible operators. As before, some comments including references end this chapter, which, as alluded to above, might be slightly improved.
The study of the behaviour of the essential spectra under perturbations is the subject of the third chapter of this book. The first section of this chapter recalls the definitions of Weyl, Browder and Riesz operators, as well as some of the direct extensions of this classes, as for instance upper- and lower- semi-Weyl operators. They are accompanied by various spectral properties of those classes and their stability under some perturbations. The next section presents characterizations of Weyl and Browder operators, some of them in terms of hyper-kernels or hyper-ranges. The chapter also contains information about semi-\(B\)-Browder spectra, Drazin spectra, meromorphic operators, algebraic operators, essentially left and right Drazin invertibility, and the concept of regularity in a unital Banach algebra. At the end of this chapter, the spectral properties of the Drazin inverse are investigated, and, as usual, comments concerning bibliographical references are included.
Polaroid operators, that is, those operators whose isolated points of the spectrum are poles of the resolvent functions, form the subject of the fourth chapter. They are associated with the more general classes of left and right polaroid operators, and also with the particular case of hereditary polaroid operators. An example of a hereditary polaroid operator is provided by the class of the so-called paranormal operators. Some connections with isometries and with Toeplitz operators on Hardy spaces are also presented in the corresponding sections of this chapter.
Browder-type theorems are discussed in the fifth chapter. Roughly speaking, when the Browder spectrum equals the Weyl spectrum (in general, the first one may be strictly included in the second), the corresponding operator is said to satisfy Browder’s theorem. Such operators can be characterized in term of their quasi-nilpotent parts. There is also an approximate point version of Browder’s theorem called \(a\)-Browder theorem.
The next chapter, the sixth and the last, is dedicated to a presentation of the Weyl-type theorems. An operator satisfies Weyl’s theorem when, after eliminating the Weyl spectrum from its spectrum, what remains consists of isolated points of the spectrum of finite multiplicity. Among several related properties, Weyl-type theorems for polaroid operators and Weyl’s theorem for Toeplitz operators are discussed.
The book is well written, with clear and convincing arguments, and with a rich collection of facts appearing for the first time in a monograph, exhibited on more than 500 pages. Taking care to have a self-contained text, some introductory parts overlap with the corresponding ones from the previous monograph of the author [loc.cit.].
The author himself has many contributions in this area; the bibliography, including 307 titles, mentions 46 of his articles, written either alone or with a long list of co-authors.
One should observe that the text inadvertently contains some omissions. For instance, on p.104, to define correctly the operator \(T\) on line –7, one should assume that \(\Omega\subset\mathbb{C}\).
Another omission detected by the reviewer is related to Definition 2.10, used already by other authors. In this definition, the expression “for every neighborhood” should be replaced by “for every connected (open) neighborhood”; otherwise, simple counterexamples may be produced, using the points in the boundary of an operator not having the SVEP. In any case, the book is an interesting and needed contribution, due to the large interest for this subject among many active mathematicians.

MSC:

47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A53 (Semi-) Fredholm operators; index theories
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