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Introduction to the theory of singular integral operators with shift. (English) Zbl 0811.47049

Mathematics and its Applications (Dordrecht). 289. Dordrecht: Kluwer Academic Publishers. xvi, 288 p. (1994).
The book is devoted to the full construction of the Noether theory of the singular integral equations with shift (SIES) and boundary value problems for analytic functions with shift (BPAFS). In the simplest case this (SIES) equation has the following form \[ T\varphi\equiv a_ 1\varphi+ a_ 2 W\varphi+ a_ 3 S\varphi+ a_ 4 WS\varphi=f, \] where \(a_ j\) are coefficients, \(S\) is the operator of singular integration with a Cauchy kernel, \(S\varphi= (\pi i)^{-1} \int_ \Gamma (\tau- t)^{-1} \varphi(\tau) d\tau\), \(\Gamma\) is some contour and \(W\) is the shift operator on \(\Gamma\): \(W\varphi= \varphi[ d(t)]\). The main spaces considered are \(L_ p (\Gamma)\), \(p\geq 1\), \(H^ \lambda (\Gamma)\), \(0<\lambda< 1\), and analogous spaces with weight.
Fifteen years after the publication of the book of one of the authors [G. S. Litvinchuk, “Boundary value problems and singular integral equations with shift”, (Russian), Nauka, Moscow (1977; Zbl 0462.30029), Chinese translation: Beijing (1982)] not less than 300 papers were published on the Noether theory of different operators concerned with this item; namely, on the Noether theory of SIES and integral operators of convolution type (with periodic coefficients), their discrete analogues, on the theory of pseudo-differential operators with shift, etc. Essential results were obtained in the theory of equations and operators with a non-Carleman shift. For illustration of new achievements we give some classification by the authors on the following examples.
Let \(\alpha(t)\) be a shift preserving the orientation on the contour \(\Gamma\) (a simple closed curve) and having at least one periodic (in particulary fixed) point on \(\Gamma\) and let \(M(\alpha, k)\) be the set of all periodic points with multiplicity \(k\) of the shift \(\alpha(t)\). The following four cases are possible:
1) \(M(\alpha, k)\equiv\Gamma\) (a Carleman shift with multiplicity \(k\));
2) \(M(\alpha, k)\) is a finite set \(k=1\);
3) \(M(\alpha, k)\) is a finite set and \(k>1\);
4) \(M(\alpha, k)\) is an arbitrary non-empty set not coinciding with the whole contour \(\Gamma\).
The case 1) was studied in the mentioned book by G. S. Letvinchuk in detail, but the investigations to cases 2)–4) on that moment was not yet completed. Here the authors give the full theory on the new cases for non-Carleman shift and a new point of view on the case of Carleman shift in a general approach for all cases 1)–4). Of course, the authors also consider more complicated objects than those mentioned above: continuous or discontinuous coefficients, closed or unclosed contours, shift \(\alpha (t)\) preserving or changing orientation on \(\Gamma\), and general algebras generated by operators of multiplication by continuous functions, the shift operator and the operator of singular integration, etc.
This book is a remarkable illustration of new results both as a new methodical approach to the Noether theory of singular integral operators with (and classically without) shift. The general results of this book have many applications in different domains: theory of elasticity, dynamic systems, etc. They may be applied for the investigation of many concrete classes of integral operators.
The book may be recommended to scientific investigators and to students. Contents.
Introduction.
Chapter 1. Background information.
§1. On Noetherian operators.
§2. On the operator of singular integration.
§3. On the shift function and shift operator.
§4. On \(C^*\)-algebras.
Chapter 2. Noetherity criterion and formula for the index of a singular integral functional operator of first order in the continuous case.
§1. Criterion of Noetherity for singular integral functional operators of first order with orientation preserving shift.
§2. The calculation of the index of a singular integral functional operator of the first order with a shift preserving the orientation.
§3. The Noetherity criterion and the index formula for a singular integral functional operator of the first order with a shift changing the orientation.
§4. References and a survey of similar or closed results.
Chapter 3. The Noether theory of a singular integral functional operator of finite order in the continuous case.
§1. The Noetherity criterion and the index formula for a system of singular integral equation with a Cauchy kernel and continuous coefficients on a closed contour.
§2. Theorems concerning decreasing the order of functional and singular integral operators.
§3. Noetherity criterion and a formula for the index for systems of singular integral equations with a Carleman shift.
§4. An invertibility criterion for a matrix functional operator with a non-Carleman shift.
§5. Noether theory for singular integral functional operators of superior order.
§6. References and survey of similar results.
Chapter 4. The Noether theory of singular integral functional operators with continuous coefficients on a non-closed contour.
§1. The Noetherity criterion and the index formula for singular integral operators with continuous coefficients on a non-closed contour.
§2. The Noetherity criterion and the index formula for singular integral functional operators with continuous coefficients on a non-closed contour.
§3. Systems of singular integral operators with shift on a non-closed contour.
§4. References and a survey of closely related results.
Chapter 5. The Noether theory in algebras of singular integral functional operators.
§1. \(C^*\)-algebras of singular integral operators.
§2. \(C^*\)-algebras of singular integral operators with a Carleman shift.
§3. \(C^*\)-algebras of singular integral operators with non-Carleman shift which has periodic points.
§4. Further development of a local method for studying the Noetherity of bounded linear operators of non-local type and its applications. Commentaries to the literature.
References. Subject Indices.

MSC:

47G10 Integral operators
47B38 Linear operators on function spaces (general)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A53 (Semi-) Fredholm operators; index theories
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45E05 Integral equations with kernels of Cauchy type
45-02 Research exposition (monographs, survey articles) pertaining to integral equations
30E25 Boundary value problems in the complex plane

Citations:

Zbl 0462.30029
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