Equations With Involutive Operators,9780817641573

Equations With Involutive Operators

by ;
ISBN13: 9780817641573
ISBN10: 0817641572
Format: Hardcover
Pub. Date: 2001-07-01
Publisher(s): Birkhauser
List Price: $139.99

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Summary

Equations with Involutive Operators demonstrates an important interplay between abstract and concrete operator theory. The focus is on the investigation of a number of equations, which, while seemingly different, are all unified by the same idea: they are all realizations of some operator equations in Banach spaces. One permeating theme in these equations involves the role of the Fredholm property.The text is carefully written, self-contained, and covers a broad range of topics and results. Key ideas are developed in a step-by step approach, beginning with required background and historical material, and culminating in the final chapters with state-of-the art topics. Experts in operator theory, integral equations, and function theory as well as students in these areas will find open problems for further investigations. The book will also be useful to engineers using operator theory and integral equation techniques. Good examples, bibliography and index make this text a valuable classroom or reference resource.

Table of Contents

Introduction xv
Notation xxi
On Fredholmness of Singular Type Operators
1(50)
Fredholm operators
1(13)
Basics on Fredholm operators
1(4)
Fredholmness of operators with projectors
5(4)
Fredholmness of matrix operators
9(5)
Singular integral operators with piecewise continuous coefficients in the space Lp(Γ)
14(12)
Singular integral operators in the space Lp(Gamma;)
14(2)
The notion of p-index
16(5)
Fredholmness criterion and the index formula
21(3)
Approximation of non-Fredholm operators by Fredholm ones
24(1)
On compactness of some ``composite'' singular operators
25(1)
On Fredholmness of convolution type operators
26(22)
Fourier transforms and the Wiener algebra
26(4)
General preliminaries on convolution operators
30(4)
Theorems on Fredholmness
34(2)
On compactness of some convolution type operators
36(5)
On Fredholmness of convolution type operators with ``variable'' coefficients
41(2)
Discrete convolution type equations
43(2)
On Hankel operators
45(3)
Bibliographic notes to Chapter 1
48(3)
On Fredholmness of Other Singular-type Operators
51(60)
On operators with homogeneous kernels; the one-dimensional case
51(17)
Connection with convolution operators; Lp-boundedness
52(1)
On Fredholmness of the operators λI -- K
53(12)
The Carleman equation and notion of the standard α-lemniscate
65(3)
Operators with homogeneous kernels; the multi-dimensional case
68(16)
Lp-boundedness
69(3)
On spherical harmonics
72(3)
Formal reduction to a system of one-dimensional equations with homogeneous kernels
75(2)
Justification of the reduction and the main result for rotation invariant homogeneous kernels
77(5)
Some cases of non-rotation invariant kernels and other types of homogeneity
82(2)
Convolution-type operators with discontinuous symbols
84(25)
Setting of the problem
85(1)
Lizorkin space
86(1)
Fractional integration of Liouville and Bessel type
87(3)
The case of a continuous symbol
90(3)
The case of a discontinuous coefficient G(x)
93(1)
Fourier transform of a certain function
94(1)
On a special group of operators
95(3)
The model operator with a discontinuous symbol
98(2)
The case G(x) ≡ 1
100(1)
The general case of a discontinuous symbol
101(8)
Bibliographic notes to Chapter 2
109(2)
Functional and Singular Integral Equations with Carleman Shifts in the Case of Continuous Coefficients
111(42)
Carleman and generalized Carleman shifts
111(17)
On some properties of shifts
111(3)
On α(t)-factorization of functions
114(4)
The winding number of functions, invariant and anti-invariant with respect to shift
118(1)
On α(t)-factorization in the case of the shift α(t) = teiw on the unit circle
119(5)
Logarithmic means and factorization
124(3)
Factorization with the shift x + h on the real line
127(1)
A functional equation with shift
128(14)
Connection between a functional equation with shift and an algebraic system
129(2)
Functional equations in the degenerate case
131(3)
Functional equations with the shift α(t) = teiw
134(5)
Some generalizations
139(3)
Singular integral equations with Carleman shift on a closed curve; the case of continuous coefficients
142(5)
Accompanying and associated operators
142(2)
Fredholmness theorem; the case of preservation of the orientation
144(1)
Fredholmness theorem; the case of change of the orientation
145(1)
On regularizers of singular operators with shift
146(1)
Singular integral equations with Carleman shift on an open curve; the case of continuous coefficients
147(4)
The passage to a ``composite'' singular integral operator
147(2)
Representation of a ``composite'' singular integral operator as a composition of usual singular operators
149(1)
Fredholmness theorem for the operator (12.1)
150(1)
Bibliographic notes to Chapter 3
151(2)
Two-term Equations (A + QB)y = f with an Involutive Operator Q; an Abstract Approach and Applications
153(70)
Fredholmness of an abstract equation with an involutive operator; non-matrix approach
154(8)
Prompting ideas
154(1)
A system of axioms. Examples
155(2)
Fredholmness theorem
157(4)
Invertibility theorem
161(1)
Application to singular integral equations with complex conjugate unknowns
162(3)
Anti-quasicommutation of the operators Q and S
162(1)
The case of a closed curve
163(1)
The case of an open curve
164(1)
Applications to integral equations on the real line with reflection or inversion
165(14)
Convolution type equations with reflection
165(2)
The case of weighted spaces
167(5)
Singular convolution operators with reflection
172(1)
Convolution type operators with a discontinuous symbol and reflection
173(3)
Equations with homogeneous kernels and the shift 1/x
176(2)
Discrete analogue of convolution equations with reflection
178(1)
Application to singular integral equations with Carleman shift on an open curve; the case of discontinuous coefficients
179(9)
The normed operator
180(1)
Reduction of the normed operator to a ``composite'' singular operator
180(2)
Representation of a ``composite'' singular operator as a composition of usual singular operators and the theorem on Fredholmness of the ``composite'' operator
182(4)
Theorem on Fredholmness of the normed operator
186(1)
Theorem on Fredholmness in the general case
186(2)
Singular integral equations with a fractional linear shift in the space Lp with a special weight
188(5)
The choice of the weighted space and construction of the involutive operator
189(2)
The case of shift preserving the orientation (D < 0) < 0)
191(1)
The case of shift changing the orientation (D > 0)
192(1)
Fredholmness of abstract equations with a generalized involutive operator (non-matrix approach)
193(13)
System of axioms and the theorem on Fredholmness
193(4)
On deficiency numbers of the operator K = A + QB
197(2)
Application to discrete Wiener-Hopf equations with oscillating coefficients
199(3)
Application to paired discrete convolution type equations with oscillating coefficients
202(1)
The case of irrational oscillation
203(3)
Abstract equations with algebraic operators
206(13)
On a partition of unity
207(1)
Algebraic operators. Equations with such operators in the case of constant coefficients
208(4)
The nilpotent case
212(1)
Regularization of equations with quasi-algebraic operators in the case of operator coefficients
213(2)
The characteristic part of equations with algebraic operators
215(3)
Application to a functional equation with the Fourier transform
218(1)
Bibliographic notes to Chapter 4
219(4)
Equations with Several Generalized Involutive Operators. Matrix Abstract Approach and Applications
223(52)
Fredholmness of abstract equations with generalized involutive operators (matrix approach)
223(21)
The case of one generalized involutive operator
224(7)
A scheme of investigation of equations with two generalized involutive operators (reduction)
231(8)
Connections between matrix and non-matrix approaches
239(5)
Singular integral equations with a finite group of shifts in the case of continuous coefficients
244(9)
The equation with one generalized Carleman shift
245(1)
Classification of a finite group of shifts on closed or open curves
246(2)
Singular integral equations on a closed curve with two shifts
248(5)
Singular integral equations with a finite group of shifts (the case of piecewise continuous coefficients)
253(8)
The case of a closed curve
252(4)
The case of one shift, changing the orientation on a closed curve
256(2)
The case of an open curve
258(1)
Singular integral equations with shift and complex conjugation
259(2)
Convolution type equations with shifts and complex conjugation
261(12)
Discrete convolution operators with almost stabilizing coefficients
261(4)
Fredholmness of discrete convolution operators with oscillation and reflection
265(2)
Convolution type equations with reflection and complex conjugation
267(3)
Integral equations with homogeneous kernels involving terms with inversion and complex conjugation
270(3)
Bibliographic notes to Chapter 5
273(2)
Application of the Abstract Approach to Singular Equation on the Real Line with Fractional Linear Shift
275(64)
Singular integral operators perturbed by integral operators with homogeneous kernels
277(10)
Some necessary conditions
278(1)
Reduction to a system of paired convolution equations
279(5)
Systems of singular integral equations perturbed by integrals with homogeneous kernels
284(3)
Singular integral operators with a fractional linear Carleman shift in the weighted space Lγp(R1)
287(24)
Setting of the problem and introduction of the involutive operator Qν
288(1)
Some connections between the involutive operator Qν and the singular integral operators S, Sα and Sα
289(5)
Reduction of singular integral equations with a fractional linear shift to a system of perturbed singular equations without shift
294(3)
The case of preservation of the orientation (D < 0)
297(7)
The case of change of the orientation (D > 0)
304(1)
The investigation of equation (B)
305(4)
The case of a nonfractional linear shift on R1
309(2)
Equations including operators with homogeneous kernels, the singular integral operators and the inversion shift
311(5)
Potential type operators on the real 1 ine with a fractional linear Carleman shift
316(10)
On normal solvability of potential type operators with discontinuous characteristics
317(2)
The case of linear shift (the case of reflection)
319(2)
The case of fractional linear shift
321(5)
Generalized Carleman fractional linear shifts on the real line
326(7)
Iterations of fractional linear shifts and its properties
326(4)
Some additional properties of fractional linear shifts
330(1)
On a finite group of fractional linear transformations
331(2)
Singular integral equations with a generalized Carleman fractional linear shift
333(4)
Bibliographic Notes to Chapter 6
337(2)
Application to Hankel Type and Multidimensional Integral Equations
339(56)
Convolution integral equations of Hankel type
339(22)
Formulation of the main result
340(3)
The scheme of the proof and some basic ideas
343(4)
Proof of Theorem 34.1. The case p = 2, γ = 0
347(2)
Proof of Theorem 34.1. The general case
349(10)
The case of the equation corresponding to the diffraction problem
359(2)
Some multidimensional singular type equations with shifts
361(32)
Some properties of linear involutive transformations in Rn
361(5)
Wiener-Hopf operators with reflection in sectors on the plane
366(4)
Convolution operators with Carleman linear transform
370(9)
Equations with homogeneous kernels and the inversion shift in Rn
379(6)
Functional equations with shifts in different variables
385(3)
On a functional equation on the torus and factorization with shift
388(5)
Bibliographic Notes to Chapter 7
393(2)
Bibliography 395(24)
Index 419(6)
List of Symbols 425

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