Equations With Involutive Operators

, by Karapetiants, Nikolai; Samko, Stefan G.

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ISBN: 9780817641573 | 0817641572
Cover: Hardcover
Copyright: 7/1/2001

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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.

Introduction

Notation

xxi

On Fredholmness of Singular Type Operators

(50)

Fredholm operators

(13)

Basics on Fredholm operators

(4)

Fredholmness of operators with projectors

(4)

Fredholmness of matrix operators

(5)

Singular integral operators with piecewise continuous coefficients in the space Lp(Γ)

(12)

Singular integral operators in the space Lp(Gamma;)

(2)

The notion of p-index

(5)

Fredholmness criterion and the index formula

(3)

Approximation of non-Fredholm operators by Fredholm ones

(1)

On compactness of some ``composite'' singular operators

(1)

On Fredholmness of convolution type operators

(22)

Fourier transforms and the Wiener algebra

(4)

General preliminaries on convolution operators

(4)

Theorems on Fredholmness

(2)

On compactness of some convolution type operators

(5)

On Fredholmness of convolution type operators with ``variable'' coefficients

(2)

Discrete convolution type equations

(2)

On Hankel operators

(3)

Bibliographic notes to Chapter 1

(3)

On Fredholmness of Other Singular-type Operators

(60)

On operators with homogeneous kernels; the one-dimensional case

(17)

Connection with convolution operators; Lp-boundedness

(1)

On Fredholmness of the operators λI -- K

(12)

The Carleman equation and notion of the standard α-lemniscate

(3)

Operators with homogeneous kernels; the multi-dimensional case

(16)

Lp-boundedness

(3)

On spherical harmonics

(3)

Formal reduction to a system of one-dimensional equations with homogeneous kernels

(2)

Justification of the reduction and the main result for rotation invariant homogeneous kernels

(5)

Some cases of non-rotation invariant kernels and other types of homogeneity

(2)

Convolution-type operators with discontinuous symbols

(25)

Setting of the problem

(1)

Lizorkin space

(1)

Fractional integration of Liouville and Bessel type

(3)

The case of a continuous symbol

(3)

The case of a discontinuous coefficient G(x)

(1)

Fourier transform of a certain function

(1)

On a special group of operators

(3)

The model operator with a discontinuous symbol

(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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Equations With Involutive Operators

Summary

Table of Contents

Supplemental Materials