Equations With Involutive Operators

by Karapetiants, Nikolai; Samko, Stefan G.

• ISBN13: 9780817641573
• ISBN10: 0817641572
• Format: Hardcover
• Copyright: 2001-07-01
• Publisher: Birkhauser

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