"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…mehr
"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.
Demonstrates a fascinating interplay between abstract and concrete operator theory. 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. Self-contained, covering a broad range of topics and results. Initial chapters present required background material and lead to coverage of state-of-the art topics. Good examples, bibliography, index, and suggestions of open problems. For students and specialists of operator theory, integral equations, and function theory, as well as engineers using integral equation techniques.
Inhaltsangabe
1 On Fredholmness of Singular Type Operators.- 1 Fredholm operators.- 2 Singular integral operators with piecewise continuous coefficients in the space Lp(?).- 3 On Fredholmness of convolution type operators.- 4 Bibliographic notes to Chapter 1.- 2 On Fredholmness of Other Singular-type Operators.- 5 On operators with homogeneous kernels; the one-dimensional case.- 6 Operators with homogeneous kernels; the multi-dimensional case.- 7 Convolution-type operators with discontinuous symbols.- 8 Bibliographic notes to Chapter 2.- 3 Functional and Singular Integral Equations with Carleman Shifts in the Case of Continuous Coefficients.- 9 Carleman and generalized Carleman shifts1ll.- 10 A functional equation with shift.- 11 Singular integral equations with Carleman shift on a closed curve; the case of continuous coefficients.- 12 Singular integral equations with Carleman shift on an open curve; the case of continuous coefficients.- 13 Bibliographic notes to Chapter 3.- 4 Two-term Equations (A + QB)? = f with an Involutive Operator Q; an Abstract Approach and Applications.- 14 Fredholmness of an abstract equation with an involutive operator; non-matrix approach.- 15 Application to singular integral equations with complex conjugate unknowns.- 16 Applications to integral equations on the real line with reflection or inversion.- 17 Application to singular integral equations with Carleman shift on an open curve; the case of discontinuous coefficients.- 18 Singular integral equations with a fractional linear shift in the space Lp with a special weight.- 19 Fredholmness of abstract equations with a generalized involutive operator (non-matrix approach).- 20 Abstract equations with algebraic operators.- 21 Bibliographic notes to Chapter 4.- 5 Equations with Several GeneralizedInvolutive Operators. Matrix Abstract Approach and Applications.- 22 Fredholmness of abstract equations with generalized involutive operators (matrix approach).- 23 Singular integral equations with a finite group of shifts in the case of continuous coefficients.- 24 Singular integral equations with a finite group of shifts (the case of piecewise continuous coefficients).- 25 Convolution type equations with shifts and complex conjugation.- 26 Bibliographic notes to Chapter 5.- 6 Application of the Abstract Approach to Singular Equation on the Real Line with Fractional Linear Shift.- 27 Singular integral operators perturbed by integral operators with homogeneous kernels.- 28 Singular integral operators with a fractional linear Carleman shift in the weighted space Lp?(Rl).- 29 Equations including operators with homogeneous kernels, the singular integral operators and the inversion shift.- 30. Potential type operators on the real 1ine with a fractional linear Carleman shift.- 31 Generalized Carleman fractional linear shifts on the real line . ..- 32 Singular integral equations with a generalized Carleman fractional linear shift.- 33. Bibliographic Notes to Chapter 6.- 7 Application to Hankel Type and Multidimensional Integral Equations.- 34 Convolution integral equations of Hankel type.- 35 Some multidimensional singular type equations with shifts.- 36 Bibliographic Notes to Chapter 7.- List of Symbols.
1 On Fredholmness of Singular Type Operators.- 1 Fredholm operators.- 2 Singular integral operators with piecewise continuous coefficients in the space Lp(?).- 3 On Fredholmness of convolution type operators.- 4 Bibliographic notes to Chapter 1.- 2 On Fredholmness of Other Singular-type Operators.- 5 On operators with homogeneous kernels; the one-dimensional case.- 6 Operators with homogeneous kernels; the multi-dimensional case.- 7 Convolution-type operators with discontinuous symbols.- 8 Bibliographic notes to Chapter 2.- 3 Functional and Singular Integral Equations with Carleman Shifts in the Case of Continuous Coefficients.- 9 Carleman and generalized Carleman shifts1ll.- 10 A functional equation with shift.- 11 Singular integral equations with Carleman shift on a closed curve; the case of continuous coefficients.- 12 Singular integral equations with Carleman shift on an open curve; the case of continuous coefficients.- 13 Bibliographic notes to Chapter 3.- 4 Two-term Equations (A + QB)? = f with an Involutive Operator Q; an Abstract Approach and Applications.- 14 Fredholmness of an abstract equation with an involutive operator; non-matrix approach.- 15 Application to singular integral equations with complex conjugate unknowns.- 16 Applications to integral equations on the real line with reflection or inversion.- 17 Application to singular integral equations with Carleman shift on an open curve; the case of discontinuous coefficients.- 18 Singular integral equations with a fractional linear shift in the space Lp with a special weight.- 19 Fredholmness of abstract equations with a generalized involutive operator (non-matrix approach).- 20 Abstract equations with algebraic operators.- 21 Bibliographic notes to Chapter 4.- 5 Equations with Several GeneralizedInvolutive Operators. Matrix Abstract Approach and Applications.- 22 Fredholmness of abstract equations with generalized involutive operators (matrix approach).- 23 Singular integral equations with a finite group of shifts in the case of continuous coefficients.- 24 Singular integral equations with a finite group of shifts (the case of piecewise continuous coefficients).- 25 Convolution type equations with shifts and complex conjugation.- 26 Bibliographic notes to Chapter 5.- 6 Application of the Abstract Approach to Singular Equation on the Real Line with Fractional Linear Shift.- 27 Singular integral operators perturbed by integral operators with homogeneous kernels.- 28 Singular integral operators with a fractional linear Carleman shift in the weighted space Lp?(Rl).- 29 Equations including operators with homogeneous kernels, the singular integral operators and the inversion shift.- 30. Potential type operators on the real 1ine with a fractional linear Carleman shift.- 31 Generalized Carleman fractional linear shifts on the real line . ..- 32 Singular integral equations with a generalized Carleman fractional linear shift.- 33. Bibliographic Notes to Chapter 6.- 7 Application to Hankel Type and Multidimensional Integral Equations.- 34 Convolution integral equations of Hankel type.- 35 Some multidimensional singular type equations with shifts.- 36 Bibliographic Notes to Chapter 7.- List of Symbols.
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