One-Dimensional Linear Singular Integral Equations

1 The operator of singular integration.- 1.1 Notations, definitions and auxiliary statements.- 1.2 The boundedness of the operator S? in the space Lp(?) with ? being a simple curve.- 1.3 Nonsimple curves.- 1.4 Integral operators in weighted Lp spaces.- 1.5 Unbounded curves.- 1.6 The operator of singular integration in spaces of Hölder continuous functions.- 1.7 The operator S?*.- 1.8 Exercises.- Comments and references.- 2 One-sided invertible operators.- 2.1 Direct sum of subspaces.- 2.2 The direct complement.- 2.3 Linear operators. Notations and simplest classes.- 2.4 Projectors connected with the operator of singular integration.- 2.5 One-sided invertible operators.- 2.6 Singular integral operators and related operators.- 2.7 Examples of one-sided invertible singular integral operators.- 2.8 Two lemmas on the spectrum of an element in a subalgebra of a Banach algebra.- 2.9 Subalgebras of a Banach algebra generated by one element.- 2.10 Exercises.- Comments and references.- 3 Singular integral operators with continuous coefficients.- 3.1 The index of a continuous function.- 3.2 Singular integral operators with rational coefficients.- 3.3 Factorization of functions.- 3.4 The canonical factorization in a commutative Banach algebra.- 3.5 Proof of the factorization theorem.- 3.6 The local factorization principle.- 3.7 Operators with continuous coefficients.- 3.8 Approximate solutions of singular integral equations.- 3.9 Generalized factorizations of continuous functions.- 3.10 Operators with continuous coefficients (continuation).- 3.11 Additional facts and generalizations.- 3.12 Operators with degenerating coefficients.- 3.13 A generalization of singular integral operators with continuous coefficients.- 3.14 Solution of Wiener-Hopf equations.- 3.15 Someapplications.- 3.16 Exercises.- Comments and references.- 4 Fredholm operators.- 4.1 Normally solvable operators.- 4.2 The restriction of normally solvable operators.- 4.3 Perturbation of normally solvable operators.- 4.4 The normal solvability of the adjoint operator.- 4.5 Generalized invertible operators.- 4.6 Fredholm operators.- 4.7 Regularization of operators. Applications to singular integral operators.- 4.8 Index and trace.- 4.9 Functions of Fredholm operators and their index.- 4.10 The structure of the set of Fredholm operators.- 4.11 The Dependence of kerX and imX on the operator X.- 4.12 The continuity of the function kx.- 4.13 The case of a Hilbert space.- 4.14 The normal solvability of multiplication by a matrix function.- 4.15 ?±-operators.- 4.16 One-sided regularization of operators.- 4.17 Projections of invertible operators.- 4.18 Exercises.- Comments and references.- 5 Local Principles and their first applications.- 5.1 Localizing classes.- 5.2 Multipliers on $$ \mathop l\limits^ \sim _p $$.- 5.3 paired equations with continuous coefficients on $$ \mathop l\limits^ \sim _p $$.- 5.4 Operators of local type.- 5.5 Exercises.- Comments and references.- References.