Introduction to the Theory of Singular Integral Operators with Shift, 1994 Mathematics and Its Applications Series, Vol. 289

Background information. Noetherity criterion and a formula for the index of a singular integral functional operator of first order in the continuous case. The Noether theory of a singular integral functional operator of finite order in the continuous case. The Noether theory of singular integral functional operators with continuous coefficients on a non-closed contour. The Noether theory in algebras of singular integral functional operators.

1. Background information.- 2.Noetherity criterion and a formula for the index of a singular integral functional operator of first order in the continuous case.- 3. The Noether theory of a singular integral functional operator of finite order in the continuous case.- 4. The Noether theory of singular integral functional operators with continuous coefficients on a non-closed contour.- 5. The Noether theory in algebras of singular integral functional operators.- References.

This book is devoted to the Fredholm theory of singular integral operators with shift Lp, 1_ spaces. Fredholm criteria are derived and the indices of the Fredholm operators are computed. Simultaneously, a theory of continuous invertibility for functional operators is constructed, and its relation to the theory of dynamical systems is discussed. Also, a new systematic approach to the Fredholm theory of classical singular integral equations with Cauchy kernel is proposed.