Introductlon to the theory of singular lntegral operators .,th shlft I by V,ctor G. Kravchenko and G.orgll S. Lltvlnchuk. p. cm. --(Mathematlcs and ltS appl icatlons ; v. 289) Includes blbllographlcal references and index.

A quaternionic partial differential equation is shown to be a generalisation of the traditional Riccati equation and its relationship with the Schrödinger equation is established. Various approaches to the problem of finding particular solutions to this equation are explored, and the generalisations of two theorems of Euler on the Riccati equation, which correspond to this partial differential equation, are stated and proved.

No abstract

Key words Singular integral operators with shift, operator factorization, factorization of matrix functions MSC (2000) Primary: 47G10; Secondary: 47A68 Dedicated to Professor Frank-Olme Speck on the occasion of his 60th birthdayThis paper deals with what we call modified singular integral operators. When dealing with (pure) singular integral operators on the unit circle with coefficients belonging to a decomposing algebra of continuous functions it is known that a factorization of the symbol induces a factorization of the original operator, which is a representation of the operator as a product of three singular integral operators where the outer operators in that representation are invertible.The main purpose of this paper is to obtain a similar operator factorization for the case of singular integral operators with a backward shift and to extract from there some consequences for their Fredholm characteristics. At the end of the paper it is shown that the operator factorization is also possible for other classes of singular integral operators, namely those including either a conjugation operator or a composition of a conjugation with a forward shift operator.

A method is developed for investigating the Noetherian property of operators of a particular kind, prototypes of which are singular integral operators with a shift (SIOS). The algebra of symbols of SIOS is constructed. Questions of invertibility of the symbols are considered. Algorithms are obtained for verifying conditions for the Noetherian property and for computing the index of SIOS.Bibliography: 31 titles.* Editor's note. The reader will recognize that the Noetherian operators of the Russian literature are just the Fredholm operators of the Western literature.