## Introduction to the Theory of Singular Integral Operators with Shiftproblem (0. 2) was the same u that of problem (0. 1). Incidentally, later on Mandzhavidze and Khvedclidze (I) and Simonenko (I) achieved a direct reduction of problem (0. 2) to problem (0. 1) with the help of conformal mappings. Apparenlly, the first paper in which SIES were considered was the paper by Vekua (2) published in 1948. Vekua verified that the equation (0. 3) where (1; C(f), 5 is the operator of 'ingular integration with a Cauchy kernel (Srp)(!) "" (." i)-I fr(T - t)-lrp(T)dT, W is the shift operator (WrpHt) = rp{a(t, in the case 01 = - (13,0, = 0., could be reduced to problem (0. 2). We note thai, in problem (0. 2), the shift ott) need not be a Carlemao shift, . ei., it is oot necessary that a . . (t):::: t for some integer 11 2, where ai(l) "" o(ok_dt)), 0(1(1):::: !. For the first time, the condition 0, (1) == 1 appeared in BPAFS theory in connection with the study of the problem (0. 4) by Carle man (2) who, in particular, showed that problem (0. 4) Wall a natural generalization of the problem on the existence of an a. utomorphic function belonging to a certain group of Fucs. Thus, the paper by Vckua (2) is also the fint paper in which a singular integral equation with a non.Carieman 5hifl is on c sidered." |

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### Contents

Background information | 1 |

Noetherity criterion and a formula for the index of a singular integral | 37 |

2 The calculation of the index of a singular integral functional operator of the first | 60 |

Copyright | |

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### Other editions - View all

Introduction to the Theory of Singular Integral Operators with Shift Viktor G. Kravchenko,Georgii S. Litvinchuk No preview available - 2012 |

Introduction to the Theory of Singular Integral Operators with Shift Viktor G. Kravchenko,Georgii S. Litvinchuk No preview available - 2014 |

### Common terms and phrases

a-reducible a-solution abstract scheme algebra analogously arbitrary Banach Banach space belongs bounded operator C*-algebra C*-subalgebra Carleman shift Cauchy index Cauchy kernel changes the orientation Chapter closed contour Cnxn(T cokernel compact operator Consequently consider construction continuous functions continuously invertible operator Corollary corresponding defined Definition denote diffeomorphism equality equation essential kernel exists finite number fixed points follows Fredholm operator fulfilled holds homeomorphism homotopy index formula inequalities integral functional operator isomorphism Karlovich and Kravchenko Kveselava-Vekua Lemma Litvinchuk matrix a(t matrix functions Noether theory Noetherian operator Noetherity criterion non-Carleman shift non-closed norm normal form obtain operator A(a operator Ac operator Ta,b orientation-preserving shift paired operator periodic points point t0 preserves the orientation problem proved satisfying the conditions Section sequence set of periodic shift a(t shift operator SIFO singular integral functional singular integral operator solution solvable space LP(T subalgebra subspaces Suppose symbol unit circle zero