Algebras of Singular Integral Operators on Rearrangement-Invariant Spaces and Nikolski Ideals

摘要

We construct a presymbol for the Banach algebra Alg(Ω, S)generated by the Cauchy singular integral operator S and the operatorsof multiplication by functions in a Banach subalgebra Ω of L∞.This presymbol is a homomorphism Alg(Ω,S)→Ω⊕Ω whose kernelcoincides with the commutator ideal of Alg(Ω,S). In terms of the presymbol,necessary conditions for Fredholmness of an operator in Alg(Ω,S)are proved. All operators are considered on reflexive rearrangement-invariantspaces with nontrivial Boyd indices over the unit circle.