We construct a presymbol for the Banach algebra Alg(Ω, S) generated by the Cauchy singular integral operator S and the operators of multiplication by functions in a Banach subalgebra Ω of L^∞. This presymbol is a homomorphism Alg(Ω,S)→Ω⊕Ω whose kernel coincides 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-invariant spaces with nontrivial Boyd indices over the unit circle.

The author is partially supported by F.C.T. (Portugal) grants POCTI 34222/MAT/2000 and PRAXIS XXI/BPD/22006/99.