Functional Analysis and Applications Seminar 

11/10/2002, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Instituto Superior Técnico, U.T.L.

Algebras of Singular Integral Operators on Rearrangement-InvariantSpaces and Nikolski Ideals

We construct a presymbol for the Banach algebra $\operatorname{alg}(A, S)$ generated by the Cauchy singular integral operator $S$ and the operators of multiplication by functions in a Banach subalgebra $A$ of essentially bounded functions. This presymbol mapping is a homomorphism of $\operatorname{alg}(A,S)$ onto $A+A$ whose kernel coincides with the commutator ideal of $\operatorname{alg}(A,S)$. In terms of the presymbol, necessary conditions for Fredholmness of an operator in $\operatorname{alg}(A,S)$ are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.