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Published: 
December 8, 2002

Keywords: 
Douglas algebra, Nikolski ideal, singular integral operator, Fredholmness, rearrangementinvariant space

Subject: 
Primary 47B35, 47B38, 47A53; Secondary 46E30

Abstract:

We construct a presymbol for the Banach algebra $\alg(\Omega, S)$
generated by the Cauchy singular integral operator $S$ and the operators
of multiplication by functions in a Banach subalgebra $\Omega$ of $L^\infty$.
This presymbol is a homomorphism
$\alg(\Omega,S)\to\Omega\oplus\Omega$ whose kernel
coincides with the commutator ideal of $\alg(\Omega,S)$. In terms of the presymbol,
necessary conditions for Fredholmness of an operator in $\alg(\Omega,S)$
are proved. All operators are considered on reflexive rearrangementinvariant
spaces with nontrivial Boyd indices over the unit circle.

Acknowledgments:
The author is partially supported by F.C.T. (Portugal) grants POCTI 34222/MAT/2000 and PRAXIS XXI/BPD/22006/99.
Author information:
Departamento de Matem\'{a}tica, Instituto Superior T\'{e}cnico, Av. Rovisco Pais, 1049001 Lisboa, Portugal
akarlov@math.ist.utl.pt
http://www.math.ist.utl.pt/~akarlov
 