 

Alexei Yu. Karlovich
Algebras of Singular Integral Operators on RearrangementInvariant Spaces and Nikolski Ideals


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(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 rearrangementinvariant
spaces with nontrivial Boyd indices over the unit circle.


Acknowledgements
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 Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049001 Lisboa, Portugal
akarlov@math.ist.utl.pt
http://www.math.ist.utl.pt/~akarlov

