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New York Journal of Mathematics 8 (2002), 215-234.

Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals

Alexei Yu. Karlovich

Published: December 8, 2002
Keywords: Douglas algebra, Nikolski ideal, singular integral operator, Fredholmness, rearrangement-invariant space
Subject: Primary 47B35, 47B38, 47A53; Secondary 46E30


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 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.

Author information:
Departamento de Matem\'{a}tica, Instituto Superior T\'{e}cnico, Av. Rovisco Pais, 1049--001 Lisboa, Portugal