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{\LARGE Compactness of some commutators\\[1ex] on rearrangement-invariant spaces}
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\centerline{\large \large A.~Karlovich }
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Commutators $aS_\Gamma-S_\Gamma aI$ and $W_\alpha S_\Gamma-S_\Gamma W_\alpha$ 
of the Cauchy singular integral operator $S_\Gamma$ with the operator of 
multiplication $aI$ by a function $a\in L^\infty(\Gamma)$ and with the shift 
operator $W_\alpha$  defined by $W_\alpha f=f\circ\alpha$ play an 
important role in  the Fredholm theory of singular integral operators 
with shifts. Compactness criteria for the first commutator on 
$L^2({\bf T})$ over the unit circle ${\bf T}$ are well known for a long 
time, while  compactness criteria for the second commutator on 
$L^2({\bf T})$ were obtained recently (1995) by P. Muhly and J. Xia 
when $\alpha$ is an orientation-preserving bi-Lipschitz homeomorphism of ${\bf T}$ 
onto itself.

We obtain analogous results for the case of the unit segment ${\bf J}:=[0,1]$.
Using interpolation of compact operators, we extend these results on 
general rearrangement-invariant spaces. Note that these spaces include 
classical Lebesgue, Orlicz, and Lorentz spaces.

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{\bf Theorem 1.}
{\it
Suppose $\Gamma$ is either the unit circle or the unit segment 
and $a\in L^\infty(\Gamma)$. The operator $aS_\Gamma-S_\Gamma aI$ 
is compact on a rearrangement-invariant space $X(\Gamma)$ 
with nontrivial Boyd indices if and only if $a$ has vanishing 
mean oscillation on $\Gamma$.
}

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{\bf Theorem 2.}
{\it
Let $\Gamma$ be either the unit circle or the unit segment.
Suppose $\alpha$ is an orientation-preserving bi-Lipschitz homeomorphism
of $\Gamma$ onto itself. 
If $\Gamma=[0,1]$ then suppose in addition that $\alpha(0)=0$ 
and $\alpha(1)=1$.  The operator $W_\alpha S_\Gamma-S_\Gamma W_\alpha$ 
is compact on a rearrangement-invariant space $X(\Gamma)$ 
with nontrivial Boyd indices if and only if $\alpha'$ has 
vanishing mean oscillation on $\Gamma$.
}

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Passage from ${\bf T}$ to ${\bf J}$ (``cutting'') involves an 
operator $N$ with two fixed singularities at the endpoints of 
half-circles. The main difficulty here consists of the 
proof of compactness for the commutator of $R$ with the 
multiplication operators. Note that the results of
Theorems 1 and 2 for ${\bf J}$ are new even in the case
of Lebesgue spaces $L^p({\bf J}), 1<p<\infty$.

Theorems 1 and 2 are obtained in collaboration with Yuri Karlovich.
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\small{Departemento de Matem\'atica, \\ Instituto Superior T\'ecnico,\\ Av. Rovisco Pais 1,\\ 1049-001, Lisboa, Portugal}
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