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\title{One-sided invertibility of functional operators
in rearrangement-invariant spaces}
\author{Alexei Karlovich, IST, Lisbon, Portugal}
\date{~}
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Let $\Gamma$ be a closed Jordan smooth curve and let $\alpha$
be a {\it diffeomorphism} of $\Gamma$ onto itself which {\it preserves} 
or
{\it changes} the orientation of $\Gamma$. Suppose the set $\Lambda$ of 
periodic points of $\alpha$ is arbitrary {\it non-empty} set. This talk 
is devoted to criteria of one-sided invertibility of the functional operators
\[
A=aI-bW,
\]
where $a$ and $b$ are continuous functions, $I$ is the identity
operator, $W$ is the shift operator: 
\[
(Wf)(t)=f[\alpha(t)],\quad t\in\Gamma,
\]
in  {\it reflexive rearrangement-invariant spaces of fundamental
type $X(\Gamma)$ with nontrivial Boyd indices}. 
These spaces
generalize classic Lebesgue, Orlicz, and Lorentz spaces.
As a corollary, the spectrum of the weighted shift operator
$gW$ with continuous coefficient $g$ is calculated. In particular,
the spectrum of $W$ in the space $X(\Gamma)$ with distinct Boyd indices
is ``massive'', i.e., it has a non-zero two-dimensional
Lebesgue measure. 

This talk is based on a joint work with Yu.~I.~Karlovich.
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