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{\bf 
Alexei Karlovich

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Invertibility of functional operators
\\
with non-Carleman shift
\\
in rearrangement-invariant spaces
}
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Let $\omega$ be a diffeomorphism (shift) of $[0,1]$ 
onto itself such that $\omega(0)=0$ and $\omega(1)=1$, 
but $\omega(t)\ne t$ for $t\in(0,1)$. 
This talk is devoted to criteria for the two- and 
one-sided invertibility of functional operators
\[
A=aI-bW_\omega,
\]
where $a$ and $b$ are continuous functions on $[0,1]$, 
$I$ is the identity operator, $W_\omega$ is the shift 
operator: $W_\omega f=f\circ \omega$, in a reflexive 
rearrangement-invariant space $X(0,1)$ which Boyd 
indices $\alpha_X, \beta_X$ and Zippin (fundamantal)
indices $p_X,q_X$ satisfy
\[
0<\alpha_X=p_X\le q_X=\beta_X<1.
\]
These spaces are a wide
generalization of classic Lebesgue, Orlicz, and Lorentz 
spaces.

Moreover, we prove criteria for the two-sided invertibility
of $A$ in Lebesgue spaces $L^p(0,1),1<p<\infty$, under
weaker assumptions on $a,b$ and $\omega'$. We assume that
$a,b$ and $\omega'$ are continuous on $(0,1)$ and slowly
oscillate at $0$ and $1$, but they may not have one-sided limits
at $0$ and $1$.

A part of these results are obtained jointly with Amarino Lebre
and Yuri Karlovich.
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