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\Year{2001}
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\newcommand{\cA}{{\cal A}}
\newcommand{\cG}{{\cal G}}
\newcommand{\cV}{{\cal V}}
\newcommand{\cW}{{\cal W}}
\newcommand{\fB}{{\frak B}}
\newcommand{\fL}{{\frak L}}
\newcommand{\I}{{\Bbb I}}
\newcommand{\V}{{\Bbb V}}
\newcommand{\eps}{{\varepsilon}}
\newcommand{\vka}{{\varkappa_h}}        % limit of a test sequence
\newcommand{\wa}{{\widetilde{\alpha}}}  % for shift \alpha
\newcommand{\wb}{{\widetilde{\beta}}}   % for inverse shift \beta
\newcommand{\uniform}{{\ \rightrightarrows\ }}
\newcommand{\slim}{{s\!-\!\!\lim\limits_{n\to\infty}}}
\newcommand{\Ker}{{\rm Ker}\,}
\newcommand{\Coker}{{\rm Coker}\,}
\newcommand{\supp}{{\rm supp}\,}
\newcommand{\rang}{{\rm rank}\,}
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\begin{document}
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\Title{Invertibility of Functional Operators with Slowly Oscillating\\[2ex] Non-Carleman Shifts}
\Shorttitle{Invertibility of Functional Operators with Slowly Oscillating Shifts}
\By{{\sc Alexei~Karlovich, Yuri~Karlovich, Amarino~Lebre}}
\Names{A.~Karlovich, Yu.~Karlovich, A.~Lebre}
\Email{akarlov@math.ist.utl.pt} \maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%                   Please insert now the article body.                       %
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\begin{abstract}
We prove criteria for the invertibility of the binomial functional
operator
\[
A=aI-bW_\alpha
\]
in the Lebesgue spaces $L^p(0,1)$, $1<p<\infty$, where $a$ and $b$
are continuous functions on $(0,1)$, $I$ is the identity operator,
$W_\alpha$ is the shift operator, $W_\alpha f=f\circ\alpha$,
generated by a non-Carleman shift $\alpha:[0,1]\to[0,1]$ which
has only two fixed points $0$ and $1$. We suppose that
$\log\alpha'$ is bounded and continuous on $(0,1)$ and that
$a,b,\alpha'$ slowly oscillate at $0$ and $1$. The main difficulty
connected with slow oscillation is overcome by using the method of
limit operators.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{-0.5cm}
\newsection{Introduction}
%%%
Let $\alpha$ be an orientation preserving homeomorphism of $[0,1]$
onto itself which has only two fixed points $0$ and $1$. So,
$\alpha(0)=0$ and $\alpha(1)=1$, but $\alpha(t)\ne t$ for $t\in \I:=(0,1)$.
The function $\alpha$ is referred to as a shift function ({\it shift}).

Denote by $\beta:=\alpha_{-1}$ the inverse function to $\alpha$. Since
$\alpha$ and $\beta$ strictly monotonically increase on $[0,1]$,
their derivatives exist and are positive almost everywhere on $\I$.
If $\alpha$ and $\beta$ are absolutely continuous on $[0,1]$ and
$\log\alpha'\in L^\infty:=L^\infty(\I)$, then the shift operator
$W_\alpha$ defined by
\[
(W_\alpha\varphi)(t):=\varphi[\alpha(t)],
\quad
t\in \I,
\]
is bounded in the Lebesgue space $L^p:=L^p(\I)$ for every
$p\in[1,\infty]$. Moreover, it is invertible and its inverse is
given by $W_\alpha^{-1}=W_{\beta}$.

In what follows we always suppose that $p\in(1,\infty)$ and that
$q=p/(p-1)$ is the conjugate exponent to $p$.

In the space $L^p$ we consider the binomial functional operator with shift
%%%
\begin{equation}\label{eq:operator}
A:=aI-bW_\alpha
\end{equation}
%%%
where $I$ the identity operator, and the coefficients $a,b$ are bounded and
continuous ($BC$) functions on $\I$.

The investigation of two- and one-sided invertibility of functional
operators (in particular, of \reff{eq:operator}) in various functional spaces
plays an important role in the theory of functional differential operators
(see, e.g., \cite{A95,AL94,Kur99}), the theory of singular integral operators,
convolution type operators, and pseudodifferential operators with shifts
and/or oscillating coefficients (see \cite{ABL98,KL94,L77,L00} and
references therein), the theory of dynamical systems \cite{CL99}, etc.

In the case of ``nice'' shifts and coefficients the following invertibility
criterion was actually proved in \cite{K74}
(see also \cite[Ch.~2, Th.~$4^\prime$]{KL94}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:nice}
Suppose $a,b$ are continuous functions on $[0,1]$ and $\alpha$ is an
orientation preserving diffeomorphism of $[0,1]$ onto itself.
The operator \reff{eq:operator} is invertible in the Lebesgue space
$L^p,p\in(1,\infty)$, if and only if
%%%
\begin{eqnarray*}
&\mbox{either}&
\min_{t\in[0,1]}|a(t)|>0,
\quad
|a(j)|>|b(j)|\Big(\alpha'(j)\Big)^{-1/p},
\quad
j\in\{0,1\};
\\
&\mbox{or}&
\min_{t\in[0,1]}|b(t)|>0,
\quad
|a(j)|<|b(j)|\Big(\alpha'(j)\Big)^{-1/p},
\quad
j\in\{0,1\}.
\end{eqnarray*}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Under the same assumptions on the smoothness of shifts and coefficients,
this theorem was generalized in \cite{KK76} to the case of an arbitrary
nonempty set
of periodic points of shifts. Further, the one-sided invertibility of
\reff{eq:operator} in Lebesgue spaces and in more general
rearrangement-invariant spaces was studied in \cite{AK89,K01,KK01,M85}
(see also the surveys \cite{K95,KKL90}).

In contrast to the classical assumptions \cite{K95,KKL90,KL94,L77,L00},
we do not assume the existence of one-sided limits of the coefficients and
one-sided derivatives of the shift at the fixed points $0$ and $1$.
In this paper we prove invertibility criteria for the operator
\reff{eq:operator} with slowly oscillating data in Lebesgue spaces $L^p$.

For a function $\varphi\in BC$, consider its oscillation on a set $J\subset\I$:
\[
\omega(\varphi,J):=\sup_{x,y\in J}|\varphi(x)-\varphi(y)|.
\]
Following \cite{S77}, a function $\varphi\in BC$
is called a {\it slowly oscillating function at} $0$ if for every $\lambda\in \I$,
\[
\lim_{r\to 0}\omega(\varphi,[\lambda r,r])=0,
\quad\Big(\mbox{or equivalently,}\quad
\lim_{r\to 0}\omega(\varphi,[r/2,r])=0
\Big).
\]
In this case we will write $\varphi\in SO$. A function $\varphi\in BC$
is called slowly oscillating at $1$ if the function
$t\mapsto \varphi(1-t)$ belongs to $SO$.
We say that a non-Carleman shift $\alpha$ is a
{\it slowly oscillating shift at} $j\in\{0,1\}$ if $\log \alpha'\in BC$ and
$\alpha'$ is a slowly oscillating function at $j$, or, equivalently
(see Lemma~\rref{le:SO-equivalence}), if $\log\alpha'\in BC$ and
\[
\lim_{t\to j}(t-j)\frac{d}{dt}\left(\frac{\alpha(t)-\alpha(j)}{t-j}\right)=0.
\]
We say that the operator \reff{eq:operator} has {\it  slowly
oscillating data $\{a,b;\alpha\}$ at the endpoint} $j\in\{0,1\}$ if the
coefficients $a,b$ are slowly oscillating functions at $j$ and $\alpha$
is a slowly oscillating shift at $j$.

Operators with slowly oscillating (Carleman) shifts were studied for the first
time in \cite{KL01}. In that paper singular integral operators with piecewise
continuous coefficients and slowly oscillating Carleman backward shifts were
interpreted as Mellin pseudodifferential operators with slowly oscillating
symbols. Further, by applying the method of limit operators (see also
\cite{BKR00,RRS98}), the investigation of Mellin pseudodifferential
operators was reduced to the investigation of Mellin convolutions. Studying
Mellin pseudodifferential operators requires that the shift and the coefficients
of the initial operator to be infinitely differentiable (apart from the fixed points). 
These assumptions seem to be redundant, and it is natural to exclude
them. For this purpose,  in this paper we apply the method of limit operators
directly to the operator \reff{eq:operator}. This allows us to overcome the main
difficulty connected  with slow oscillation and to get necessary conditions
for the invertibility of operators \reff{eq:operator} at the fixed points.

The paper is organized as follows. In Section 2 we assume that
$a,b\in BC$ and that $\alpha'$ is invertible in $BC$. Under these
assumptions we formulate sufficient conditions for the two- and
one-sided invertibility of the operator \reff{eq:operator}. We
also  prove that under respective assumptions the right (left)
invertible operator \reff{eq:operator} is not invertible because
its kernel (cokernel) has dimension infinity. Clearly, these
conditions for strict one-sided invertibility give us necessary
conditions for two-sided invertibility. But there is a gap
between sufficient and necessary conditions for the two-sided
invertibility at the fixed points. This gap is connected with the fact
that we do not assume the existence of the one-sided limits at $0$
and $1$ for the coefficients $a,b$ and for the derivative of the
shift $\alpha$.

To fill this gap we have to assume that the operator
\reff{eq:operator} has slowly oscillating data $\{a,b;\alpha\}$ at
$0$ and $1$. In Section 3 we uncover some properties of slowly
oscillating functions and shifts. The method of limit operators
allows us to reduce the investigation of the invertibility of the
operator \reff{eq:operator} with ``bad'' (slowly oscillating) data
to the investigation of the invertibility of a family of
operators (so called limit operators) of the same form but with
``nice'' coefficients (continuous on $[0,1]$) and ``nice'' shifts
(diffeomorphisms of $[0,1]$ onto itself).  On the basis of the results
from Section 3, in Section 4 we calculate the limit operators for the
operator \reff{eq:operator}. We emphasize that these
limit operators already satisfy the conditions of
Theorem~\rref{th:nice}.

In Section 5, following the scheme of \cite{K01} and using the 
results of Sections 2 and~4, we prove our main result:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:main}
Suppose the operator \reff{eq:operator} has slowly oscillating data
$\{a,b;\alpha\}$ at the two endpoints $0$ and $1$.
The operator \reff{eq:operator} is invertible in the Lebesgue
space $L^p,p\in(1,\infty)$,  if and only if
\begin{eqnarray*}
&\mbox{either}& \inf_{t\in\I}|a(t)|>0,
\quad
\liminf_{t\to j}\left( |a(t)|-|b(t)|\Big(\alpha'(t)\Big)^{-1/p}\right)>0,
\quad j\in\{0,1\};
\\
&\mbox{or}& \inf_{t\in\I}|b(t)|>0,
\quad
\limsup_{t\to j}\left(|a(t)|-|b(t)|\Big(\alpha'(t)\Big)^{-1/p}\right)<0,
\quad j\in\{0,1\}.
\end{eqnarray*}
\end{theorem}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsection{Sufficient conditions for one-sided invertibility}
\vspace*{-1cm}
\newsubsection{Sufficient conditions for invertibility}
%%%
For a Banach algebra ${\frak A}$ denote by $\cG{\frak A}$ the group of all
invertible elements in ${\frak A}$. We denote by $C$ the set of all 
continuous functions $f:\I\to\C$. For $f\in C$, put
\[
f_*(j):=\liminf_{t\to j}|f(t)|,\quad
f^*(j):=\limsup_{t\to j}|f(t)|,
\quad j\in\{0,1\}.
\]
Denote by $B$ the set of all bounded functions which are defined
everywhere on $\I$ (in contrast to $L^\infty$). Obviously, $BC=C\cap B$.
Clearly, if $f\in BC$ then the upper limits $f^*(0)$ and $f^*(1)$
are finite. It is easy to see that a function $f\in BC$ belongs to
$\cG BC$ if and only if $f(t)\ne 0$ for every $t\in\I$ and  the
lower limits $f_*(0)$ and $f_*(1)$ are positive.

Fix an arbitrary point $x\in\I$. Let $l$
be a half-open segment with endpoints $x$ and $\alpha(x)$ such that
$x\in l$ but $\alpha(x)\not\in l$. Notice that either $x<\alpha(x)$ and then
$1$ is the attracting point of $\alpha$, or $\alpha(x)<x$ and then $0$ is the
attracting point of $\alpha$ (see, e.g., \cite[Ch.~1, Section~3]{KL94}).
Put $\alpha_0(t)=t$ and $\alpha_n(t)=\alpha[\alpha_{n-1}(t)]$
for $n\in\Z, t\in[0,1]$. Assume for definiteness that $0$ is the repelling 
point and $1$ is the attracting point for the shift $\alpha$, that is,
\[
\lim_{n\to\infty}\alpha_{-n}(t)=0,
\quad
\lim_{n\to\infty}\alpha_n(t)=1,
\quad
t\in\I.
\]
The opposite case, when $1$ is the repelling point and $0$ is the attracting 
point, can be considered absolutely analogously.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:sp-Lp}
Suppose $g\in BC$ and $\alpha'\in {\cal G}BC$. The spectral radius of
the weighted shift operator $gW_\alpha$ satisfies the estimate
%%%
\begin{equation}\label{eq:sp-Lp-1}
r(gW_\alpha)\le\max\Big\{M^*(0),M^*(1)\Big\},
\end{equation}
%%%
where $M(t):=g(t)(\alpha'(t))^{-1/p}$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
This proof is a slight generalization of the proof given in
\cite[Theorem~37.1]{L77}, (see also \cite[Ch.~2, Lemmas~6-7]{KL94}
and \cite[Theorem~2.2]{A95}).

Since  $\I$ is invariant with respect to the shift $\alpha_n$
for arbitrary $n\in\N$, we have
%%%
\begin{eqnarray}
\label{eq:sp-Lp-2}
\|(gW_\alpha)^n\varphi\|_p
&=&
\left(\int_0^1
\left|\prod_{k=0}^{n-1}g[\alpha_{k-n}(t)]\right|^p|\varphi(t)|^p|\alpha'_{-n}(t)|\, dt
\right)^{1/p}
\\
&\le&
\sup_{t\in \I}
\left(
\left|\prod_{k=0}^{n-1}g[\alpha_{k-n}(t)]\right||\alpha'_{-n}(t)|^{1/p}
\right)\|\varphi\|_p.
\nonumber
\end{eqnarray}
%%%
Clearly,
%%%
\begin{equation}\label{eq:sp-Lp-3}
\left|\prod_{k=0}^{n-1}g[\alpha_{k-n}(t)]\right||\alpha'_{-n}(t)|^{1/p}
=
\prod_{k=1}^n\frac{|g[\alpha_{-k}(t)]|}{(\alpha'[\alpha_{-k}(t)])^{1/p}}.
\end{equation}
%%%
Since $g$ is bounded and $\alpha'$ is bounded away from zero on $ \I$, the
quantities $M^*(0)$ and $M^*(1)$  are finite.
Fix $\eps>0$. From the definition of $M^*(j),j\in\{0,1\}$, it follows that
there exist points $t_1$ and $t_2$ such that
%%%
\begin{eqnarray}
\label{eq:sp-Lp-4}
&&
\frac{|g(t)|}{(\alpha'(t))^{1/p}}<M^*(0)+\eps,\ t\in(0,t_1),
\quad
\frac{|g(t)|}{(\alpha'(t))^{1/p}}<M^*(1)+\eps,\ t\in(t_2,1).
\end{eqnarray}
%%%

Since $1$ is the attracting point of the shift $\alpha$, there
exists a number
$k_0\in\N$ such that $\alpha_{k_0}(t_1)\in (t_2,1)$. Put
\[
L:=\sup_{t\in  \I}\frac{|g(t)|}{(\alpha'(t))^{1/p}},
\quad
Q:=\sup_{t\in(0,t_1)\cup(\alpha_{k_0}(t_1),1)}
\frac{|g(t)|}{(\alpha'(t))^{1/p}}.
\]
Then we have for every $t\in \I$,
%%%
\begin{equation}\label{eq:sp-Lp-6}
\prod_{k=1}^n\frac{|g[\alpha_{-k}(t)]|}{(\alpha'[\alpha_{-k}(t)])^{1/p}}
\le L^{k_0}Q^{n-k_0}.
\end{equation}
%%%
On the other hand, we see from \reff{eq:sp-Lp-4} that
%%%
\begin{equation}\label{eq:sp-Lp-7}
Q\le\max\Big\{M^*(0),M^*(1)\Big\}+\eps.
\end{equation}
%%%
Combining \reff{eq:sp-Lp-2}, \reff{eq:sp-Lp-3}, \reff{eq:sp-Lp-6},
\reff{eq:sp-Lp-7}, we obtain
\[
\|(gW_\alpha)^n\|_{\fB(L^p)}
\le
L^{k_0}
\left(
\max\Big\{M^*(0),M^*(1)\Big\}+\eps
\right)^{n-k_0}.
\]
Applying the spectral radius formula %for the spectral radius
to the latter
inequality, we get
\[
r(gW_\alpha)
\le
\max\Big\{M^*(0),M^*(1)\Big\}+\eps.
\]
Since $\eps$ is arbitrary, this inequality gives us
\reff{eq:sp-Lp-1}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For further generalizations of the formula for the spectral radius of
a weighted shift operator see \cite[Ch.~1, Section~5]{A95}.

 From Lemma~\rref{le:sp-Lp}, using standard arguments (see, e.g.,
\cite[Section~3]{KK01}), we get the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:2sided}
Suppose $a,b\in BC$ and $\alpha'\in\cG BC$.

{\rm (a)} If $b\in{\cal G}BC$ and
\[
\max\left\{
\left(\frac{a}{b}(\alpha'_{-1})^{-1/p}\right)^*(0),
\left(\frac{a}{b}(\alpha'_{-1})^{-1/p}\right)^*(1)
\right\}<1,
\]
then the operator \reff{eq:operator} is invertible, and
\[
A^{-1}=-W_\alpha^{-1}\sum_{n=0}^\infty (b^{-1}aW_\alpha^{-1})^nb^{-1}I.
\]

{\rm (b)} If $a\in{\cal G}BC$ and
\[
\max\left\{
\left(\frac{b}{a}(\alpha')^{-1/p}\right)^*(0),
\left(\frac{b}{a}(\alpha')^{-1/p}\right)^*(1)
\right\}<1,
\]
%%%
then the operator \reff{eq:operator} is invertible, and
\[
A^{-1}=\sum_{n=0}^\infty (a^{-1}bW_\alpha)^na^{-1}I.
\]
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Sufficient conditions for right invertibility}
%%%
Put
\[
\gamma_\pm:=\bigcup_{k=1}^\infty\alpha_{\pm k}(l).
\]
Consider functions $\chi_0,\chi_1\in C[0,1]$ such that
$\chi_0(t)+\chi_1(t)=1$ for $t\in[0,1]$ and $\chi_0(t)=1$ for 
$t\in\gamma_-$, $\chi_1(t)=1$ for $t\in\gamma_+$.

Using slight modifications of arguments in \cite[Lemma~3.5]{K01}
(see also \cite[Lemma~4]{K84}, \cite{Kit}), from Lemma~\rref{le:2sided}
we can get the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:right}
If $a,b,\alpha'\in{\cal G}BC$ and
%%%
\begin{equation}\label{eq:right-1}
\max\left\{
\left(\frac{a}{b}(\alpha_{-1}')^{-1/p}\right)^*(1),
\left(\frac{b}{a}(\alpha')^{-1/p}\right)^*(0)
\right\}<1,
\end{equation}
%%%
then the operator \reff{eq:operator} is right invertible. One of its right inverses is
\[
A^{(r)}=\sum_{n=0}^\infty (a^{-1}bW_\alpha)^na^{-1}\chi_0I
-
\sum_{n=1}^\infty (W_\alpha^{-1}b^{-1}aI)^na^{-1}\chi_1I.
\]
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:kernel}
Under the assumptions of Lemma~\rref{le:right}, $\dim\Ker A=\infty$.
\end{lemma}
\vspace{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We literally repeat the proof of \cite[Lemma~3.6]{K01} (see also \cite[Lemma~3]{K84}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{co:right}
If $a,b,\alpha'\in {\cal G}BC$ and inequality \reff{eq:right-1} holds,
then the operator \reff{eq:operator} is right invertible but not invertible.
\vspace{3mm}
\end{corollary}

Corollary~\rref{co:right} immediately follows
from Lemmas~\rref{le:right} and~\rref{le:kernel}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Sufficient conditions for left invertibility}
%%%
By analogy with Lemma~\rref{le:right} (see also
\cite[Lemma~3.8]{K01}, \cite[Lemma~4]{K84}, \cite{Kit})
it is easy to prove the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:left}
If $a,b,\alpha'\in{\cal G}BC$ and
%%%
\begin{equation}\label{eq:left-1}
\max\left\{
\left(\frac{a}{b}(\alpha_{-1}')^{-1/p}\right)^*(0),
\left(\frac{b}{a}(\alpha')^{-1/p}\right)^*(1)
\right\}<1,
\end{equation}
%%%
then the operator \reff{eq:operator} is left invertible. One of its left inverses is
\[
A^{(l)}=\sum_{n=0}^\infty\chi_1 (a^{-1}bW_\alpha)^na^{-1}I
-
\sum_{n=1}^\infty \chi_0(W_\alpha^{-1}b^{-1}aI)^na^{-1}I.
\]
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To get more information on the cokernel of the operator \reff{eq:operator},
we need the following auxiliary proposition.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:cokernel}
Suppose   $g,\alpha'\in{\cal G}BC$.

{\rm (a)}
If $\left(\frac{1}{g}(\alpha_{-1}')^{-1/p}\right)^*(0)<1$, then
%%%
\begin{equation}\label{eq:cok-2}
\lim_{n\to +\infty}\inf_{t\in l}
\left(
\left|
\prod_{k=0}^{n-1}g[\alpha_{k-n}(t)]
\right||\alpha_{-n}'(t)|^{1/p}
\right)=+\infty.
\end{equation}

{\rm (b)} If
$\Big({g}\cdot(\alpha')^{-1/p}\Big)^*(1)<1$, then
\[
\lim_{n\to +\infty}\inf_{t\in l}
\left(
\left|
\prod_{k=1}^{n}g^{-1}[\alpha_{n-k}(t)]
\right||\alpha_{n}'(t)|^{1/p}
\right)=+\infty.
\]
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
{\rm (a)} The hypothesis is equivalent to the inequality
\[
\Big((g\circ\alpha)\cdot (\alpha')^{-1/p}\Big)_*(0)> 1.
\]
Fix $\eps>0$ such that
\[
\Big((g\circ\alpha)\cdot(\alpha')^{-1/p}\Big)_*(0)> 1+\eps.
\]
 From the definition of the lower limit it follows that there exists a
point $t_0\in \I$ such that
\[
\frac{|g[\alpha(t)]|}{(\alpha'(t))^{1/p}}>1+\eps
\quad\mbox{for all}\quad
t\in (0,t_0).
\]
Since zero is the repelling point of the shift $\alpha$, there exists a
number $k_0\in\N$ such that $\alpha_{-k}(l)\subset(0,t_0)$ for all $k>k_0$. Hence,
%%%
\begin{equation}\label{eq:cok-5}
\frac{|g[\alpha_{1-k}(t)]|}{(\alpha'[\alpha_{-k}(t)])^{1/p}}>1+\eps
\quad\mbox{for all}\quad t\in l\quad\mbox{and}\quad k>k_0.
\end{equation}
%%%
On the other hand, in view of \reff{eq:sp-Lp-3},
%%%
\begin{eqnarray}
\label{eq:cok-6}
&&
\left|
\prod_{k=0}^{n-1}g[\alpha_{k-n}(t)]
\right||\alpha_{-n}'(t)|^{1/p}
=
\left(\prod_{k=1}^n\frac{|g[\alpha_{1-k}(t)]|}{(\alpha'[\alpha_{-k}(t)])^{1/p}}\right)
\cdot
\left|\frac{g[\alpha_{-n}(t)]}{g(t)}\right|.
\end{eqnarray}
%%%
Set
%%%
\begin{equation}\label{eq:cok-7}
M_0:=\inf_{t\in \I}\frac{|g[\alpha(t)]|}{(\alpha'(t))^{1/p}}>0,
\quad
N:=\inf_{t\in \I}|g(t)|/\sup_{t\in \I}|g(t)|>0.
\end{equation}
%%%
Combining \reff{eq:cok-5}--\reff{eq:cok-7}, we get for sufficiently large $n$,
%%%
\begin{eqnarray*}
%\label{eq:cok-8}
\inf_{t\in l}\left(\left|
\prod_{k=0}^{n-1}g[\alpha_{k-n}(t)]
\right||\alpha_{-n}'(t)|^{1/p}\right)
\!\!\!\!&\ge&
N\cdot\prod_{k=1}^n\left(
\inf_{t\in l}\frac{|g[\alpha_{1-k}(t)]|}{(\alpha'[\alpha_{-k}(t)])^{1/p}}
\right)
\\
&\ge&
NM_0^{k_0}(1+\eps)^{n-k_0}.
\end{eqnarray*}
%%%
The latter inequality implies \reff{eq:cok-2}.
%
Statement (b) is proved by analogy.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:cokernel}
Under the assumptions of Lemma~\rref{le:left}, $\dim\Coker A=\infty$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
The idea of the proof is borrowed from \cite[Lemma~3]{K84}.
Let $d\in L^p$ and $\supp d\subset l$. Let us prove that if
$d(t)\ne 0$ a. e. on $l$, then $d\not\in{\rm Im}\,A$.

Assume the contrary, that is, assume the equation
%%%
\begin{equation}\label{eq:equation-1}
(Af)(t)=a(t)f(t)-b(t)f[\alpha(t)]=d(t),
\quad t\in\I.
\end{equation}
%%%
has a solution $f\in L^p$. Since $\supp d\subset l$, we have
for every $t\in l$ and $n\in\N$,
%%%
\begin{eqnarray}
\label{eq:cokernel-1}
f[\alpha_{-n}(t)] &=&
\left(\prod_{k=1}^n g[\alpha_{-k}(t)]\right) f(t),
\\
\label{eq:cokernel-2}
f[\alpha_n(t)] &=&
\left(\prod_{k=0}^{n-1} g^{-1}[\alpha_{k}(t)]\right) \left(f(t)-\frac{d(t)}{a(t)}\right),
\end{eqnarray}
%%%
where $g:=b/a$. From \reff{eq:cokernel-1} we see that
\[
f(t)
=
\left(\prod_{k=1}^n g[\alpha_{n-k}(t)]\right)f[\alpha_n(t)]
=
\left(\prod_{k=0}^{n-1} g[\alpha_{k}(t)]\right)f[\alpha_n(t)],
\quad
t\in\alpha_{-n}(l).
\]
Let $\chi_u$ stand for the characteristic function of a set $u\subset\I$.
For every $n\in\N$,
%%%
\begin{eqnarray*}
\|f\|_p &\ge& \|f\chi_{\alpha_{-n}(l)}\|_p
=
\left(
\int_{\alpha_{-n}(l)}
\left|
\prod_{k=0}^{n-1} g[\alpha_{k}(t)]
\right|^p|f[\alpha_n(t)]|^p dt
\right)^{1/p}
\\
&=&
\left(
\int_{l}
\left|
\prod_{k=0}^{n-1} g[\alpha_{k-n}(t)]
\right|^p|f(t)|^p |\alpha_{-n}'(t)|dt
\right)^{1/p}
\\
&\ge&
\inf_{t\in l}
\left(
\left|
\prod_{k=0}^{n-1}g[\alpha_{k-n}(t)]
\right||\alpha_{-n}'(t)|^{1/p}
\right)
\|f\chi_l\|_p.
\end{eqnarray*}
%%%
In view of Proposition~\rref{pr:cokernel}(a), $f\not\in L^p$
whenever $f(t)\ne 0$ a. e. on $l$.

Analogously, from \reff{eq:cokernel-2} we see that
\[
f(t)
=
\left(\prod_{k=1}^{n} g^{-1}[\alpha_{-k}(t)]\right)\varphi[\alpha_{-n}(t)],
\quad
t\in\alpha_{n}(l),
\]
where $\varphi:=f-d/a$. Then for every $n\in\N$,
\[
\|f\|_p
\ge
\inf_{t\in l}
\left(
\left|
\prod_{k=1}^{n}g^{-1}[\alpha_{n-k}(t)]
\right||\alpha_{n}'(t)|^{1/p}
\right)
\|\varphi\chi_l\|_p.
\]
In view of Proposition~\rref{pr:cokernel}(b), $f\not\in L^p$
whenever $\varphi(t)\ne 0$ a. e. on $l$. Thus, if
\reff{eq:equation-1} holds and $d(t)\ne 0$ a. e. on $l$, then
$f\not\in L^p$, and we are led to a contradiction.

Since $d\not\in {\rm Im}\, A$ is arbitrary, it is easily seen that
$\dim\Coker A=\infty$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{co:left}
If $a,b,\alpha'\in {\cal G}BC$ and inequality \reff{eq:left-1} holds,
then the operator \reff{eq:operator} is left invertible but not invertible.
\end{corollary}
\vspace{3mm}

Corollary~\rref{co:left} immediately follows from
Lemmas~\rref{le:left} and~\rref{le:cokernel}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsection{Slowly oscillating functions and shifts}
\vspace*{-1cm}
\newsubsection{Some properties of slowly oscillating functions}
%%%
Let $C^1$ be the set of all continuously differentiable functions on $\I$.

Below we formulate three sufficient conditions which guarantee slow oscillation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:SO-difference}
Suppose $\varphi\in SO$ and $\psi\in BC$. If
\[
\lim_{t\to 0}\Big(\varphi(t)-\psi(t)\Big)=0,
\]
then $\psi\in SO$.
\end{proposition}
\vspace{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The proof is straightforward.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:SO-suf}
If $\varphi\in BC\cap C^{1}$ and
%%%
\begin{equation}\label{eq:SO-suf-1}
\lim_{t\to 0}t\varphi'(t)=0,
\end{equation}
%%%
then $\varphi\in SO$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Fix $\eps>0$. From \reff{eq:SO-suf-1} we infer that there exists a
$\delta(\eps)>0$ such that
%%%
\begin{equation}\label{eq:SO-suf-2}
|t\varphi'(t)|<\eps
\quad\mbox{whenever}\quad t\in(0,\delta(\eps)).
\end{equation}
%%%
Fix $\lambda\in \I$ and assume that $r\in(0,\delta(\eps))$.
Consider $x,y\in[\lambda r,r]$ and assume for definiteness that
$x\le y$. Then since $\varphi\in C^1$, we obtain
%%%
\begin{equation}
\label{eq:SO-suf-3}
|\varphi(x)-\varphi(y)|
\le
\int_x^y|t\varphi'(t)|\frac{dt}{t}
\le
\left(\sup_{t\in(0,r]}|t\varphi'(t)|\right)
\log\frac{1}{\lambda}.
\end{equation}
%%%
  From \reff{eq:SO-suf-2} and \reff{eq:SO-suf-3} we get for
$r\in(0,\delta(\eps))$,
\[
\omega(\varphi,[\lambda r,r])
\le
\left(\sup_{t\in(0,r]}|t\varphi'(t)|\right)
\log\frac{1}{\lambda}
\le
\eps\log\frac{1}{\lambda}.
\]
Hence, $\varphi\in SO$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:SO-nec}
Let $\varphi\in BC\cap C^1$ and $\psi(t):=t\varphi'(t)$, $t\in\I$.
If $\varphi$ and $\psi$ slowly oscillate at zero, then
%%%
\begin{equation}\label{eq:SO-nec-1}
\lim_{t\to 0}\psi(t)=0.
\end{equation}
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
First let $\varphi$ and hence $\psi$ be real-valued functions.
Contrary to \reff{eq:SO-nec-1} assume that there are a constant
$c>0$ and a sequence $r_n\to 0$ such that
%%%
\begin{equation}\label{eq:SO-nec-2}
\mbox{either}\quad
\psi(r_n)\geq 2c>0
\quad\quad\mbox{or}\quad\quad
\psi(r_n)\leq- 2c<0
\end{equation}
for $n\in\N$.
Since $\psi$ slowly oscillates at zero, there exists an $n_0\in\N$
such that $\omega(\psi,[r_n/2,r_n])\leq c$ for all $n>n_0$.
Hence, by \reff{eq:SO-nec-2},
\[
\mbox{either}\quad
\inf\Big\{\psi(t):t\in [r_n/2,r_n] \Big\} \geq c
\quad {\rm or} \quad
\sup\Big\{\psi(t):t\in [r_n/2,r_n] \Big\}\leq -c.
\]
Then
\begin{eqnarray*}
&\mbox{either}&
\varphi(r_n)-\varphi(r_n/2)=
\int_{r_n/2}^{r_n}\psi(t) \frac{dt}{t}\geq c\log 2 \quad
{\rm for}\;{\rm all}\quad n>n_0,
\\
&\mbox{or}&
\varphi(r_n)-\varphi(r_n/2)=
\int_{r_n/2}^{r_n}\psi(t) \frac{dt}{t}\leq -c\log 2 \quad {\rm
for}\;{\rm all}\quad n>n_0.
\end{eqnarray*}
In both cases $\varphi\notin SO$, and this contradiction
proves \reff{eq:SO-nec-1}.

Passage to the real and imaginary parts of $\varphi$ and $\psi$
completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:Sarason}
If $\varphi\in SO$, then
the function
\[
\zeta(t):=\frac{1}{t}\int_0^t\varphi(s)ds, \quad t\in\I,
\]
belongs to $SO$ too. Moreover, $\lim\limits_{t\to
0}\,(\varphi(t)-\zeta(t))=0$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
It follows from \cite[Lemmas~3 and~4]{S77} that $\zeta\in SO$.
Since $t\zeta'(t)=\varphi(t)-\zeta(t)$ for $t\in\I$, the function
$\psi:\, t\mapsto t\zeta'(t)$ belongs to $SO$ too. It remains to
apply Proposition~\rref{pr:SO-nec} with $\varphi=\zeta$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5555
\newpage

We say that $h=\{h_n\}_{n=1}^\infty\subset \I$ is a {\it test
sequence (of numbers)} if $\lim\limits_{n\to\infty}h_n=0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:SO-uniform}
Suppose that the continuous functions ${\cal F}:\I\times\I\to\I$ and
$f_j:\I\to\I$, $j=1,2$, satisfy
%%%
\begin{equation}\label{eq:SO-uniform-1}
xf_1(y)\le{\cal F}(x,y)\le xf_2(y),
\quad x,y\in\I.
\end{equation}
%%%
Let $\varphi:\I\to\C$ be a slowly oscillating function at zero. If for some test
sequence $h=\{h_n\}_{n=1}^\infty$ the limit
%%%
\begin{equation}\label{eq:SO-uniform-2}
\lim_{n\to\infty}\varphi(h_n)=:\varphi_h
\end{equation}
%%%
exists, then for every $y\in\I$ the limit
$\lim\limits_{n\to\infty}\varphi({\cal F}(h_n,y))$
also exists. Moreover,
%%%
\begin{equation}\label{eq:SO-uniform-3}
\lim_{n\to\infty}\varphi({\cal F}(h_n,y))=\varphi_h,
\end{equation}
%%%
and the convergence is uniform on every segment $J\subset\I$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Fix $\eps>0$. Since $\varphi\in SO$, there exists a $\delta=\delta(\eps)\in(0,1)$
such that
%%%
\begin{equation}\label{eq:SO-uniform-5}
\omega(\varphi,[\lambda r,r])<\frac{\eps}{2}
\quad\mbox{for all} \quad r\in(0,\delta).
\end{equation}
%%%
On the other hand, \reff{eq:SO-uniform-2} implies the existence of a number
$n_1(\eps)\in\N$ such that
%%%
\begin{equation}\label{eq:SO-uniform-6}
|\varphi(h_n)-\varphi_h|<\frac{\eps}{2}
\quad\mbox{for all}\quad n\ge n_1(\eps).
\end{equation}
%%%
For a given segment $J\subset\I$,
put $\lambda:=\min\limits_{y\in J} f_1(y)\in\I$ and set $r_n=h_n$.
Then from the property $f_2(y)<1$ and from \reff{eq:SO-uniform-1} we get
%%%
\begin{equation}\label{eq:SO-uniform-8}
\lambda r_n\le h_n\le r_n,
\quad
\lambda r_n\le {\cal F}(h_n,y)\le r_n
\quad\mbox{for all}\quad y\in J.
\end{equation}
%%%
Since $h=\{h_n\}_{n=1}^\infty$ is a test sequence, there exists a number
$n_2(\eps)\ge n_1(\eps)$ such that $r_n\in(0,\delta)$ for all
$n\ge n_2(\eps)$. Hence, from \reff{eq:SO-uniform-5} and
\reff{eq:SO-uniform-8} we deduce that for every $n\ge n_2(\eps)$,
%%%
\begin{equation}\label{eq:SO-uniform-9}
|\varphi({\cal F}(h_n,y))-\varphi(h_n)|\le\omega(\varphi,[\lambda r_n,r_n])
<\frac{\eps}{2}
\quad\mbox{for all}\quad y\in J.
\end{equation}
%%%
Combining \reff{eq:SO-uniform-6} and \reff{eq:SO-uniform-9}, we
conclude that for every $y\in J$ and every $n\ge n_2(\eps)$,
\[
|\varphi({\cal F}(h_n,y))-\varphi_h|
\le
|\varphi({\cal F}(h_n,y))-\varphi(h_n)|
+
|\varphi(h_n)-\varphi_h|
<\frac{\eps}{2}+\frac{\eps}{2}=\eps,
\]
that is, \reff{eq:SO-uniform-3} holds and the convergence
is uniform on $J\subset\I$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Some properties of slowly oscillating shifts}
%%%
For a shift $\alpha$ and $j\in\{0,1\}$, put
\[
\wa_j(t):=\frac{\alpha(t)-\alpha(j)}{t-j},\quad t\in \I.
\]
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:shift}
If $\alpha'\in\cG B$ and $j\in\{0,1\}$, then the following properties hold:

{\rm (a)}
for every $t\in \I$,
%%%
\begin{equation}\label{eq:shift-1}
m_\alpha t\le\alpha(t)\le M_\alpha t,
\quad
m_\alpha(1-t)\le1-\alpha(t)\le M_\alpha(1-t),
\end{equation}
%%%
where
%%%
\begin{equation}\label{eq:shift-2}
0<m_\alpha:=\inf_{t\in \I}\alpha'(t),
\quad
M_\alpha:=\sup_{t\in \I}\alpha'(t)<+\infty;
\end{equation}
%%%

{\rm (b)} the functions $\wa_j$  belong to $\cG BC$;

{\rm (c)} $\wa_j$ slowly oscillates at $j$ if and only if $\wb_j$
slowly oscillates at $j$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Statement (a) follows from the mean-value theorem and the fact that $\alpha(0)=0,
\alpha(1)=1$. Statement (b) follows from statement (a).

(c) Consider the case $j=0$. Since $\alpha'\in\cG B$ if and only
if $\beta'=\alpha_{-1}'\in\cG B$, we obtain, in view of statement
(b), that $\wb_0\in\cG BC$. Further, if $s,t\in \I$, then
%%%
\begin{equation}\label{eq:shift-4}
|\wb_0(s)-\wb_0(t)|= \left|\frac{\beta(s)}{s}\right| \cdot
\left|\frac{\beta(t)}{t}\right| \cdot \left|
\frac{s}{\beta(s)}-\frac{t}{\beta(t)} \right|
\le
L\left|
\frac{s}{\beta(s)}-\frac{t}{\beta(t)}
\right|,
\end{equation}
%%%
where
\[
L:=\left(\sup\limits_{t\in \I}|\wb_0(t)|\right)^2<+\infty.
\]

Let $\lambda\in \I$ and suppose $r$ is sufficiently small. Taking 
into account that $\alpha$ is a homeomorphism, we have
%%%
\begin{equation}\label{eq:shift-5}
\sup_{s,t\in[\lambda r,r]}
\left|
\frac{s}{\beta(s)}-\frac{t}{\beta(t)}
\right|
=
\sup_{x,y\in[\beta(\lambda r),\beta(r)]}
\left|
\frac{\alpha(x)}{x}-\frac{\alpha(y)}{y}
\right|.
\end{equation}
%%%
 From statement (a) we get
%%%
\begin{equation}\label{eq:shift-6}
[\beta(\lambda r),\beta(r)]\subset[m_\beta\lambda r, M_\beta r]=:[\lambda_1 r_1,r_1],
\end{equation}
%%%
where $m_\beta$ and $M_\beta$ are defined by \reff{eq:shift-2},
$\lambda_1:=\frac{m_\beta}{M_\beta}\lambda\in(0,\lambda)\subset
\I, \ r_1:=M_\beta r$. From \reff{eq:shift-4}--\reff{eq:shift-6}
we deduce that
%%%
\begin{equation}\label{eq:shift-7}
\omega(\wb_0,[\lambda r,r])
\le
L\left(\sup_{x,y\in[\lambda_1 r_1,r_1]}|\wa_0(x)-\wa_0(y)|\right)
=
L\omega(\wa_0,[\lambda_1 r_1,r_1]).
\end{equation}
%%%
If $\wa_0$ slowly oscillates at zero, then
$\omega(\wa_0,[\lambda_1 r_1,r_1])\to 0$ as $r_1\to 0$. In this
case \reff{eq:shift-7} implies that $r=r_1/M_\beta\to 0$ and
$\omega(\wb_0,[\lambda r,r])\to 0$, i.e., $\wb_0$ slowly
oscillates at zero. For the functions $\wa_1$ and $\wb_1$
statement (c) is proved by analogy.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:SO-wa}
Suppose $\alpha'\in BC$ and $j\in\{0,1\}$. If $\alpha'$ is a
slowly oscillating function at $j$, then $\wa_j$ is a slowly
oscillating function at $j$ too.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Consider the case $j=1$. If $\alpha'$ slowly oscillates at 1, then
$\alpha'(1-t)$ slowly
\newpage\noindent
oscillates at zero. Hence, by Proposition~\rref{pr:Sarason}, the
function
\vspace*{-0.05cm}
\[
\frac{1-\alpha(1-t)}{t}=\frac{1}{t}\int_0^t\alpha'(1-s)ds
\]
belongs to $SO$ together with $\alpha'(1-t)$. This is equivalent
to the slow oscillation of $\wa_1$ at 1. Applying
Proposition~\rref{pr:Sarason} with $\varphi=\alpha'$, we manage
the case $j=0$.
\end{proof}

One can construct an example of a shift $\alpha$ such that
$\log\alpha'\in BC$ and $\wa_j$ is a slowly oscillating function
at $j$, but $\alpha'$ does not slowly oscillate at $j$.

The next lemma gives us equivalent characterizations of slowly
oscillating shifts.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:SO-equivalence}
Suppose $\log\alpha'\in BC$ and $j\in\{0,1\}$. The following
statements are equivalent:

{\rm (a)}
$\alpha'$ is a slowly oscillating function at $j$;

{\rm (b)}
$\beta'$ is a slowly oscillating function at $j$;

{\rm (c)} $\displaystyle \lim\limits_{t\to j}(t-j)\wa_j'(t)=0$;

{\rm (d)} $\displaystyle \lim\limits_{t\to j}(t-j)\wb_j'(t)=0$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
The equivalence (a) $\Leftrightarrow$ (b) is proved similarly to
Proposition~\rref{pr:shift}(c) making use of
\[
|\beta'(s)-\beta'(t)|=
\frac{|\alpha'(\beta(s))-\alpha'(\beta(t))|}{\alpha'(\beta(s))\alpha'(\beta(t))},
\quad s,t\in\I.
\]

Let us prove the equivalence (a) $\Leftrightarrow$ (c) in the case
$j=1$. Taking into account Lemma~\rref{le:SO-wa}, we deduce that
$\alpha',\wa_1$ slowly oscillate at $1$. Therefore, the function
%%%
\begin{equation}\label{eq:SO-equivalence-1}
t\gamma'(t)=\alpha'(1-t)-
\gamma(t),
\quad\mbox{where}\quad
\gamma(t):=(1-\alpha(1-t))/t,
\end{equation}
%%%
slowly oscillates at zero. Due to Proposition~\rref{pr:shift}(b),
$\wa_1\in\cG BC$. Thus, by Proposition~\rref{pr:SO-nec},
%%%
\begin{equation}\label{eq:SO-equivalence-2}
\lim_{t\to 1}(t-1)\wa_1'(t)
=
\lim_{t\to 0}t\gamma'(t)
=0.
\end{equation}
%%%
Conversely, if \reff{eq:SO-equivalence-2} holds, then by
Proposition~\rref{pr:SO-suf}, $\gamma$ slowly oscillates at zero.
Hence, from \reff{eq:SO-equivalence-1}--\reff{eq:SO-equivalence-2}
and Proposition~\rref{pr:SO-difference} we deduce that
$\alpha'(1-t)$ slowly oscillates at zero. This means that
$\alpha'$ slowly oscillates at 1.

The case $j=0$ and the equivalence (b) $\Leftrightarrow$ (d) 
are treated analogously.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Lemma~\rref{le:SO-wa} and the second assertion of
Proposition~\rref{pr:Sarason} imply the following.
\begin{corollary}\label{co:equivalence}
Suppose $\alpha$ is a slowly oscillating shift at $j\in\{0,1\}$.

{\rm (a)}
If for some sequence $\{h_n\}_{n=1}^\infty\subset\I$ which tends to $j$
one of the limits
\[
\lim_{n\to\infty}\wa_j(h_n), \quad \lim_{n\to\infty}\alpha'(h_n)
\]
exists, then the other limit exists too. Moreover,
\[
\lim_{n\to\infty}\wa_j(h_n)
=
\lim_{n\to\infty}\alpha'(h_n).
\]
\newpage
\vspace*{-0.6cm}
\[
\hspace{-1cm} {\rm (b)} \quad\quad\quad\quad \liminf\limits_{t\to
j}\wa_j(t)
=
\liminf\limits_{t\to j}\alpha'(t), \quad \limsup\limits_{t\to
j}\wa_j(t)
=
\limsup\limits_{t\to j}\alpha'(t).
\]
\end{corollary}

Modifying Example 2.3 in \cite{KL01} one can construct an example of a
slowly oscillating non-Carleman shift with
\[
\liminf_{t\to j}\alpha'(t)<\limsup_{t\to j}\alpha'(t),
\quad j\in\{0,1\}.
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsection{Limit operators for a functional operator}
\vspace*{-1cm}
\newsubsection{Abstract approach}
%%%
Let $X$ be a Banach space and $X^*$ be its dual space. Let $\fB(X)$
denote
the Banach algebra of all bounded linear operators on $X$.
We say that an operator $V\in\fB(X)$ is a {\it pseudoisometry}
if $V\in\cG\fB(X)$ and $\|V\|_{\fB(X)}=1/\|V^{-1}\|_{\fB(X)}$.
Any sequence $\cV=\{V_n\}_{n=1}^\infty$ of pseudoisometries is
referred to as a {\it test sequence (of operators)}. Fix some
set $\V\subset\fB(X)$ of pseudoisometries.

Let $A\in\fB(X)$ and $\cV=\{V_n\}_{n=1}^\infty\subset\V$. If the
strong limits 
%%%
\begin{eqnarray}\label{eq:LO-defi}
&&A_\cV:=\slim V_n^{-1}AV_n \quad\mbox{in } \fB(X), \
A_{\cV^*}:=\slim (V_n^{-1}AV_n)^* \quad\mbox{in } \fB(X^*)
\end{eqnarray}
%%%
exist, then always $(A_\cV)^*=A_{{\cV}^*}$, and we will refer to the
operator $A_\cV$ as the {\it limit operator} for the operator $A$
with respect to the test sequence $\cV$. Note that usually the
limit operator $A_\cV$ is defined independently of the existence
of the strong limit $A_{{\cV}^*}$ (see, e.g., \cite{BKR00},
\cite{RRS98}), while we need the existence of both limits
\reff{eq:LO-defi} for our purposes. If the limit operator $A_\cV$
exists, then it is uniquely determined by $A$ and $\cV$, which
justifies the notation $A_\cV$. Further, if $A_\cV$ is the limit
operator of $A$ with respect to $\cV$ and if $\cW$ is a
subsequence of $\cV$, then the limit operator $A_{\cW}$ of $A$
with respect to $\cW$ also exists and coincides with $A_\cV$.

In the next proposition we summarize some properties of limit operators.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:LO-properties}
Let $\cV\subset\V$ be a test sequence. If $A,B\in\fB(X)$, $\alpha \in\C$, 
and the limit operators $A_\cV,B_\cV$ exist, then 
$(\alpha A)_\cV, (A+B)_\cV,(AB)_\cV$ also exist and
\[
(\alpha A)_\cV=\alpha A_\cV,
\quad
(A+B)_\cV=A_\cV+B_\cV,
\quad
(A B)_\cV=A_\cV B_\cV.
\]
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The proof is straightforward
(see also \cite{BKR00}, \cite{RRS98} and references therein).

The following lemma is a useful tool for the investigation of the invertibility
in Banach algebras of operators.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:tool}
If an operator $A\in\fB(X)$ is invertible, then for every test sequence
$\cV=\{V_n\}_{n=1}^\infty\subset\V$ the limit operator $A_\cV$ is also 
invertible.
\end{lemma}
\vspace{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This lemma follows from Lemma~\cite[Ch.~3, Lemma~1.1]{GF74} due
to the existence of the two limits \reff{eq:LO-defi} and the
equality $(A_\cV)^*=A_{\cV^*}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Realization}
%%%
We will construct a set of pseudoisometries on the space
$L^p$. First, let us start from the simpler case of the half-line
$\R_+:=(0,\infty)$. Let $\overline{\R}_+:=[0,\infty]$.
For $x\in\R_+$, the multiplicative shift operator
$\widetilde{V}_x$ is  defined on the space $L^p(\R_+)$ by
\[
(\widetilde{V}_xf)(y):=f(y/x),\quad y\in\R_+.
\]
Clearly, $\widetilde{V}_x\in\cG\fB(L^p(\R_+))$ and
$\widetilde{V}_x^{-1}=\widetilde{V}_{1/x}$. Moreover,
$\|\widetilde{V}_x\|_{\fB(L^p(\R_+))}=x^{1/p}$.
Hence, $\widetilde{V}_x$ is a pseudoisometry for every $x\in\R_+$.

Consider a function
$\theta:\overline{\R}_+\to[0,1]$ such that $\theta(0)=0$,
$\theta(\infty)=1$, and the derivative $\theta'$ exists and is
positive for every $x\in\R_+$. Then the transformation $\theta$
preserves the natural orientation and is invertible. Denote by
$\theta_{-1}$ its inverse, which maps $[0,1]$ onto
$\overline{\R}_+$.

It is easy to see that the transformation $G:L^p\to L^p(\R_+)$
defined by
%%%
\begin{equation}\label{eq:C}
(G\varphi)(y):=c(y)\varphi(\theta(y)),
\quad
c(y):=(\theta'(y))^{1/p},
\quad
y\in\R_+,
\end{equation}
%%%
is an isometric isomorphism with the inverse
$G^{-1}:L^p(\R_+)\to L^p$ given by
%%%
\begin{equation}\label{eq:C-inverse}
(G^{-1}\psi)(t):=\frac{\psi(\theta_{-1}(t))}{c(\theta_{-1}(t))},
\quad t\in \I.
\end{equation}
%%%
Hence,
%%%
\begin{equation}\label{eq:V}
V_x:=G^{-1}\widetilde{V}_xG\in\cG\fB(L^p),
\quad
x\in\R_+,
\end{equation}
%%%
and
\[
V_x^{-1}=V_{1/x},
\quad
\|V_x\|_{\fB(L^p)}=\|\widetilde{V}_x\|_{\fB(L^p(\R_+))}=x^{1/p}.
\]
So, $V_x$ is a pseudoisometry on $L^p$.
Consider the set of pseudoisometries
\[
\V:=\Big\{
V_x\in\fB(L^p),\ x\in \I
\Big\}.
\]
Clearly, we can identify any test sequence
$h=\{h_n\}_{n=1}^\infty\subset \I$
with the sequence of pseudoisometries
$\cV_h=\{V_{h_n}\}_{n=1}^\infty\subset\V$.

For every $x\in\R_+$, put $\gamma_x(y):=y/x,\;
y\in\overline{\R}_+$, and
%%%
\begin{equation}\label{eq:wx}
w_x:=\theta\circ\gamma_x\circ\theta_{-1}\ :\ [0,1]\to[0,1].
\end{equation}
%%%
Clearly, $w_x$ is a homeomorphism with the inverse
%%%
\begin{equation}\label{eq:wx-inverse}
(w_x)_{-1}=\theta\circ\gamma_{1/x}\circ\theta_{-1}=w_{1/x}.
\end{equation}
%%%
Put
%%%
\begin{equation}\label{eq:dx}
d_x:=\frac{c\circ \theta_{-1}\circ w_x}{c\circ\theta_{-1}},
\quad
W_x:=W_{w_x}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:Vx}
For every $x\in\R_+$, we have
%%%
\begin{equation}\label{eq:Vx-1}
V_x=d_xW_x=W_x\frac{1}{d_{1/x}}I.
\end{equation}
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $\varphi\in L^p$. From \reff{eq:C}--\reff{eq:wx} we get
%%%
\[
V_x\varphi= G^{-1}\widetilde{V}_xG\varphi
=
\left( \frac{c\circ\gamma_x\circ\theta_{-1}}{c\circ\theta_{-1}}
\right)\cdot (\varphi\circ\theta\circ\gamma_x\circ\theta_{-1}) =
d_xW_x\varphi.
\]
%%%
Using \reff{eq:wx-inverse} and \reff{eq:dx} we immediately obtain
$d_x\circ w_{1/x}={1}/{d_{1/x}}$, which gives the second part of
\reff{eq:Vx-1}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:composit}
{\rm (a)} If $a\in L^\infty$, then for every $x\in\R_+$,
%%%
\begin{equation}\label{eq:composit-1}
V_x^{-1}aV_x=(a\circ w_{1/x})I,
\quad
(V_x^{-1}aV_x)^*=(\overline{a}\circ w_{1/x})I.
\end{equation}
%%%

{\rm (b)} If $\alpha'\in \cG L^\infty$, then for every $x\in\R_+$,
%%%
\begin{equation}\label{eq:composit-2}
V_x^{-1}W_\alpha V_x
=
\frac{d_{1/x}}{d_{1/x}\circ\alpha_x}W_{\alpha_x},\quad
(V_x^{-1}W_\alpha V_x)^*
=
\beta_x'W_{\beta_x}\frac{d_{1/x}}{d_{1/x}\circ\alpha_x}I,
\end{equation}
where
\begin{equation}\label{eq:der-bound-0}
\alpha_x:=w_x\circ\alpha\circ w_{1/x}, \quad
\beta_x:=w_x\circ\beta\circ w_{1/x}.
\end{equation}
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
(a) If $a\in L^\infty$, then $aI\in\fB(L^p)$. Hence,
$V_x^{-1}aV_x\in\fB(L^p)$ for every $x\in\R_+$. From
\reff{eq:Vx-1} we get
\[
V_x^{-1}aV_x
=
d_{1/x}W_{1/x}aW_x\frac{1}{d_{1/x}}I
=
d_{1/x}(a\circ w_{1/x})\frac{1}{d_{1/x}}I
=
(a\circ w_{1/x})I.
\]
Further,
\[
(V_x^{-1}a V_x)^*=\Big((a\circ w_{1/x})I\Big)^*=(\overline{a}\circ w_{1/x})I,
\]
and so statement (a) is proved.

(b) Clearly, $W_{1/x}W_\alpha W_x=W_{\alpha_x}$. Using this
identity and \reff{eq:Vx-1}, we get the first equality in
\reff{eq:composit-2}:
\[
V_x^{-1}W_\alpha V_x
=
d_{1/x}W_{1/x}W_\alpha W_x\frac{1}{d_{1/x}}I
=
d_{1/x}W_{\alpha_x}\frac{1}{d_{1/x}}I
=
\frac{d_{1/x}}{d_{1/x}\circ\alpha_x}W_{\alpha_x}.
\]
Then passing to adjoint operators in that formula, taking into
account that $d_{1/x}$ is real-valued, and using the identity
$(W_\alpha)^* =\beta'W_\beta$, $\beta=\alpha_{-1}$, we obtain the
second equality in \reff{eq:composit-2}.
Statement (b) and Proposition~\rref{pr:composit} are proved.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In what follows we take the concrete function
%%%
\begin{equation}\label{eq:theta}
\theta(y):=\frac{y}{1+y},
\quad
y\in\overline{\R}_+.
\end{equation}
%%%
Then
%%%
\begin{equation}\label{eq:theta-inverse}
\theta_{-1}(t)=\frac{t}{1-t},
\quad t\in[0,1],
\end{equation}
%%%
and for every $x\in\R_+$,
%%%
\begin{eqnarray}\label{eq:wx-concrete}
&&
w_x(t)
=
\frac{\gamma_x(\theta_{-1}(t))}{1+\gamma_x(\theta_{-1}(t))}
=
\frac{\theta_{-1}(t)}{x+\theta_{-1}(t)}
=
\frac{t}{x-xt+t},\quad t\in[0,1].
\end{eqnarray}
%%%
Analogously,
%%%
\begin{equation}\label{eq:wx-inverse-concrete}
w_{1/x}(t)=
\frac{xt}{1+xt-t}=:F(x,t),
\quad t\in[0,1].
\end{equation}
%%%
Clearly,
%%%
\begin{equation}\label{eq:Fest}
xt\le F(x,t)\le\frac{xt}{1-t},
\quad x,t\in \I.
\end{equation}
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:der-bound}
If $\alpha'\in\cG B$, then for the functions \reff{eq:der-bound-0}
we have
\[
\sup_{x\in\R_+}\sup_{t\in \I}\alpha'_x(t)<+\infty,
\quad
\sup_{x\in\R_+}\sup_{t\in \I}\beta'_x(t)<+\infty.
\]
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $x\in\R_+$. Clearly, for every $t\in \I$,
\[
\alpha_x'(t)
=
w_x'[\alpha(w_{1/x}(t))]\cdot\alpha'(w_{1/x}(t))\cdot w_{1/x}'(t)
=
\alpha'(w_{1/x}(t))\frac{w_x'[\alpha(w_{1/x}(t))]}{w_x'(w_{1/x}(t))}.
\]
Since $\alpha'\in\cG B$, we have for every $x\in\R_+$,
%%%
\begin{equation}\label{eq:der-bound-1}
\alpha_x'(t)\le M_\alpha \,\sup_{\tau\in \I}\,
\frac{w_x'[\alpha(\tau)]}{w_x'(\tau)}, \quad t\in \I,
\end{equation}
%%%
where $M_\alpha$ is defined in \reff{eq:shift-2}. From \reff{eq:wx-concrete}
we get
\[
w_x'(t)=\frac{x}{(x-xt+t)^2},
\quad t\in \I.
\]
Then for $x\in\R_+$ and $\tau\in \I$ we obtain
%%%
\begin{equation}\label{eq:der-bound-2}
\frac{w_x'[\alpha(\tau)]}{w_x'(\tau)}
=
\left( \frac{(1-\tau)x+\tau}{(1-\alpha(\tau))x+\alpha(\tau)}
\right)^2.
\end{equation}
%%%
 From Proposition~\rref{pr:shift}(a) we get
%%%
\begin{equation}\label{eq:der-bound-3}
(1-\alpha(\tau))x+\alpha(\tau) \ge m_\alpha (1-\tau)x+m_\alpha
\tau.
\end{equation}
%%%
Combining \reff{eq:der-bound-1}--\reff{eq:der-bound-3}, we arrive at the estimate
\[
\sup_{x\in\R_+}\sup_{t\in \I}\alpha_x'(t)\le \frac{M_\alpha}{m_\alpha^2}<+\infty.
\]
Since $\alpha'\in\cG B$ implies $\beta'\in\cG B$, from the
estimate just proved we obtain
\[
\sup_{x\in\R_+}\sup_{t\in \I}\beta_x'(t)\le \frac{M_\beta}{m_\beta^2}<+\infty.
\]
The lemma is proved.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Limit operators for the multiplication operator}
%%%
We shall use the symbol $\uniform$ to indicate uniform
convergence of sequences in a given segment. We shall use $J$ to
denote an arbitrary segment in $\I$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:slim-mult}
Suppose a sequence $\varphi_n\in BC$ converges pointwise to a
function $\varphi\in BC$. If this convergence is uniform on
every segment $J\subset \I$, then
\[
\slim \varphi_n I=\varphi I.
\]
\end{lemma}

The proof is standard.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:LO-mult}
If $a\in SO$ and for some test sequence $h=\{h_n\}_{n=1}^\infty$
the limit
%%%
\begin{equation}\label{eq:LO-mult-1}
\lim_{n\to\infty} a(h_n)=:a_h
\end{equation}
%%%
exists,
then for the multiplication operator $aI$ the limit operator
$(aI)_{\cV_h}$ with respect to the test sequence of operators
%%%
\begin{equation}\label{eq:LO-mult-3}
\cV_h:=
\left\{
V_{h_n}\in\fB(L^p)\ : \ \lim_{n\to\infty}h_n=0,
\quad
\lim_{n\to\infty} a(h_n)=a_h
\right\}\subset\V
\end{equation}
%%%
exists and this limit operator is given by $(aI)_{\cV_h}=a_hI$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
 From Proposition~\rref{pr:composit}(a) and
\reff{eq:wx-inverse-concrete} we get
%%%
\begin{equation}\label{eq:LO-mult-4}
V_{h_n}^{-1}aV_{h_n}=a(F(h_n,\cdot))I,
\quad
(V_{h_n}^{-1}aV_{h_n})^*=\overline{a(F(h_n,\cdot))}I.
\end{equation}
%%%
Since $a\in SO$, the function $\overline{a}$ is slowly
oscillating too.
Taking into account \reff{eq:LO-mult-1} and \reff{eq:Fest}, from
Lemma~\rref{le:SO-uniform} we obtain
%%%
\begin{equation}\label{eq:LO-mult-5}
a(F(h_n,t))\uniform a_h,
\quad
\overline{a(F(h_n,t))}\uniform\overline{a_h},
\quad t\in J.
\end{equation}
%%%
 From \reff{eq:LO-mult-4}--\reff{eq:LO-mult-5} and
Lemma~\rref{le:slim-mult} we get
\[
\slim V_{h_n}^{-1}aV_{h_n}=a_h I\in\fB(L^p),
\quad
\slim (V_{h_n}^{-1}aV_{h_n})^*=\overline{a_h} I\in\fB(L^q).
\]
Hence, $a_hI$ is the limit operator for $aI$ with respect to the test sequence
\reff{eq:LO-mult-3}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Limit operators for the shift operator}
%%%
In view of Proposition~\rref{pr:composit}(b), to calculate a limit
operator for the shift operator $W_\alpha$,  we need a result on the strong
convergence of shift operators.
The following lemma can be easily obtained  from \cite[Theorem~1]{DS98}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:shift-convergence}
Suppose $\alpha^{(n)}:[0,1]\to[0,1]$ are
orientation preserving homeomorphisms 
such that $\alpha^{(n)}$ and $\alpha^{(n)}_{-1}$ are absolutely continuous on $[0,1]$
and $\log(\alpha^{(n)})'\in L^\infty$ for every $n\in\N\cup\{0\}$. If
\begin{itemize}
\item[{\rm (i)}]
for some $A>0$ and all $n\in\N\cup\{0\}$,
\[
\left\|
\Big(\alpha_{-1}^{(n)}\Big)'
\right\|_{\infty}\le A,
\]

\item[{\rm (ii)}]
$\alpha^{(n)}\to\alpha^{(0)}$ almost everywhere on $[0,1]$ as $n\to\infty$,
\end{itemize}
then the sequence of shift operators $W_{\alpha^{(n)}}\in\fB(L^p)$
strongly converges to the shift operator $W_{\alpha^{(0)}}\in\fB(L^p)$.
\end{lemma}
\vspace{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Using \reff{eq:der-bound-0} and
\reff{eq:wx-concrete}--\reff{eq:wx-inverse-concrete},
we get for $x\in \I$ and $t\in[0,1]$,
%%%
\begin{eqnarray}
\label{eq:alpha_x}
\alpha_x(t)
&=&
\frac{\alpha[F(x,t)]}{x-x\alpha[F(x,t)]+\alpha[F(x,t)]},
\\
\label{eq:beta_x}
 \beta_x(t) &=&
\frac{\beta[F(x,t)]}{x-x\beta[F(x,t)]+\beta[F(x,t)]}.
\end{eqnarray}
%%%
Further, from \reff{eq:C} and \reff{eq:theta}--\reff{eq:theta-inverse} 
we obtain
\[
(c\circ\theta_{-1})(t)
=
(1+\theta_{-1}(t))^{-2/p}
=
\left(
1+\frac{t}{1-t}
\right)^{-2/p}
=
(1-t)^{2/p},
\quad
t\in[0,1].
\]
Hence, from \reff{eq:dx} and \reff{eq:wx-inverse-concrete} it
follows that
\[
d_{1/x}(t)=\frac{(c\circ \theta_{-1}\circ w_{1/x})(t)}{(c\circ
\theta_{-1})(t)}=(1+xt-t)^{-2/p},
\]
and then
%%%
\begin{equation}\label{eq:fx}
f_x(t) := \left(
\frac{d_{1/x}}{d_{1/x}\circ\alpha_x} \right)(t)
=
\left(\frac{1+x\alpha_x(t)-\alpha_x(t)}{1+xt-t}\right)^{2/p},
\quad t\in[0,1).
\end{equation}

In what follows we shall denote $\wa(t):=\wa_0(t)=\alpha(t)/t$,
$\wb(t):=\beta(t)/t$ for $t\in\I$.

Now we  apply Lemma~\rref{le:SO-uniform} to shift functions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:SO-shift}
Suppose $\wa\in SO$.
If for some test sequence $h=\{h_n\}_{n=1}^\infty$ the limit
%%%
\begin{equation}\label{eq:SO-shift-1}
\lim_{n\to\infty}\wa(h_n)=:\vka
\end{equation}
%%%
exists, then we have the following:

{\rm (a)}
for every $t\in[0,1]$, the sequence $\alpha_{h_n}(t)$
converges to the function
\[
\alpha_h(t):=\frac{\vka t}{1+\vka t-t};
\]
%%%

{\rm (b)}
for every $t\in[0,1)$, the sequence $f_{h_n}(t)$
converges to the function
\[
f_h(t):=(1+\vka t-t)^{-2/p}.
\]
In {\rm (a)--(b)} the convergence is uniform with respect to $t$
on every segment $J\subset \I$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Statement (a) is trivial for the fixed points of $\alpha$ 
(= endpoints). Consider the case of inner points of $[0,1]$. Due to 
\reff{eq:alpha_x},
%%%
\begin{equation}\label{eq:SO-shift-2}
\alpha_{h_n}(t)=
\frac{\alpha[F(h_n,t)]/h_n}
{1-h_n\Big(\alpha[F(h_n,t)]/h_n\Big)+\alpha[F(h_n,t)]/h_n},
 \quad n\in\N, \quad t\in \I.
\end{equation}
%%%
Taking into account
\reff{eq:Fest} and \reff{eq:SO-shift-1}, we deduce from Lemma~\rref{le:SO-uniform}
that
%%%
\begin{equation}\label{eq:SO-shift-3}
\frac{\alpha[F(h_n,t)]}{F(h_n,t)}=\wa[F(h_n,t)] \uniform\vka,\quad
t\in J.
\end{equation}
%%%
On the other hand,
%%%
\begin{equation}\label{eq:SO-shift-4}
\frac{F(h_n,t)}{h_n}=\frac{t}{1+h_nt-t}\uniform\frac{t}{1-t},
\quad t\in J.
\end{equation}
%%%
 From \reff{eq:SO-shift-3} and \reff{eq:SO-shift-4} we get
%%%
\begin{equation}\label{eq:SO-shift-5}
\frac{\alpha[F(h_n,t)]}{h_n}\uniform\frac{\vka t}{1-t},
\quad t\in J.
\end{equation}
%%%
Since, by \reff{eq:SO-shift-5}, the denominator of the fraction in
\reff{eq:SO-shift-2} uniformly converges on $J$ to the function
$1+\vka t/(1-t)$ and since it is uniformly bounded away from zero 
on $J$, we deduce from \reff{eq:SO-shift-2} and \reff{eq:SO-shift-5} that
%%%
\begin{equation}\label{eq:SO-shift-7}
\alpha_{h_n}(t)\uniform \frac{\vka t/(1-t)}{1+\vka t/(1-t)}=
\alpha_h(t), \quad t\in J.
\end{equation}
%%%
Statement (a) is proved.

(b) According to \reff{eq:fx},
%%%
\begin{equation}
\label{eq:SO-shift-8} f_{h_n}(t) = \left(
\frac{1+h_n\alpha_{h_n}(t)-\alpha_{h_n}(t)}{1+h_nt-t}
\right)^{2/p}, \quad t\in[0,1).
\end{equation}
%%%
Since $\alpha_{h_n}(0)=0$, we get $f_{h_n}(0)=f_h(0)=1$.
%%%
As the sequence $1+h_n t-t$ is uniformly bounded away from zero
for all $t\in J$ and all sufficiently large $n$, we infer from
\reff{eq:SO-shift-7}--\reff{eq:SO-shift-8} that
%%%
\begin{equation}\label{eq:SO-shift-10}
\frac{1+h_n\alpha_{h_n}(t)-\alpha_{h_n}(t)}{1+h_nt-t}
\uniform
\frac{1-\alpha_h(t)}{1-t}=\frac{1}{1+\vka t-t},
\quad t\in J.
\end{equation}
%%%
Statement (b) directly follows from \reff{eq:SO-shift-8} and
\reff{eq:SO-shift-10}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:LO-shift-prime}
Suppose $\wa\in SO$ and $\alpha'\in\cG B$. If for some test sequence
$h=\{h_n\}_{n=1}^\infty$ the limit \reff{eq:SO-shift-1} exists, then
%%%
\begin{equation}\label{eq:LO-shift-1}
\slim V_{h_n}^{-1}W_\alpha V_{h_n}=f_h W_{\alpha_h}.
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
 From the first equality in \reff{eq:composit-2} and \reff{eq:fx} we deduce that
%%%
\begin{equation}\label{eq:LO-shift-4}
V_{h_n}^{-1}W_\alpha V_{h_n}
=
f_{h_n}W_{\alpha_{h_n}}.
\end{equation}
%%%
 From Lemma~\rref{le:SO-shift}(b) and Lemma~\rref{le:slim-mult} we get
%%%
\begin{equation}\label{eq:LO-shift-5}
\slim f_{h_n}I=f_h I.
\end{equation}
%%%
Since $\wa\in SO$, we infer from Lemma~\rref{le:SO-shift}(a) that
%%%
\begin{equation}\label{eq:LO-shift-8}
\lim_{n\to\infty}\alpha_{h_n}(t)=\alpha_h(t),
\quad t\in[0,1].
\end{equation}
%%%
Since $\alpha'\in\cG B$, from Lemma~\rref{le:der-bound} we deduce that
for some $L>0$,
%%%
\begin{equation}\label{eq:LO-shift-6}
\sup_{t\in \I}\beta_{h_n}'(t)\le L
\quad\mbox{for all}\quad n\in\N.
\end{equation}
%%%
Further, for $\beta_h=(\alpha_h)_{-1}$, we have
%%%
\begin{equation}\label{eq:LO-shift-7}
\sup_{t\in \I}\beta_h'(t)
=
\sup_{t\in \I}\frac{\vka}{(\vka-\vka t+t)^2}
\le
\vka\Big(\min\{1,\vka\}\Big)^{-2}.
\end{equation}
%%%
  From \reff{eq:LO-shift-8}--\reff{eq:LO-shift-7} and
Lemma~\rref{le:shift-convergence} we obtain
%%%
\begin{equation}\label{eq:LO-shift-9}
\slim W_{\alpha_{h_n}}=W_{\alpha_h}.
\end{equation}
%%%
Combining \reff{eq:LO-shift-4}--\reff{eq:LO-shift-5} and \reff{eq:LO-shift-9}, 
we arrive at \reff{eq:LO-shift-1}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Under the assumptions of Theorem~\rref{th:LO-shift-prime} we cannot 
prove the existence of the strong limit
\[
\slim(V_{h_n}^{-1}W_\alpha V_{h_n})^*.
\]
To prove this we need more restrictions on the shift function
$\alpha$. More precisely, we have to assume that $\alpha'\in SO$
(see Lemma~\rref{le:SO-shift3} and Theorem~\rref{th:LO-shift}). As
mentioned before, this implies that $\wa\in SO$, but in general 
$\alpha'\notin SO$ if $\wa\in SO$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:SO-shift2}
Suppose $\wa\in SO$ and $\alpha'\in\cG B$. If for some test
sequence $h=\{h_n\}_{n=1}^\infty$ the limit \reff{eq:SO-shift-1}
exists, then for every $t\in[0,1]$ the sequence $\beta_{h_n}(t)$
converges to the function
%%%
\begin{equation}\label{eq:SO-shift2-0}
\beta_h(t):=\frac{t}{\vka-\vka t+t}.
\end{equation}
%%%
Moreover, this convergence is uniform with respect to $t$ on every
segment $J\subset \I$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof} Lemma~\rref{le:SO-shift2} is reduced to Lemma~\rref{le:SO-shift}(a).
Since $\alpha'\in \cG B$, from Proposition~\rref{pr:shift}(b) it
follows that $\wa\in \cG BC$. Hence $\vka>0$ and
\reff{eq:SO-shift-1} implies that
%%%
\begin{equation}\label{eq:SO-shift2-1}
\lim_{n\to\infty}\wb(\alpha(h_n))
=
\lim_{n\to\infty}\frac{\beta(\alpha(h_n))}{\alpha(h_n)}
=
\lim_{n\to\infty}\frac{h_n}{\alpha(h_n)}
=
\frac{1}{\vka}.
\end{equation}
%%%
Due to Proposition~\rref{pr:shift}(a),
%%%
\begin{equation}\label{eq:SO-shift2-2}
m_{\beta}\alpha(h_n)\le \beta(\alpha(h_n))\le
M_{\beta}\alpha(h_n).
\end{equation}
%%%
By Proposition \rref{pr:shift}(c), $\wb\in SO$. Taking into
account that
$\widetilde{\beta}(h_n)=\widetilde{\beta}\Big[\beta(\alpha(h_n))\Big]$,
we deduce from \reff{eq:SO-shift2-1}--\reff{eq:SO-shift2-2} and
Lemma~\rref{le:SO-uniform} with ${\cal F}(x,y)=\beta(x)$ and the
test sequence $h=\{\alpha(h_n)\}_{n=1}^{\infty}$ that
%%%
\begin{equation}\label{eq:SO-shift2-3}
\lim_{n\to \infty}\widetilde{\beta}(h_n)=\lim_{n\to
\infty}\widetilde{\beta}(\alpha(h_n))=\frac{1}{\vka}.
\end{equation}
%%%
Now Lemma \rref{le:SO-shift2} follows from Lemma
\rref{le:SO-shift}(a) with $\beta$ in place of $\alpha$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:SO-shift3}
Suppose $\alpha$ is a slowly oscillating shift at zero. If for
some test sequence $h=\{h_n\}_{n=1}^\infty$ the limit
\reff{eq:SO-shift-1} exists, then for every $t\in \I$ the
sequence of the derivatives $\beta_{h_n}'(t)$ converges to the
derivative $\beta_h'(t)$. Moreover, this convergence is uniform
with respect to $t$ on every segment $J\subset \I$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
This lemma is proved by analogy with Lemmas~\rref{le:SO-shift} and
\rref{le:SO-shift2}. By Lemma~\rref{le:SO-equivalence}, $\beta'\in
SO$.  From Lemma~\rref{le:SO-wa} it follows that $\wb\in SO$.
Then, by \reff{eq:SO-shift-1}, for the test sequence
$\{\alpha(h_n)\}_{n=1}^{\infty}$ we have \reff{eq:SO-shift2-1},
which implies \reff{eq:SO-shift2-3} (see the proof of
Lemma~\rref{le:SO-shift2}). Furthermore, since
$\beta$ is a slowly oscillating shift at zero, we derive from
Corollary~\rref{co:equivalence} and \reff{eq:SO-shift2-3} that
%%%
\begin{equation}\label{eq:SO-shift3-1}
\lim_{n\to\infty}\beta'(h_n)
=
\lim_{n\to\infty}\widetilde{\beta}(h_n)
=
\frac{1}{\vka}.
\end{equation}
%%%
Then we deduce from \reff{eq:SO-shift3-1}, \reff{eq:Fest}, and
Lemma~\rref{le:SO-uniform} that
%%%
\begin{equation}\label{eq:SO-shift3-2}
\wb[F(h_n,t)]\uniform\frac{1}{\vka}, \quad
\beta'[F(h_n,t)]\uniform\frac{1}{\vka} \quad\mbox{for every}\quad
J\subset \I.
\end{equation}
%%%
 From \reff{eq:wx-inverse-concrete} and \reff{eq:beta_x} it follows
that
%%%
\begin{equation}\label{eq:SO-shift3-3}
\beta_{h_n}'(t)=\frac{\beta'[F(h_n,t)]}{t^2\Big(h_n/F(h_n,t)+(1-h_n)\wb[F(
h_n,t)]\Big)^2}.
\end{equation}
%%%
Combining \reff{eq:SO-shift-4}, \reff{eq:SO-shift2-0}, \reff{eq:SO-shift3-2},
and \reff{eq:SO-shift3-3}, we infer that
\[
\beta'_{h_n}(t)\uniform\frac{1/\vka}{t^2\Big((1-t)/t+1/\vka\Big)^2}
=\frac{\vka}{(\vka-\vka t+t)^2}=\beta_h'(t)
\]
on every $J\subset \I$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

On the basis of Lemmas~\rref{le:shift-convergence}--\rref{le:SO-shift},
Theorem~\rref{th:LO-shift-prime}, and
Lemmas~\rref{le:SO-shift2}--\rref{le:SO-shift3}
we can calculate the limit operators for the shift operator $W_\alpha$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:LO-shift}
Suppose $\alpha$ is a slowly oscillating shift at zero.
If for some test sequence $h=\{h_n\}_{n=1}^\infty$ the limit
%%%
\begin{equation}\label{eq:LO-shift-10}
\lim_{n\to\infty}\alpha'(h_n)=:\vka
\end{equation}
%%%
exists, then for the shift operator $W_\alpha$ the limit
operator $(W_\alpha)_{\cV_h}$ with respect to the test sequence of
operators
%%%
\begin{equation}\label{eq:LO-shift-11}
\cV_h:=\Big\{V_{h_n}\in\fB(L^p)\ :\
\lim_{n\to\infty} h_n=0,
\
\lim_{n\to\infty}\alpha'(h_n)=\vka
\Big\}
\subset\V
\end{equation}
%%%
exists and this limit operator is given by
$(W_\alpha)_{\cV_h}=f_h W_{\alpha_h}$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
 From  Lemma~\rref{le:SO-wa} it follows that $\wa\in SO$.
Corollary~\rref{co:equivalence} implies that
\[
\lim_{n\to\infty}\wa(h_n)
=
\lim_{n\to\infty}\alpha'(h_n)=\vka.
\]
Then due to Theorem~\rref{th:LO-shift-prime} we obtain \reff{eq:LO-shift-1}.
 From the second equality in \reff{eq:composit-2} and from \reff{eq:fx} we 
deduce that
%%%
\begin{equation}\label{eq:LO-shift-13}
(V_{h_n}^{-1}W_\alpha V_{h_n})^*
=
\beta_{h_n}' W_{\beta_{h_n}} f_{h_n} I.
\end{equation}
%%%
  From Lemmas~\rref{le:SO-shift}(b), \rref{le:SO-shift3}, and
\rref{le:slim-mult} we infer that
%%%
\begin{eqnarray}\label{eq:LO-shift-14}
&&
\slim f_{h_n}I
=
f_hI \quad\Big(\in\fB(L^q)\Big),
\quad
\slim \beta_{h_n}'I
=
\beta_h'I \quad\Big(\in\fB(L^q)\Big).
\end{eqnarray}
%%%

As in the proof of Theorem~\rref{th:LO-shift-prime}, from Lemmas~\rref{le:der-bound},
~\rref{le:shift-convergence}, and~\rref{le:SO-shift2} one can get
%%%
\begin{equation}\label{eq:LO-shift-15}
\slim W_{\beta_{h_n}}=W_{\beta_h} \quad\Big(\in\fB(L^q)\Big).
\end{equation}
%%%
Combining \reff{eq:LO-shift-13}--\reff{eq:LO-shift-15}, we obtain
%%%
\begin{equation}\label{eq:LO-shift-16}
\slim(V_{h_n}^{-1}W_\alpha V_{h_n})^*
=
\slim\beta_{h_n}' W_{\beta_{h_n}} f_{h_n}I
=
\beta_h' W_{\beta_h} f_h I.
\end{equation}
%%%
According to \reff{eq:LO-shift-1} and \reff{eq:LO-shift-16},
$f_hW_{\alpha_h}$ is the limit operator for $W_\alpha$ with
respect to the test sequence \reff{eq:LO-shift-11}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Limit operators for the functional operator}
%%%
Combining Proposition~\rref{pr:LO-properties}, Theorems~\rref{th:LO-mult}
and \rref{th:LO-shift}, we get the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:LO-FO}
Suppose the operator \reff{eq:operator} has slowly oscillating data
$\{a,b;\alpha\}$ at zero. If for some test sequence $h=\{h_n\}_{n=1}^\infty$
the limits
%%%
\[
\lim_{n\to\infty}a(h_n)=:a_h,
\quad
\lim_{n\to\infty}b(h_n)=:b_h,
\quad
\lim_{n\to\infty}\alpha'(h_n)=:\vka
\]
%%%
exist, then for the operator \reff{eq:operator} the
limit operator $A_{\cV_h}$ with respect to the test sequence of operators
\[
\cV_h:=\left\{ V_{h_n}\in\fB(L^p)\
:
\begin{array}{ll}
\lim\limits_{n\to\infty} h_n=0,
&
\lim\limits_{n\to\infty}a(h_n)=a_h,
\\[2ex]
\lim\limits_{n\to \infty}b(h_n)=b_h,
&
\lim\limits_{n\to\infty}\alpha'(h_n)=\varkappa_h
\end{array}
\right\}\subset\V
\]
exists and this limit operator is given by
$A_{\cV_h}=a_h I-b_h f_hW_{\alpha_h}$, where
%%%
\begin{equation}\label{eq:alpha-f}
f_h(t):=(1+\vka t-t)^{-2/p},
\quad
\alpha_h(t):=\frac{\vka t}{1+\vka t-t},
\quad t\in[0,1].
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsection{Necessary conditions for invertibility}
\vspace*{-0.7cm}
\newsubsection{Necessary conditions at the fixed points}
%%%
For the operator \reff{eq:operator} with $a,b\in BC$ and $\alpha'\in{\cal G}BC$,
define the real-valued function
\[
\eta(t):=|a(t)|-|b(t)|\Big(\alpha'(t)\Big)^{-1/p},\quad t\in\I,
\]
and the real numbers
\[
\eta_0(j):=\limsup_{t\to j}\eta(t),
\quad
\eta_1(j):=\liminf_{t\to j}\eta(t),
\quad j\in \{0,1\}.
\]

The results of Section 4 and Theorem~\rref{th:nice} imply the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:necessity}
Suppose the operator \reff{eq:operator} has slowly oscillating data
$\{a,b;\alpha\}$ at the endpoint $j\in\{0,1\}$.
If the operator \reff{eq:operator} is invertible, then
$0\notin [\eta_1(j),\eta_0(j)]$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let us start with the case $j=0$. From the definition of
the operator \reff{eq:operator} with
slowly oscillating data at zero it follows that $\eta\in SO$.
As $\eta\in BC$, for every $\delta\in[\eta_1(0),\eta_0(0)]$
there exists some test sequence $g=\{g_k\}_{k=1}^\infty$ such that
\[
\lim_{k\to\infty}\eta(g_k)=\delta.
\]
Since the functions $a,b,\alpha'\in SO$ are bounded, from the bounded
sequences
\[
\{a(g_k)\}_{k=1}^{\infty},
\quad
\{b(g_k)\}_{k=1}^{\infty},
\quad
\{\alpha'(g_k)\}_{k=1}^{\infty}
\]
we can consecutively extract convergent subsequences
\[
\{a(g_{k_n})\}_{n=1}^{\infty},
\quad
\{b(g_{k_n})\}_{n=1}^{\infty},
\quad
\{\alpha'(g_{k_n})\}_{n=1}^{\infty}.
\]
Put $h_n=g_{k_n},\ n\in\N$. Clearly, $h=\{h_n\}_{n=1}^\infty$
is a subsequence of the test sequence $g=\{g_k\}_{k=1}^\infty$
and the limits
%%%
\begin{equation}\label{eq:necessity1-1}
\lim_{n\to\infty}a(h_n)=:a_h,
\quad
\lim_{n\to\infty}b(h_n)=:b_h,
\quad
\lim_{n\to\infty}\alpha'(h_n)=:\vka
\end{equation}
%%%
exist. Moreover,
\[
\delta=|a_h|-|b_h|\vka^{-1/p}.
\]
Thus, by Theorem~\rref{th:LO-FO}, for the functional operator 
\reff{eq:operator} the limit operator 
$A_h:=A_{\cV_h}=a_h I-b_h f_hW_{\alpha_h}$ exists, where
$a_h$ and $b_h$ are given by \reff{eq:necessity1-1} and $\alpha_h$
and $f_h$ are defined in \reff{eq:alpha-f}. If the operator
\reff{eq:operator} is invertible, then the operator $A_h$ is
invertible too, due to Lemma~\rref{le:tool}. In that case, from
Theorem~\rref{th:nice} we deduce that
\[
|a_h|-|b_hf_h(0)|\Big(\alpha_h'(0)\Big)^{-1/p}=
|a_h|-|b_h|\varkappa^{-1/p}_h=\delta\ne 0.
\]
Since $\delta\in[\eta_1(0),\eta_0(0)]$ is arbitrary, we have
proved that $0\notin[\eta_1(0),\eta_0(0)]$.

It remains to prove the statement in  the case $j=1$.
Consider the reflection operator $R\in\fB(L^p)$ defined by
\[
(R\varphi)(t):=\varphi(1-t),\quad t\in \I.
\]
Clearly, $R$ is involutive and hence, invertible. From the
invertibility of $aI-bW_\alpha$ and the obvious equality
\[
R(aI-bW_\alpha)R=
\widehat{a}I-\widehat{b}W_{\widehat{\alpha}}=:\widehat{A},
\]
where
\[
\widehat{a}(t):=a(1-t), \quad \widehat{b}(t):=b(1-t), \quad
\widehat{\alpha}(t):=1-\alpha(1-t),
\]
it follows that $\widehat{A}$ is invertible too. Since
$\widehat{a}, \widehat{b}\in SO$ and
\[
\lim_{t\to 1}(1-t)
\frac{d}{dt}
\left(\frac{1-\alpha(t)}{1-t}\right)
=
\lim_{t\to 0}t
\frac{d}{dt}
\left(\frac{\widehat{\alpha}(t)}{t}\right)=0,
\]
it follows that the operator $\widehat{A}$ has slowly
oscillating data at zero. From the part just proved we get
\[
0\notin\left[
\liminf_{t\to 0}\widehat{\eta}(t),
\limsup_{t\to 0}\widehat{\eta}(t)
\right]=[\eta_1(1),\eta_0(1)],
\]
where
\[
\widehat{\eta}(t):=\eta(1-t)
=
|\widehat{a}(t)|-|\widehat{b}(t)|
\Big(\widehat{\alpha}'(t)\Big)^{-1/p}, \quad t\in \I,
\]
which completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The following proposition fixes relations between $\eta_0,\eta_1$
and characteristics from Section 2.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{le:aux}
Suppose $\alpha'\in{\cal G}BC$ and $a,b\in BC$.

{\rm (a)} If $a\in{\cal G}BC$ and $\eta_1(j)>0$, then
\begin{equation}\label{f5.2a}
\left( \frac{b}{a}(\alpha')^{-1/p} \right)^*(j)<1.
\end{equation}

{\rm (b)} If $b\in{\cal G}BC$, $\alpha'$ slowly oscillates at the
endpoint $j\in\{0,1\}$, and $\eta_0(j)<0$, then
%%%
\begin{equation}\label{eq:aux-1}
\left( \frac{a}{b}(\alpha_{-1}')^{-1/p} \right)^*(j)<1.
\end{equation}
%%%
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
(a) If $\eta_1(j)>0$, then there is an $\varepsilon>0$ such that
for every $t$ in a neighborhood of $j$, 
\[
|a(t)|>|b(t)|\Big(\alpha'(t)\Big)^{-1/p}+\eps
\]
or, equivalently,
\[
\frac{|b(t)|}{|a(t)|}\Big(\alpha'(t)\Big)^{-1/p}<1-\frac{\varepsilon}{|a(t)|}.
\]
Hence
\[
\left(\frac{b}{a}(\alpha')^{-1/p}\right)^*(j)\leq
1-\eps\left(\inf_{t\in \I}\frac{1}{|a(t)|}\right)<1.
\]

(b) If $\eta_0(j)<0$, then analogously to (a),
%%%
\begin{equation}\label{f5.a}
\left(\frac{a}{b}(\alpha_{-1}'\circ
\alpha)^{-1/p}\right)^*(j)=\left(\frac{a}{b}(\alpha')^{1/p}\right)^*(j)<1,\quad
j\in\{0,1\}.
\end{equation}
%%%
By Lemma~\rref{le:SO-equivalence}, the function
$\alpha_{-1}'=\beta'$ slowly oscillates at $j$ together with
$\alpha'$. Then taking into account Proposition~\rref{pr:shift}(a)
one can prove, by analogy with Lemma~\rref{le:SO-uniform}, that
%%%
\begin{equation}\label{f5.b}
\left( \frac{a}{b}(\alpha_{-1}'\circ\alpha)^{-1/p} \right)^*(j)
=
\left( \frac{a}{b}(\alpha_{-1}')^{-1/p} \right)^*(j),
\quad j\in\{0,1\}.
\end{equation}
%%%
Finally, \reff{eq:aux-1} follows from \reff{f5.a} and \reff{f5.b}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we are in a position to prove necessary conditions for the invertibility
of the operator \reff{eq:operator} at the endpoints (= fixed points of the shift
$\alpha$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{th:necessity2}
Suppose the operator \reff{eq:operator} has slowly oscillating data
$\{a,b;\alpha\}$ at both endpoints $0$ and $1$.
If the operator \reff{eq:operator} is invertible, then
%%%
\begin{equation}\label{eq:nec-fixed}
\mbox{either}\quad
\eta_1(j)>0,
\quad j\in\{0,1\}
\quad\quad
\mbox{or}
\quad
\eta_0(j)<0,
\quad
j\in\{0,1\}.
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
This statement is proved by analogy with \cite[Lemma~5.1]{K01}.
Since $\cG\fB(L^p)$ is open in the norm topology, we can without loss of
generality suppose that $a,b\in\cG BC$. Assume the
contrary. Then one of the following four conditions is fulfilled.

1. $\eta_0(0)<0<\eta_1(1)$.
In view of Proposition~\rref{le:aux}, these conditions
imply that inequality \reff{eq:left-1} is satisfied.
By Corollary~\rref{co:left}, the operator \reff{eq:operator} is not
invertible, and we arrive at a contradiction.

2. $\eta_0(1)<0<\eta_1(0)$. Analogously, by Proposition~\rref{le:aux},
these conditions imply that inequality \reff{eq:right-1} is fulfilled.
We get a contradiction again, due to Corollary~\rref{co:right}.

3. $\eta_1(0)\le 0\le\eta_0(0)$.
In this case we immediately obtain a contradiction from
Theorem~\rref{th:necessity}.

4. $\eta_1(1)\le 0\le\eta_0(1)$.
As in the previous case, we conclude that this situation is impossible
by virtue of Theorem~\rref{th:necessity}.

Thus, in each case we establish a contradiction, and this proves the theorem.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Necessary conditions at inner points}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Under assumption \reff{eq:nec-fixed} we put
\[
\kappa_-:=\left\{\begin{array}{ll}
0, & \eta_1(0)>0,\\
1, & \eta_0(0)<0,
\end{array}\right.
\quad
\kappa_+:=\left\{\begin{array}{ll}
0, & \eta_1(1)>0,\\
1, & \eta_0(1)<0.
\end{array}\right.
\]
Let $n\in\N$. Define the six projections
$\Pi_n^\pm,\Pi_n^0,\widetilde{\Pi}_n^\pm,\widetilde{\Pi}_n^0$ as the
operators of multiplication by the characteristic functions of the sets
\[
\gamma_n^\pm:=
\!\!\!\bigcup_{k=n+1}^\infty\!\!\! \alpha_{\pm k}(l),\quad
\gamma_n^0  :=
\!\!\!\bigcup_{k=-n }^n     \!\!\! \alpha_k(l),\quad
\widetilde{\gamma}_n^\pm :=
\!\!\!\!\bigcup_{k=n+1\pm\kappa_\pm}^\infty\!\!\!\! \alpha_{\pm k}(l),\quad
\widetilde{\gamma}_n^0   :=
\!\!\!\!\bigcup_{k=-n+\kappa_-}^{n+\kappa_+}\!\!\!\!\alpha_k(l).
\]
The space $L^p$ decomposes into the direct sums
%%%
\begin{eqnarray}
L^p &=& \Pi_n^-L^p \stackrel{\cdot}{+} \Pi_n^0L^p
\stackrel{\cdot}{+} \Pi_n^+L^p,
\label{eq:sum}\\
L^p &=& \widetilde{\Pi}_n^-L^p \stackrel{\cdot}{+} \widetilde{\Pi}_n^0L^p
\stackrel{\cdot}{+}\widetilde{\Pi}_n^+L^p.
\label{eq:tilde-sum}
\end{eqnarray}
%%%
We consider $A: L^p\to L^p$ as an operator
acting from the direct sum \reff{eq:tilde-sum} into the direct sum \reff{eq:sum}.
This operator can be represented as the $3\times 3$ operator matrix
%%%
\begin{equation}\label{eq:matrixA}
A=\left[\begin{array}{ccc}
\Pi_n^-A\widetilde{\Pi}_n^- &
\Pi_n^-A\widetilde{\Pi}_n^0 &
\Pi_n^-A\widetilde{\Pi}_n^+ \\
\Pi_n^0A\widetilde{\Pi}_n^- &
\Pi_n^0A\widetilde{\Pi}_n^0 &
\Pi_n^0A\widetilde{\Pi}_n^+ \\
\Pi_n^+A\widetilde{\Pi}_n^- &
\Pi_n^+A\widetilde{\Pi}_n^0 &
\Pi_n^+A\widetilde{\Pi}_n^+
\end{array}\right].
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:invert-plus-minus}
{\rm (a)}
If $\eta_0(1)\eta_1(1)>0$, then the operator
$\Pi_n^+A\widetilde{\Pi}_n^+:\widetilde{\Pi}_n^+L^p\to \Pi_n^+L^p$
is invertible for every sufficiently large $n$.

{\rm (b)}
If $\eta_0(0)\eta_1(0)>0$, then the operator
$\Pi_n^-A\widetilde{\Pi}_n^-:\widetilde{\Pi}_n^-L^p\to \Pi_n^-L^p$
is invertible for every sufficiently large $n$.
\end{lemma}
\vspace{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The statement is proved by the literal repetition of the proof in 
\cite[Lemma~6.2]{K01}. In this proof we essentially use 
Lemma~\rref{le:2sided} and Proposition~\rref{le:aux}.

As usual, denote by $[L^p(l)]^r$ the Banach space of vectors with
$r$ components from $L^p(l)$, equipped with the Euclidean norm.
Consider the operators
\[
\sigma_n:\Pi_n^0L^p\to [L^p(l)]^{2n+1},
\quad
\widetilde{\sigma}_n:\widetilde{\Pi}_n^0L^p\to [L^p(l)]^{2n+1}
\]
defined by
%%%%
\begin{eqnarray}\label{eq:sigma}
&&
(\sigma_n\varphi)(t)=\{\varphi[\alpha_k(t)]\}_{k=-n}^n,
\quad
(\widetilde{\sigma}_n\varphi)(t)=
\{\varphi[\alpha_k(t)]\}_{k=-n+\kappa_-}^{n+\kappa_+},
\quad t\in l.
\end{eqnarray}
%%%%
It is easily seen that the operators $\sigma_n$ and $\widetilde{\sigma}_n$
are invertible. Their inverses are given by
\[
\begin{array}{lll}
\sigma_n^{-1}\Big(\{\psi_k(t)\}_{k=-n}^n\Big)=\psi_k[\alpha_{-k}(t)],
& t\in\alpha_k(l), & k\in\{-n,\dots,n\},
\\
\widetilde{\sigma}_n^{-1}\Big(\{\psi_k(t)\}_{k=-n+\kappa_-}^{n+\kappa_+}\Big)
=\psi_k[\alpha_{-k}(t)], &
t\in\alpha_k(l), & k\in\{-n+\kappa_-,\dots,n+\kappa_+\},
\end{array}
\]
respectively. Then
%%%%
\begin{equation}\label{eq:operatorAn}
A_n=\sigma_n\Pi_n^0A\widetilde{\Pi}_n^0\widetilde{\sigma}_n^{-1}:
[L^p(l)]^{2n+1}\to [L^p(l)]^{2n+1}
\end{equation}
%%%%
is the operator of multiplication by the square matrix function
%%%
\begin{eqnarray}\label{eq:matrixAn}
&&
\cA_n(t)=\Big(
a[\alpha_k(t)]\delta_{k,j} - b[\alpha_k(t)]\delta_{k,j-1}
\Big)_{k=-n,\dots,n;j=-n+\kappa_-,\dots,n+\kappa_+},
\quad t\in \overline{l},
\end{eqnarray}
%%%
with entries continuous on $\overline{l}$, where $\delta_{k,j}$
is the Kronecker delta.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{le:invert-one-cr}
Under assumption \reff{eq:nec-fixed} the operator \reff{eq:operator} is
invertible if and only if the operator \reff{eq:operatorAn} is
invertible for every sufficiently large $n$.
\end{lemma}
\vspace{3mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Using Lemma~\rref{le:invert-plus-minus} this statement can be proved by
analogy with \cite[Lemma~4]{KM88}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{pr:matrix-invertibility}
{\rm (a)} If $\eta_0(0)<0$ and $\eta_0(1)<0$, then $\det\cA_n(t)\ne 0$ for every
$t\in\overline{l}$ if and only if $b[\alpha_k(t)]\ne 0$ for every $k\in\{-n,\dots,n\}$
and $t\in\overline{l}$.

{\rm (b)} If $\eta_1(0)>0$ and $\eta_1(1)>0$, then $\det\cA_n(t)\ne 0$
for every  $t\in\overline{l}$ if and only if
$a[\alpha_k(t)]\ne 0$ for every $k\in\{-n,\dots,n\}$ and $t\in\overline{l}$.
\end{proposition}
\medskip

This statement is obvious.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsubsection{Proof of Theorem~\rref{th:main}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we are able to prove the main result of this paper.
We argue by analogy with the proof of \cite[Theorem~6.8]{K01}.

\begin{proof}
{\it Sufficiency} follows from Lemma~\rref{le:2sided} and
Proposition~\rref{le:aux}.

{\it Necessity.} Suppose that the operator \reff{eq:operator} is
invertible. Then, by Theorem~\rref{th:necessity2}, either
$\eta_1(0)>0$ and $\eta_1(1)>0$, or $\eta_0(0)<0$ and $\eta_0(1)<0$.
In view of Lemma~\rref{le:invert-one-cr}, the operator
\reff{eq:operatorAn} and, hence, the operator $\cA_n(\cdot)I$
are invertible for every sufficiently large $n$.

If $\eta_0(j)<0,j\in\{0,1\}$, then, by 
Proposition~\rref{pr:matrix-invertibility}(a),
$b[\alpha_k(t)]\ne 0$ for every $k\in\{-n,\dots,n\}$, $t\in\overline{l}$
and every sufficiently large $n$. Hence, $b(t)\ne 0$ for all $t\in\I$.
Besides, from the definition of $\eta_0$ we see that $b_*(j)> 0,j\in\{0,1\}$.
Thus, $b\in\cG BC$.

Analogously, using Proposition~\rref{pr:matrix-invertibility}(b),
one can prove that $a\in\cG BC$ in the case $\eta_1(j)>0,j\in\{0,1\}$,
which completes the proof of this theorem.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{acknowledgements}
The first author is partially supported by F.C.T. (Portugal) grant
PRAXIS XXI/BPD/ 22006/99; the second author is partially supported
by CONACYT (M\'exico) grant, C\'atedra Patrimonial, nivel II, No.
990017-R2000; all the authors are partially supported by F.C.T.
(Portugal) under Project No. POCTI/34222/99-FEDER.
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\address{
A.~Yu.~Karlovich, A.~B.~Lebre\\
Departamento de Matem\'{a}tica\\
Instituto Superior T\'{e}cnico\\
Av. Rovisco Pais\\
1049 -- 001 Lisboa\\
Portugal
}
\address{
Yu.~I.~Karlovich\\
Departamento de Matem\'{a}ticas\\
CINVESTAV del I.P.N.\\
Apartado Postal 14--740\\
07000 M\'exico, D.F.\\
M\'exico
}
%\address{ }
%\address{ }

\subjclass{Primary  39B32; Secondary  47A10, 47B38}

\received{ } % filled in by the editors
\end{document}