\documentclass{kluwer} % Specifies the document style. %%% \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}[lemma]{Theorem} \newtheorem{corollary}[lemma]{Corollary} %%% \newcommand{\alg}{{\rm alg}\,} \newcommand{\diag}{{\rm diag}\,} \newcommand{\eps}{{\varepsilon}} %%% % calligraphic \newcommand{\cA}{\mathcal{A}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} %%% % bb \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\T}{\mathbb{T}} %%% % frak \newcommand{\fB}{\mathfrak{B}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \begin{article} \begin{opening} \title{ Compactness of Commutators Arising in the Fredholm Theory of Singular Integral Operators with Shifts} %\thanks{Footnote to the title with the `thanks' command.}} \author{Alexei \surname{Karlovich}\email{akarlov@math.ist.utl.pt}} \institute{Instituto Superior T\'ecnico, Portugal} \author{Yuri \surname{Karlovich}\email{karlovich@buzon.uaem.mx}} \institute{Universidad Aut\'onoma del Estado de Morelos, M\'exico} \runningauthor{Alexei Karlovich and Yuri Karlovich} \runningtitle{Compactness of Commutators} \date{April 8, 2002} \dedication{To Professor G.~S.~Litvinchuk on the occasion of his 70th birthday} \begin{ao}\\ A.~Karlovich\\ Departamento de Matem\'atica\\ Instituto Superior T\'ecnico\\ Av. Rovisco Pais 1\\ 1049-001 Lisboa, Portugal \end{ao} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} The paper is devoted to the compactness of the commutators $aS_\Gamma-S_\Gamma aI$ and $W_\alpha S_\Gamma-S_\Gamma W_\alpha$, where $S_\Gamma$ is the Cauchy singular integral operator, $a$ is a bounded measurable function, $W_\alpha$ is the shift operator given by $W_\alpha f=f\circ\alpha$, and $\alpha$ is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function $a$ or, respectively, the derivative of the shift $\alpha$ has vanishing mean oscillation. \end{abstract} \keywords{Cauchy singular integral operator, shift operator, commutator, compact operator, rearrangement-invariant space, Boyd indices, interpolation of compactness} \end{opening} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Let $\Gamma$ be the unit interval $\mathbb{I}:=(0,1)$, or an arc on the unit circle $\T$, or the unit circle $\T$. For a function $f\in L^1(\Gamma)$, the Cauchy singular integral of $f$ is given by \[ (S_\Gamma f)(t):=\frac{1}{\pi i}\: v.p.\int_\Gamma\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\Gamma. \] Let $\alpha$ be a homeomorphism ({\it shift}) of $\overline{\Gamma}$ onto itself which preserves the orientation on $\overline{\Gamma}$. Consider the shift operator $W_\alpha$ given by %%% \begin{equation}\label{eq:shift} (W_\alpha f)(t):=f[\alpha(t)], \quad t\in\Gamma. \end{equation} %%% Under appropriate assumptions, guaranteeing the boundedness of operators $S_\Gamma, W_\alpha$, and the operator $aI$ of multiplication by a measurable function $a:\Gamma\to\C$, we study the compactness of the commutators %%% \begin{equation}\label{eq:commutators} aS_\Gamma-S_\Gamma aI, \quad W_\alpha S_\Gamma-S_\Gamma W_\alpha \end{equation} %%% on rearrangement-invariant spaces $X(\Gamma)$. These commutators arise naturally in the Fredholm theory of singular integral operators with shifts \cite{KS01,KL94,L77,L00}. Moreover, Fredholm criteria for singular integral operators with shifts depend essentially on whether the commutators (\ref{eq:commutators}) are compact or not. Commutators of multiplication operators and singular integral operators are closely connected with operators of Hankel type. The compactness of both these types of operators on Lebesgue spaces was studied by many authors. We mention here \cite{H58,J78,U78} and also a survey \cite{P80}. Sufficient conditions for the compactness of commutators of multiplication operators and singular integral operators with kernels of Calder\'on-Zygmund type on $L^p(X)$, where $X$ is a space of homogeneous type, are obtained in \cite[Part II, Theorem~1.1]{KL01}, see also the references therein. The compactness of the commutator of the Cauchy singular integral operator $S_\Gamma$ and the shift operator $W_\alpha$ was studied on $L^p(\Gamma)$ by D.~Kveselava under assumptions that $\Gamma$ is a Lyapunov curve and $\alpha'$ is a H\"older continuous function (see \cite[Ch.~1, Theorem~3.1]{L77}). That theorem was extended to the case of smooth curves and $C^1$-diffeomorphisms in \cite{GD77}, its detailed proof is in \cite{G80}. All mentioned results give only sufficient conditions for the compactness of $W_\alpha S_\Gamma-S_\Gamma W_\alpha$. A compactness criterion for $W_\alpha S_\T-S_\T W_\alpha$ on $L^p(\T), 1\lambda\}=\mu\{x\in\cR:|g(x)|>\lambda\}$ for all $\lambda\ge 0$. A function norm $\rho:\cM^+ \to [0,\infty]$ is called rearrangement-invariant if $\rho(f)=\rho(g)$ for every pair of equimeasurable functions $f,g \in \cM^+_0$. In that case, the Banach function space $X(\cR,\mu)$ generated by $\rho$ is said to be a {\it rearrangement-invariant space}. The Lebesgue space $L^p(\cR,\mu)$, $1\le p\le\infty$, is the simplest example of a rearrangement-invariant space. Orlicz and Lorentz spaces are other important classical examples of rearrangement-invariant spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Interpolation of compact operators on rearrangement-invariant spaces} %%% In this subsection we state two interpolation theorems for compact operators which allow us to reduce studying the compactness on rearrange\-ment-invariant spaces to studying the compactness on the Lebesgue space $L^2(\cR,\mu)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Berezhnoi} Let $X(\cR,\mu)$ be a rearrangement-invariant space and $X'(\cR,\mu)$ its associate space. Suppose a linear operator $A$ is bounded and compact on $X(\cR,\mu)$ and is bounded on $X'(\cR,\mu)$. Then $A$ is bounded and compact on the Lebesgue space $L^2(\cR,\mu)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This result follows from Berezhnoi's interpolation theorem for compact operators on Calder\'on-Lozanovskii spaces $\varphi(X_0,X_1)$ \cite{Ber80} and Lozanovskii's result saying that $L^2(\cR,\mu)$ is isometrically isomorphic to the Calder\'on-Lozanovskii space $\varphi(X(\cR,\mu),X'(\cR,\mu))$ with $\varphi(t)=\sqrt{t},\, t\in\R_+:=(0,\infty)$ \cite[Theorem~5]{Loz1969}, see also \cite{Cald1964}. One can find more information on Calde\-r\'on-Lozanovskii spaces in \cite{BK91} and \cite[Ch. 15]{Mal89}. We need a notion of the Boyd indices of rearrangement-invariant spaces to state the second interpolation theorem. By the Luxemburg representation theorem \cite[Ch.~2, Theorem~4.10]{BeSh}, there is a unique rearrangement-invariant function norm $\overline{\rho}$ over $[0,\mu(\cR)]$ with the Lebesgue measure $m$ such that $\rho(f) = \overline{\rho}(f^*)$ for all $f\in \cM_0^+$, where $f^*$ is the non-increasing rearrangement of $f$ (see, e.g., \cite[p.~39]{BeSh}). The rearrangement-invariant space over $([0,\mu(\cR)],m)$ generated by $\overline{\rho}$ is called the Luxemburg representation of $X(\cR,\mu)$ and is denoted by $\overline{X}$. For each $s\in\R_+$, let $E_s$ denote the dilation operator defined on $\cM_0([0,\mu(\cR)],m)$ by \[ (E_s f)(t):= \left\{ \begin{array}{ll} f(st), & st\in[0,\mu(\cR)]\\ 0, & st\not\in[0,\mu(\cR)] \end{array} \right. , \quad t\in [0,\mu(\cR)]. \] For every $s\in\R_+$, the operator $E_{1/s}$ is bounded on $\overline{X}$ \cite[p.~165]{BeSh}, its norm is denoted by $h_X(s)$. The function $h_X:\R_+\to\R_+$ is submultiplicative and non-decreasing. From \cite[Ch.~2, Theorem~1.3]{kps} it follows that the limits \[ \underline{\sigma}_X:=\lim_{s\to 0}\frac{\log h_X(s)}{\log s}, \quad \overline{\sigma}_X:=\lim_{s\to \infty}\frac{\log h_X(s)}{\log s} \] exist and $\underline{\sigma}_X\le\overline{\sigma}_X$. The numbers $\underline{\sigma}_X$ and $\overline{\sigma}_X$ are called the {\it lower and upper Boyd indices} of the rearrangement-invariant space $X(\cR,\mu)$, respectively. For an arbitrary rearrangement-invariant space, its Boyd indices belong to $[0,1]$. The Boyd indices of the rearrangement-invariant spaces $X(\cR,\mu)$ and $X'(\cR,\mu)$ are connected via the following identity: %%% \begin{equation}\label{eq:duality} \underline{\sigma}_X+\overline{\sigma}_{X'}= \underline{\sigma}_{X'}+\overline{\sigma}_X=1. \end{equation} %%% One can find the proofs of these and other properties of the Boyd indices in \cite{BeSh,LT,Mal85}. For the Lebesgue spaces $L^p(\cR,\mu), 1\le p\le\infty$, the Boyd indices coincide and equal $1/p$. We will say that the Boyd indices are nontrivial if $\underline{\sigma}_X, \overline{\sigma}_X\in(0,1)$. In the case of Orlicz spaces the latter condition is equivalent to the reflexivity of the space (see, e.g., \cite{Mal85}). An example of Young functions which generate reflexive Orlicz spaces whose Boyd indices do not coincide is given, for instance, in \cite[p.~93]{Mal89}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Sharpley} Let $10$. For such $\alpha$, the shift operator $W_\alpha$ given by (\ref{eq:shift}) is bounded and continuously invertible on every Lebesgue space $L^p(\Gamma), 1\le p\le\infty$. Its inverse equals $W_\alpha^{-1}=W_{\alpha_{-1}}$. Therefore, by the Calder\'on-Mitjagin interpolation theorem (see, e.g., \cite[Theorem~2.a.10]{LT}), the operator $W_\alpha$ is bounded and continuously invertible on an arbitrary rearrangement-invariant space $X(\Gamma)$ whenever $\alpha$ and $\alpha_{-1}$ are absolutely continuous and $\alpha'\in\cG L^\infty(\Gamma)$. The following theorem gives a criterion for the boundedness of the Cauchy singular integral operator on rearrangement invariant spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:boundedness-S} Let $\Gamma$ be the unit circle $\T$ or the unit interval $\mathbb{I}$. The Cauchy singular integral operator $S_\Gamma$ is bounded on a rearrangement-invariant space $X(\Gamma)$ if and only if $X(\Gamma)$ has nontrivial Boyd indices. \end{theorem} The idea of the proof of Theorem~\ref{th:boundedness-S} goes back to D.~Boyd, who considered the case of the real line (for the proof, see, e.g., \cite[Ch.~3, Theorem~5.18]{BeSh}). The proofs for the unit interval and for the unit circle are contained in \cite[Ch.~2, Section~8.6]{kps}. Finally, if $a\in L^\infty(\Gamma)$, $\alpha$ is a bi-Lipschitz homeomorphism of $\overline{\Gamma}$ onto itself, and the Boyd indices of $X(\Gamma)$ are nontrivial, then the commutators (\ref{eq:commutators}) are bounded on $X(\Gamma)$. In particular, they are bounded on every Lebesgue space $L^p(\Gamma)$ for $p\in(1,\infty)$. Thus, we can apply Corollary~\ref{co:interpolation} to study the compactness of the commutators (\ref{eq:commutators}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Vanishing mean oscillation} In this subsection we mainly follow \cite{S77}. The length of an interval $I\subset\R$ will be denoted by $|I|$. For $f$ in $L^1(I)$, the average of $f$ over $I$ will be denoted by $I(f)$: \[ I(f):=\frac{1}{|I|}\int_I f(t)dt. \] Considering all subintervals $J$ of $I$, one can introduce the quantity \[ M(f,I):=\lim_{s\to 0+}\sup_{J\subset I,\, |J|\le s}\frac{1}{|J|}\int_J|f(t)-J(f)|dt. \] The space $VMO(I)$ is defined to be the class of functions $f$ in $L^1(I)$ that satisfy $M(f,I)=0$ ($VMO$ means {\it ``vanishing mean oscillation''}). For $f$ an integrable function defined in an open interval containing the point $\lambda$ of $\R$, the integral gap of $f$ is given by \[ \gamma_\lambda(f):=\limsup_{\delta\to 0} \left|\frac{1}{\delta}\int_\lambda^{\lambda+\delta}f(t)dt -\frac{1}{\delta}\int_{\lambda-\delta}^{\lambda}f(t)dt\right|. \] An elementary estimate shows that if $f\in VMO(I)$, then $\gamma_\lambda(f)=0$ for each interior point $\lambda$ of $I$. Analogous definitions can obviously be made for functions given on an arc of the unit circle $\T$ or on the whole unit circle $\T$. We suppose that the unit circle is counter-clockwise oriented. For $t_1,t_2\in\T$ we denote by $(t_1,t_2)$ the arc of $\T$ with the starting point $t_1$ and the terminating point $t_2$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:glying} Let $t_1,t_2\in\T$ and $t_1\ne t_2$. Suppose a function $f:\T\to\C$ satisfies the properties $f|_{(t_1,t_2)}\in VMO(t_1,t_2), f|_{(t_2,t_1)}\in VMO(t_2,t_1)$ and $\gamma_{t_j}(f)=0$ for $j\in\{1,2\}$. Then $f\in VMO(\T)$. \end{lemma} This statement is proved by the repetition of the proof of \cite[Lemma~2]{S77} with minor modifications. We remark that %%% \begin{equation}\label{eq:QC-1} QC(\T):=(C(\T)+H^\infty)\cap(C(\T)+\overline{H^\infty}) = L^\infty(\T)\cap VMO(\T) \end{equation} %%% is a nonseparable $C^*$-subalgebra of $L^\infty(\T)$ properly containing the\\ $C^*$-algebra $C(\T)$ of all continuous functions on $\T$ (see \cite[Section~4]{S75}). We refer to \cite{BS90}, \cite{N86}, and \cite{S78} for these and other properties of the algebra $QC(\T)$ of quasicontinuous functions on $\T$. The following statement is probably known (cf. \cite[Ch.~VIII, Problem~6.13]{Tor86}, where a BMO-analogue was stated). For convenience of the readers we give its proof. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:Lerner} Suppose $I$ is a finite interval of $\R$, or an arc of $\T$, or $\T$. If $f\in VMO(I)$, then $|f|^\alpha$ belongs to $VMO(I)$ for every $\alpha\in(0,1]$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} Assume for definiteness that $I$ is a finite interval of $\R$. Other cases are considered analogously. By \cite[Ch.~VIII, Corollary~1.2]{Tor86}, for any $F\in L^1(I)$ and every $J\subset I$, %%% \begin{equation}\label{eq:Lerner-1} \frac{1}{|J|}\int_J|F(t)-J(F)|dt\le 2\inf_{c\in\C}\frac{1}{|J|}\int_J|F(t)-c|dt. \end{equation} %%% Applying (\ref{eq:Lerner-1}), the well-known inequality \[ ||a|^\alpha-|b|^\alpha|\le |a-b|^\alpha, \quad\alpha\in(0,1],\quad a,b\in\C, \] and H\"older's inequality, %to the function $|f-c|^\alpha$, we get %%% \begin{eqnarray*} && \frac{1}{|J|}\int_J||f(t)|^\alpha-J(|f|^\alpha)|dt \le 2\inf_{c\in\C}\frac{1}{|J|}\int_J||f(t)|^\alpha-|c|^\alpha|dt \\ && \le 2\inf_{c\in\C}\frac{1}{|J|}\int_J|f(t)-c|^\alpha dt \le 2\left(\inf_{c\in\C}\frac{1}{|J|}\int_J|f(t)-c|dt\right)^\alpha \\ && \le 2\left(\frac{1}{|J|}\int_J|f(t)-J(f)|dt\right)^\alpha. \label{eq:Lerner-2} \end{eqnarray*} %%% So, $M(|f|^\alpha,I)\le 2(M(f,I))^\alpha$, which implies the required statement. \end{pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Compactness of commutators on $L^2$} \subsection{Case of the unit circle} For a Banach space $X$, let $\fB(X)$ be the Banach algebra of all bounded linear operators on $X$ and $\cK(X)$ the ideal of all compact operators on $X$. For $C,D\in\fB(X)$, put $[C,D]:=CD-DC$. Combining the compactness criteria from \cite{H58} and Sarason's representations of quasicontinuous functions (\ref{eq:QC-1}) immediately gives the following criterion (see also \cite[Section~2]{P80}). \begin{theorem}\label{th:T-aS-Sa} Suppose $a\in L^\infty(\T)$. The operator $aS_\T-S_\T aI$ is compact on $L^2(\T)$ if and only if $a\in VMO(\T)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The next statement follows from \cite[Theorem 4.1 and Proposition 4.5]{MX95}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:T-WS-SW} Suppose $\alpha$ is an orientation-preserving bi-Lipschitz homeomorphism of\ $\T$ onto itself. The operator $W_\alpha S_\T-S_\T W_\alpha$ is compact on $L^2(\T)$ if and only if $\alpha'\in VMO(\T)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Operators with fixed singularities on the upper half-circle} Let $\T_+:=\{\tau\in\T:\ \arg\tau\in(0,\pi)\}$ be the upper half-circle of the unit circle $\T$. Using standard methods we prove in this subsection that the operator \[ (N_{\T_+}f)(t):=\frac{1}{\pi i}\int_{\T_+} \frac{f(\tau)}{1-\tau t}\,d\tau, \quad t\in\T_+, \] having fixed singularities at $-1$ and $1$, belongs to $\alg(I,S_{\T_+})$, the smallest Banach subalgebra of $\fB(L^2(\T_+))$ which contains the operators $I$ and $S_{\T_+}$. \newpage Consider a M\"obius transformation of $\overline{\T_+}$ onto $\overline{\R_+}$ which sends the points $1$ and $-1$ to $0$ and $+\infty$, respectively. Namely, let \[ \psi:\overline{\T_+}\to\overline{\R_+}, \quad \psi(t)=i\frac{1-t}{1+t}. \] Obviously, its inverse is \[ \xi:\overline{\R_+}\to\overline{\T_+}, \quad \xi(x)=\frac{i-x}{i+x}. \] The map $B$ given by %%% \begin{equation}\label{eq:B} (Bf)(x) = \frac{\sqrt{2}}{i+x}\,f(\xi(x)), \quad x\in\R_+, \end{equation} %%% is an isometric isomorphism of $L^2(\T_+)$ onto $L^2(\R_+)$ with the inverse %%% \begin{equation}\label{eq:B-inverse} (B^{-1}\varphi)(t) = \frac{i\sqrt{2}}{1+t}\,\varphi(\psi(t)) ,\quad t\in\T_+. \end{equation} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma} The operator $N_{\T_+}$ belongs to the algebra $\alg(I,S_{\T_+})$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} For $f\in L^2(\R_+)$, put \[ (N_{\R_+}f)(x):=\frac{1}{\pi i}\int_{\R_+}\frac{f(y)}{x+y}\,dy, \quad x\in\R_+. \] It is well known (see, e.g., \cite[Theorem~2.3]{KN75} or \cite[Proposition~2.4]{RS90}), that this operator belongs to $\alg(I,S_{\R_+})\subset\fB(L^2(\R_+))$. Using (\ref{eq:B}) and (\ref{eq:B-inverse}), one can straightforwardly check that $BS_{\T_+}B^{-1}=S_{\R_+}$ and $BN_{\T_+}B^{-1}=-N_{\R_+}$. Thus, $N_{\T_+}$ is bounded on $L^2(\T_+)$ and, moreover, it belongs to $\alg(I,S_{\T_+})$. \end{pf} This lemma immediately implies the following. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{corollary}\label{co:fixed-sing} Suppose $a\in L^\infty(\T_+)$. If $aS_{\T_+}-S_{\T_+}aI$ is compact on $L^2(\T_+)$, then $aN_{\T_+}-N_{\T_+}aI$ is compact on $L^2(\T_+)$ too. \end{corollary} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Compactness of commutators with multiplication operators. Case of the upper half-circle} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\T_-:=\{\tau\in\T:\ \arg\tau\in(\pi,2\pi)\}$ be the lower half-circle of the unit circle $\T$. For a function $a\in L^\infty(\T_+)$, we consider the function %%% \begin{equation}\label{eq:A} A(\tau):= \left\{ \begin{array}{cc} a(\tau), &\tau\in\T_+,\\ a(\overline{\tau}), &\tau\in\T_- \end{array} \right. \end{equation} %%% given on the whole circle $\T$. The functions $a$ and $A$ have an analogous behavior in the neighborhoods of $\pm 1$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:sewing} A function $a$ belongs to $L^\infty(\T_+)\cap VMO(\T_+)$ if and only if the function $A$ given by {\rm (\ref{eq:A})} belongs to $L^\infty(\T)\cap VMO(\T)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} {\it Sufficiency} is obvious because $A|_{\T_+}=a$. {\it Necessity.} Since $a\in L^\infty(\T_+)$, we have $A|_{\T_\pm}\in L^\infty(\T_\pm)$. Therefore, $A\in L^\infty(\T)$. On the other hand, $A|_{\T_+}=a\in VMO(\T_+)$ if and only if $A|_{\T_-}\in VMO(\T_-)$. For $\delta\in(0,\pi)$, we set %%% \begin{eqnarray*} \T_+(1,\delta) &:=& \Big\{ \tau\in\T\::\ \arg\tau\in(0,\pi),\ |(1,\tau)|<\delta \Big\},\\ \T_-(1,\delta) &:=& \Big\{ \tau\in\T\::\ \arg\tau\in(\pi,2\pi),\ |(\tau,1)|<\delta \Big\},\\ \T_+(-1,\delta) &:=& \Big\{ \tau\in\T\::\ \arg\tau\in(0,\pi),\ |(\tau,-1)|<\delta \Big\},\\ \T_-(-1,\delta) &:=& \Big\{ \tau\in\T\::\ \arg\tau\in(\pi,2\pi),\ |(-1,\tau)|<\delta \Big\}, \end{eqnarray*} %%% where $|J|$ denotes the length of an interval $J\subset \T$. Then for $t\in\{-1,1\}$ we get %%% \begin{eqnarray*} && \frac{1}{\delta}\int_{\T_+(t,\delta)}A(\tau)|d\tau| = \frac{1}{\delta}\int_{\T_+(t,\delta)}a(\tau)|d\tau| \\ &&= \frac{1}{\delta}\int_{\T_-(t,\delta)}a(\overline{\tau})|d\tau| = \frac{1}{\delta}\int_{\T_-(t,\delta)}A(\tau)|d\tau|. \end{eqnarray*} %%% Hence, $\gamma_t(A)=0$ for $t\in\{-1,1\}$. Thus, $A\in VMO(\T)$ due to Lemma~\ref{le:glying}. Lemma is proved. \end{pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 Consider the Hilbert space $L_2^2(\T_+)$ of vector functions $\varphi=\{\varphi_j\}_{j=1}^2$ with the entries $\varphi_j\in L^2(\T_+)$ which is equipped with the norm \[ \|\varphi\|_{L_2^2(\T_+)}= \left( \|\varphi_1\|^2_{L^2(\T_+)}+\|\varphi_2\|^2_{L^2(\T_+)} \right)^{1/2}. \] It is easy to see that the operator $\sigma:L^2(\T)\to L_2^2(\T_+)$ defined by %%% \begin{equation}\label{eq:sigma} (\sigma\varphi)(\tau):= \left\{ \begin{array}{c}\varphi(\tau)\\ \overline{\tau}\,\varphi(\overline{\tau}) \end{array} \right\}, \quad\tau\in\T_+, \end{equation} %%% is bounded and continuously invertible, and its inverse is given by %%% \begin{equation}\label{eq:sigma-inverse} \left( \sigma^{-1}%{\tiny \left\{ %\!\!\! \begin{array}{c}\varphi_1\\ \varphi_2\end{array} %\!\!\! \right\}%} \right) (\tau)= \left\{ \begin{array}{cc} \varphi_1(\tau), &\tau\in\T_+,\\ \overline{\tau}\,\varphi_2(\overline{\tau}), & \tau\in\T_-. \end{array} \right. \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:cutting} Suppose $a\in L^\infty(\T_+)$ and $A$ is given by {\rm (\ref{eq:A})}. Then %%% \[ \sigma (AI)\sigma^{-1} = \left( \begin{array}{cc} aI & 0\\ 0 & aI \end{array} \right), \quad \sigma S_\T\sigma^{-1} = \left( \begin{array}{cc} S_{\T_+} & N_{\T_+}\\ -N_{\T_+} & -S_{\T_+} \end{array} \right). \] \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} The first equality is obvious. Let us prove the second one. Assume that $\varphi_1,\varphi_2\in L^2(\T_+)$ and $t\in\T$. Then from (\ref{eq:sigma-inverse}) we get \[ \left(S_\T\sigma^{-1} \left\{ \begin{array}{c} \varphi_1\\\varphi_2\end{array} \right\}\right)(t)= \frac{1}{\pi i}\int_{\T_+}\frac{\varphi_1(\tau)}{\tau-t}\,d\tau + \frac{1}{\pi i}\int_{\T_-}\frac{\overline{\tau}\,\varphi_2(\overline{\tau})}{\tau-t}\,d\tau. \] Changing variables in the second integral, we infer in view of $\overline{\tau}=1/\tau$ that %%% \begin{equation}\label{eq:cutting-1} \left(S_\T\sigma^{-1} \left\{ \begin{array}{c} \varphi_1\\\varphi_2\end{array} \right\} \right)(t)= (S_{\T_+}\varphi_1)(t)+(N_{\T_+}\varphi_2)(t), \quad t\in\T. \end{equation} Again taking into account that $\overline{t}=1/t$, we obtain for $t\in\T_+$, %%% \begin{equation}\label{eq:cutting-2} \overline{t}\,(S_{\T_+}\varphi_1)(\overline{t}) = -(N_{\T_+}\varphi_1)(t), \quad \overline{t}\,(N_{\T_+}\varphi_2)(\overline{t}) = -(S_{\T_+}\varphi_2)(t). \end{equation} %%% From (\ref{eq:sigma}) and (\ref{eq:cutting-1})--(\ref{eq:cutting-2}) we get the second equality. Lemma is proved. \end{pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:T_+aS-Sa} Suppose $a\in L^\infty(\T_+)$. The operator $aS_{\T_+}-S_{\T_+}aI$ is compact on $L^2(\T_+)$ if and only if $a\in VMO(\T_+)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} Suppose $a\in L^\infty(\T_+)$ and $A$ is given by (\ref{eq:A}). From Lemma~\ref{le:cutting} we deduce that %%% \begin{equation}\label{eq:T+aS-Sa} \sigma [AI, S_\T] \sigma^{-1} = \left( \begin{array}{cc} [ aI, S_{\T_+} ] & [ aI, N_{\T_+} ]\\[1ex] [ N_{\T_+}, aI ] & [ S_{\T_+}, aI ] \end{array} \right). \end{equation} %%% {\it Necessity.} If $[aI,S_{\T_+}]\in\cK(L^2(\T_+))$, then, due to Corollary~\ref{co:fixed-sing}, $[aI,N_{\T_+}]\in\cK(L^2(\T_+))$. Hence, (\ref{eq:T+aS-Sa}) implies that $[AI,S_\T]\in\cK(L^2(\T))$. By Theorem~\ref{th:T-aS-Sa}, $A\in VMO(\T)$. Thus, $a=A|_{\T_+}\in VMO(\T_+)$. {\it Sufficiency.} If $a\in VMO(\T_+)$, then, in view of Lemma~\ref{le:sewing}, the function $A$ defined by (\ref{eq:A}) belongs to $L^\infty(\T)\cap VMO(\T)$. Hence, by Theorem~\ref{th:T-aS-Sa}, $[AI,S_\T]\in\cK(L^2(\T))$. Taking into account (\ref{eq:T+aS-Sa}), we infer that $[aI,S_{\T_+}]\in\cK(L^2(\T_+))$. Theorem is proved. \end{pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Compactness of commutators with multiplication operators. Case of the unit interval} %%% In this subsection we ``transplant'' the commutator $aS_\mathbb{I}-S_\mathbb{I} aI$ from $\mathbb{I}$ to $\T_+$ using the $C^\infty$-diffeomorphism \[ \theta:\mathbb{I}\to\T_+, \quad \theta(x):=e^{\pi i x},\quad x\in\mathbb{I}. \] This transformation preserves the compactness of operators and the class of coefficients. This allows us to apply Theorem~\ref{th:T_+aS-Sa}. The inverse for $\theta$ is given by \[ \theta_{-1}:\T_+\to\mathbb{I}, \quad \theta_{-1}(\tau)=\frac{\log\tau}{\pi i} = \frac{\arg\tau}{\pi}, \quad \tau\in\T_+. \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:VMO-transpl} A function $f$ belongs to $L^\infty(\mathbb{I})\cap VMO(\mathbb{I})$ if and only if the function $f\circ\theta_{-1}$ belongs to $L^\infty(\T_+)\cap VMO(\T_+)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lemma~\ref{le:VMO-transpl} is proved by analogy with \cite[Proposition~2.5]{BKS94}. Clearly, the bounded linear operator $G:L^2(\T_+)\to L^2(\mathbb{I})$ given by \[ (G\varphi)(x):=\varphi(\theta(x)), \quad x\in\mathbb{I}, \] is continuously invertible and its inverse is given by \[ (G^{-1}f)(\tau)=f(\theta_{-1}(\tau)), \quad\tau\in\T_+. \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:transpl-compact} The operator $GS_{\T_+}G^{-1}-S_\mathbb{I}$ is compact on $L^2(\mathbb{I})$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This statement is proved by analogy with \cite[Ch.~1, Theorem~3.1]{L77} using that the kernel of the integral operator $GS_{\T_+}G^{-1}-S_\mathbb{I}$ has weak singularities on $\mathbb{I}$. But this is not the case if to take arbitrary smooth curves instead of the Lyapunov curves $\mathbb{I}$ and $\T_+$. Generalizations of Lemma~\ref{le:transpl-compact} to the case of arbitrary smooth curves can be found in \cite{G80} and in \cite{KR02}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:I-aS-Sa} Suppose $a\in L^\infty(\mathbb{I})$. The operator $aS_\mathbb{I}-S_\mathbb{I} aI$ is compact on $L^2(\mathbb{I})$ if and only if $a\in VMO(\mathbb{I})$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} Let $a\in L^\infty(\mathbb{I})$. From the obvious equality $aI=G(a\circ\theta_{-1})G^{-1}$ and Lemma~\ref{le:transpl-compact} we get %%% \begin{equation}\label{eq:I-aS-Sa} [aI,S_\mathbb{I}]-G[(a\circ\theta_{-1})I,S_{\T_+}]G^{-1} = [aI,S_{\mathbb{I}}-GS_{\T_+}G^{-1}] \in\cK(L^2(\mathbb{I})). \end{equation} {\it Necessity}. If $[aI,S_\mathbb{I}]\in\cK(L^2(\mathbb{I}))$, then from (\ref{eq:I-aS-Sa}) we deduce that $[(a\circ\theta_{-1})I,S_{\T_+}]\in\cK(L^2(\T_+))$. By Theorem~\ref{th:T_+aS-Sa}, $a\circ\theta_{-1}\in VMO(\T_+)$. Then, in view of Lemma~\ref{le:VMO-transpl}, $a\in VMO(\mathbb{I})$. Necessity is proved. {\it Sufficiency}. If $a\in VMO(\mathbb{I})$, then $a\circ\theta_{-1}\in VMO(\T_+)$, due to Lemma~\ref{le:VMO-transpl}. Therefore, by Theorem~\ref{th:T_+aS-Sa}, $[(a\circ\theta_{-1})I,S_{\T_+}]\in\cK(L^2(\T_+))$. Hence, taking into account (\ref{eq:I-aS-Sa}), we get $[aI,S_\mathbb{I}]\in\cK(L^2(\mathbb{I}))$. Theorem is proved. \end{pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Compactness of commutators with the shift operator} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this subsection we prove a compactness criterion for the commutator $W_\alpha S_\mathbb{I}-S_\mathbb{I} W_\alpha$, where $\alpha$ is a bi-Lipschitz homeomorphism of $[0,1]$ onto itself. The main step of the proof is constructing a shift $\beta:\T\to\T$ with such behavior in neighborhoods of $1$ and $-1$ as $\alpha$ has in neighborhoods of $0$ and $1$, respectively. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:VMO-shift-transpl} Let $\alpha$ be an orientation-preserving bi-Lipschitz homeomorphism of $[0,1]$ onto itself such that $\alpha(0)=0$ and $\alpha(1)=1$. Then %%% \begin{equation}\label{eq:VMO-shift-transpl-0} \beta(\tau):=\left\{ \begin{array}{ll} (\theta\circ\alpha\circ\theta_{-1})(\tau), &\tau\in\T_+, \\[1ex] \overline{(\theta\circ\alpha\circ\theta_{-1})(\overline{\tau})}, &\tau\in\T_-, \\[1ex] \tau, &\tau\in\{-1,1\}, \end{array} \right. \end{equation} %%% is an orientation-preserving bi-Lipschitz homeomorphism of $\T$ onto itself. Moreover, $\alpha'\in VMO(\mathbb{I})$ if and only if $\beta'\in VMO(\T)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} Obviously, $\beta$ is a homeomorphism of $\T$ onto itself which preserves the orientation on $\T$. Since $\theta$ is a $C^\infty$-diffeomorphism, $\theta$ and $\theta_{-1}$ are Lipschitz functions. Therefore $\alpha:\mathbb{I}\to\mathbb{I}$ and $\beta:\T\to\T$ are bi-Lipschitz homeomorphisms only simultaneously. Hence $\alpha'\in \cG L^\infty(\mathbb{I})$ and $\beta'\in \cG L^\infty(\T)$. Since $\theta'(x)=\pi ie^{\pi ix}$ whenever $x\in\mathbb{I}$, for $\tau\in\T_+$ we obtain %%% \begin{eqnarray} \beta'(\tau) &=& \frac{\theta'[(\alpha\circ\theta_{-1})(\tau)]} {\theta'[\theta_{-1}(\tau)]} \alpha'[\theta_{-1}(\tau)] \nonumber\\ &=& e^{\pi i[(\alpha\circ\theta_{-1})(\tau)-\theta_{-1}(\tau)]} \alpha'[\theta_{-1}(\tau)]. \label{eq:VMO-shift-transpl-1} \end{eqnarray} %%% If $\tau\in\T_-$, then in view of $\beta'(\tau)=\overline{\beta'(\overline{\tau})}$ and (\ref{eq:VMO-shift-transpl-1}), we get %%% \begin{equation}\label{eq:VMO-shift-transpl-2} \beta'(\tau)=e^{-\pi i[ (\alpha\circ\theta_{-1})(\overline{\tau}) -\theta_{-1}(\overline{\tau})]} \alpha'[\theta_{-1}(\overline{\tau})]. \end{equation} %%% From (\ref{eq:VMO-shift-transpl-1}) and (\ref{eq:VMO-shift-transpl-2}) we see that %%% \begin{equation}\label{eq:VMO-shift-transpl-3} \beta'(\tau)=\varphi_\alpha(\tau)\psi_\alpha(\tau), \quad\tau\in\T, \end{equation} %%% where %%% \begin{eqnarray*} \varphi_\alpha(\tau) &:=& \left\{ \begin{array}{ll} e^{\pi i[(\alpha\circ\theta_{-1})(\tau)-\theta_{-1}(\tau)]},& \tau\in\T_+, \\[1ex] e^{-\pi i[(\alpha\circ\theta_{-1})(\overline{\tau})- \theta_{-1}(\overline{\tau})]},& \tau\in\T_-, \\[1ex] 1, &\tau\in\{-1,1\} \end{array} \right. \\ &=& \left\{ \begin{array}{ll} e^{\pi i\left[ \alpha\left(\frac{\arg\tau}{\pi}\right)-\frac{\arg\tau}{\pi} \right]},& \tau\in\T_+, \\[1ex] e^{\pi i \left[\frac{2\pi-\arg\tau}{\pi}- \alpha\left(\frac{2\pi-\arg\tau}{\pi}\right) \right]},& \tau\in\T_-, \\[1ex] 1, &\tau\in\{-1,1\} \end{array} \right. \end{eqnarray*} %%% and \[ \psi_\alpha(\tau):= \left\{ \begin{array}{ll} \alpha'[\theta_{-1}(\tau)], &\tau\in\T_+,\\[1ex] \alpha'[\theta_{-1}(\overline{\tau})], &\tau\in\T_-. \end{array} \right. \] Clearly, $\varphi_\alpha$ is a continuous function on $\T$, which satisfies $|\varphi_\alpha(\tau)|=1$. Thus, %%% \begin{equation}\label{eq:VMO-shift-transpl-4} \varphi_\alpha,1/\varphi_\alpha\in C(\T)\subset QC(\T) =L^\infty(\T)\cap VMO(\T). \end{equation} %%% Taking into account that $QC(\T)$ is an algebra, we infer from (\ref{eq:VMO-shift-transpl-3}) and (\ref{eq:VMO-shift-transpl-4}) that $\beta'\in QC(\T)$ if and only if $\psi_\alpha\in QC(\T)$. By Lemma~\ref{le:sewing}, $\psi_\alpha\in QC(\T)$ if and only if $\alpha'\circ\theta_{-1}\in L^\infty(\T_+)\cap VMO(\T_+)$. On the other hand, the latter inclusion is equivalent to $\alpha'\in L^\infty(\mathbb{I})\cap VMO(\mathbb{I})$, in view of Lemma~\ref{le:VMO-transpl}. Thus, $\beta'\in \cG L^\infty(\T)\cap VMO(\T)$ if and only if $\alpha'\in \cG L^\infty(\mathbb{I})\cap VMO(\mathbb{I})$. Lemma is proved. \end{pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:I-WS-SW} Let $\alpha:[0,1]\to[0,1]$ be an orientation-preserving bi-Lipschitz homeomorphism such that $\alpha(0)=0$ and $\alpha(1)=1$. The operator $W_\alpha S_\mathbb{I}-S_\mathbb{I} W_\alpha$ is compact on $L^2(\mathbb{I})$ if and only if $\alpha'\in VMO(\mathbb{I})$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{pf} Since $\alpha$ is an orientation-preserving bi-Lipschitz homeomorphism, $\alpha'$ belongs to $\cG L^\infty (\mathbb{I})$. {\it Necessity}. The idea of the proof of necessity part is borrowed from \cite[Proposition~4.5]{MX95}. Since $W_\alpha^{-1}=W_{\alpha_{-1}}$, we conclude that $[W_\alpha,S_\mathbb{I}]\in\cK(L^2(\mathbb{I}))$ if and only if \[ W_{\alpha_{-1}}[W_\alpha,S_\mathbb{I}]W_{\alpha_{-1}} = [S_\mathbb{I},W_{\alpha_{-1}}] \in\cK(L^2(\mathbb{I})). \] On the other hand, since $S_\mathbb{I}$ is self-adjoint (see, e.g., \cite[Ch.~1, Theorem~7.1]{GK92} or \cite[Ch.~1, Corollary~4]{KL94}) and since $(W_{\alpha_{-1}})^*=\alpha'W_\alpha$, we have \[ [\alpha' W_\alpha,S_\mathbb{I}] = ([S_\mathbb{I},W_{\alpha_{-1}}])^* \in\cK(L^2(\mathbb{I})). \] Taking into account that, by assumption, $W_\alpha$ and $S_\mathbb{I}$ commute modulo compact operators, we infer that \[ [\alpha' I,S_\mathbb{I}]W_\alpha = [\alpha' W_\alpha,S_\mathbb{I}] - \alpha' [W_\alpha, S_\mathbb{I}] \in\cK(L^2(\mathbb{I})). \] Thus, $[\alpha'I,S_\mathbb{I}]\in\cK(L^2(\mathbb{I}))$. Finally, by Theorem~\ref{th:I-aS-Sa}, $\alpha'\in VMO(\mathbb{I})$. Necessity is proved. {\it Sufficiency}. Sufficiency is proved by analogy with Theorem~\ref{th:I-aS-Sa}. If $\alpha$ is a bi-Lipschitz homeomorphism and $\alpha'\in VMO(\mathbb{I})$, then, in view of Lemma~\ref{le:VMO-shift-transpl}, the function $\beta$ defined by (\ref{eq:VMO-shift-transpl-0}) is an orientation-preserving bi-Lipschitz homeomorphism of $\T$ onto itself such that $\beta'\in VMO(\T)$. Hence, by Theorem~\ref{th:T-WS-SW}, $[W_\beta, S_\T]\in\cK(L^2(\T))$. Therefore, $\chi_{\T_+}[W_\beta,S_\T]\chi_{\T_+}I\in\cK(L^2(\T))$. Since $\beta(\pm 1)=\pm 1$ and $\beta|_{\T_+}=\theta\circ\alpha\circ\theta_{-1}$ maps $\T_+$ onto itself, the latter inclusion is equivalent to %%% \begin{equation}\label{eq:I-WS-SW-1} W_{\theta\circ\alpha\circ\theta_{-1}} S_{\T_+} - S_{\T_+} W_{\theta\circ\alpha\circ\theta_{-1}} \in\cK(L^2(\T_+)). \end{equation} %%% Taking into account the obvious equality $G^{-1}W_\alpha G=W_{\theta\circ\alpha\circ\theta_{-1}}$, from (\ref{eq:I-WS-SW-1}) we get \begin{equation}\label{eq:I-WS-SW-2} [GW_{\theta\circ\alpha\circ\theta_{-1}} G^{-1},GS_{\T_+}G^{-1}] = [W_\alpha,GS_{\T_+}G^{-1}]\in\cK(L^2(\mathbb{I})). \end{equation} %%% From (\ref{eq:I-WS-SW-2}) and Lemma~\ref{le:transpl-compact} we get $W_\alpha S_\mathbb{I}-S_\mathbb{I} W_\alpha\in\cK(L^2(\mathbb{I}))$. Theorem is proved. \end{pf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{Compactness of commutators on \\rearrangement-invariant spaces} \subsection{Case of the unit circle} Let $X(\T)$ be a rearrangement-invariant space with nontrivial Boyd indices over the unit circle $\T$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:T-aS-Sa-X} Suppose $a\in L^\infty(\T)$. The operator $aS_\T-S_\T aI$ is compact on $X(\T)$ if and only if $a\in VMO(\T)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:T-WS-SW-X} Suppose $\alpha$ is an orientation-preserving bi-Lipschitz homeomorphism of $\T$ onto itself. Then the operator $W_\alpha S_\T-S_\T W_\alpha$ is compact on $X(\T)$ if and only if $\alpha'\in VMO(\T)$. \end{theorem} These theorems follow from Corollary~\ref{co:interpolation} with $p_0=2$ and Theorems~\ref{th:T-aS-Sa}, \ref{th:T-WS-SW}, respectively. Note that the corresponding results for Lebesgue spaces $L^p(\T)$, $1