\documentclass{ieot} % THEOREM Environments (Examples)------------------------------------------ \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Karlovich's commands \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cP}{\mathcal{P}} \newcommand{\fL}{\mathfrak{L}} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} \newcommand{\T}{\mathbb{T}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\eps}{\varepsilon} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %-------------------------------------------------------------------------- % editorial commands % \firstpage{1} \issuenumber{4} \volumeandyear{XX (200X)} %\commby{inhouse} \submitted{July 12, 2002} %\received{March 16, 2000} %\revised{June 1, 2000} %\accepted{July 22, 2000} %--------------------------------------------------------------------------- %Insert here the title, affiliations and abstract: % \title[Norms of Toeplitz and Hankel operators] {Norms of Toeplitz and Hankel operators \\on Hardy type subspaces \\of rearrangement-invariant spaces} \author[A.~Karlovich]{Alexei Yu. Karlovich} \address{ Departamento de Matem\'{a}tica,\\ Instituto Superior T\'{e}cnico,\\ Av. Rovisco Pais, 1\\ 1049--001 Lisboa,\\ Portugal} \email{akarlov@math.ist.utl.pt} \thanks{The author is partially supported by F.C.T. (Portugal) grants POCTI 34222/MAT/2000 and PRAXIS XXI/BPD/22006/99.} %----------classification, keywords, date \subjclass{Primary 47B35, 46E30; Secondary 47A30} \keywords{Toeplitz operator, Hankel operator, rearrangement-invariant space, Boyd indices, Lozanovskii factorization} \date{June 10, 2002} %%% ---------------------------------------------------------------------- \begin{abstract} We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Toeplitz and Hankel operators have attracted the mathematical community for the many deca\-des since the classical paper by Otto Toeplitz \cite{Toeplitz11}. The boundedness of the Toeplitz operator $T(a)$ generated by a symbol $a\in L^\infty$ on the Hardy space $H^2$ over the unit circle $\T$ was proved by Brown and Halmos \cite{BH63}. The norm of $T(a)$ on the Hardy space $H^2$ coincides with the norm of its symbol in $L^\infty$ (actually, this result was already in a footnote of \cite{Toeplitz11}). The problem about the norm of the Hankel operator $H(a):H^2\to H^2_-$ was solved by Nehari \cite{Nehari57}. The norm of the Hankel operator $H(a)$ coincides with the distance of its symbol $a$ to the Hardy space $H^\infty$ in the $L^\infty$-metric. Full proofs of the Brown-Halmos and Nehari theorems can be found in almost all texts on Toeplitz and Hankel operators. We here mention \cite[Theorem~2.7]{BS90}, \cite[Part~B, Theorem~4.1.4]{Nik02}, \cite[Theorem~1.8]{PKh82} for the proof of the Brown-Halmos theorem and \cite[Theorem~2.11]{BS90}, \cite[Lecture~8]{Nik86}, \cite[Part~B, Theorem~1.3.2]{Nik02}, \cite[Theorem~1.1]{PKh82}, \cite[p.~100]{Sarason78} for the proof of Nehari's theorem. Analogues of these classical theorems are true for the Hardy spaces $H^p$ and $H^p_-, 1\lambda\}= \mu\{\tau\in\T:|g(\tau)|>\lambda\} \quad\mbox{for all}\quad \lambda\ge 0. \] A function norm $\rho:\cM^+ \to [0,\infty]$ is called rearrangement-invariant if for every pair of equimeasurable functions $f,g \in \cM^+_0$ the equality $\rho(f)=\rho(g)$ holds. In that case, the Banach function space $X$ generated by $\rho$ is said to be a {\it rearrangement-invariant space}. A Banach function space $X$ is rearrangement-invariant if and only if its associate space $X'$ is rearrangement-invariant too \cite[p.~60]{BS88}. The Lebesgue space $L^p$, $1\le p\le\infty$, is the simplest example of a rearrange\-ment-invariant space. Orlicz and Lorentz spaces are other important classical examples of rearrangement-invariant spaces. For every rearrangement-invariant space $X$ (see, e.g., \cite[p.~78]{BS88}), %%% \begin{equation}\label{eq:imbedding} L^\infty\subset X\subset L^1. \end{equation} %%% The Calder\'on-Mitjagin interpolation theorem (see, e.g., \cite[Theorem~2.a.10]{LT}) implies that %%% \begin{equation}\label{eq:module} \|fg\|_X\le \|f\|_X\|g\|_\infty \quad\mbox{for all}\quad f\in X, \quad g\in L^\infty. \end{equation} %%% The construction of the associate space implies the following H\"older inequality for rearrangement-invariant spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:Hoelder} {\rm (see, e.g., \cite[Ch.~1, Theorem~2.4]{BS88}).} Let $X$ be a rearrangement-invariant space and $X'$ be its associate space. If $f\in X$ and $g\in X'$, then $fg$ is summable and \[ \|fg\|_{L^1}\le \|f\|_X\|g\|_{X'}. \] \end{lemma} H\"older's inequality is complemented by the following deep factorization theorem due to G.~Ja.~Lozanovskii \cite{L69}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Lozanovskii} {\rm (see \cite[Theorem~6]{L69} and also \cite[Theorem~1(i)]{G81}).} Let $X$ be a rearrangement-invariant space and let $X'$ be its associate space. If $\varphi\in L^1$, then there exist $\psi\in X$ and $\psi'\in X'$ such that \[ \varphi=\psi\psi' \quad\mbox{and}\quad \|\varphi\|_{L^1}=\|\psi\|_X\|\psi'\|_{X'}. \] \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Boyd indices} By the Luxemburg representation theorem \cite[Ch.~2, Theorem~4.10]{BS88}, there is a unique rearrangement-invariant function norm $\overline{\rho}$ over $[0,1]$ with the Lebesgue measure $dt$ 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]{BS88}). The rearrangement-invariant space over $([0,1],dt)$ generated by $\overline{\rho}$ is called the Luxemburg representation of $X$. For each $s\in\R_+:=(0,\infty)$, let $E_s$ denote the dilation operator defined on $\cM_0([0,1],dt)$ by \[ (E_s f)(t):= \left\{ \begin{array}{ll} f(st), & st\in[0,1]\\ 0, & st\not\in[0,1] \end{array} \right. , \quad t\in [0,1]. \] For every $s\in\R_+$, the operator $E_{1/s}$ is bounded on the Luxemburg representation of $X$ \cite[p.~165]{BS88}, 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 \[ \alpha_X:=\lim_{s\to 0}\frac{\log h_X(s)}{\log s}, \quad \beta_X:=\lim_{s\to \infty}\frac{\log h_X(s)}{\log s} \] exist and $\alpha_X\le\beta_X$. The numbers $\alpha_X$ and $\beta_X$ are called the {\it lower} and {\it upper Boyd indices} of the rearrangement-invariant space $X$, respectively \cite{Boyd69} (see also \cite{Bennett72}). For an arbitrary rearrangement-invariant space, its Boyd indices belong to $[0,1]$. The Boyd indices of the rearrangement-invariant spaces $X$ and $X'$ are connected via the following identity: %%% \begin{equation}\label{eq:duality} \alpha_X+\beta_{X'}= \alpha_{X'}+\beta_X=1. \end{equation} %%% One can find the proofs of these and other properties of the Boyd indices in \cite{BS88,Boyd69,LT,Mal85}. For the Lebesgue spaces $L^p, 1\le p\le\infty$, the Boyd indices coincide and equal $1/p$. We will say that the Boyd indices are {\it nontrivial} if $\alpha_X, \beta_X\in(0,1)$. In the case of Orlicz spaces the latter condition is equivalent to the reflexivity of the space (see, e.g., \cite{Mal85}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Multiplication operators} \subsection{Reflexivity and separability of rearrangement-invariant spaces} For $f\in X$ and $g\in X'$, write \[ (f,g):=\int_\T f\overline{g}\,dm. \] In the following statement we summarize relations between reflexivity and separability of rearrangement-invariant spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:separability} {\rm (a)} A rearrangement-invariant space $X$ is reflexive if and only if both $X$ and its associate space $X'$ are separable. {\rm (b)} A rearrangement-invariant space $X$ is separable if and only if its associate space $X'$ is canonically isometrically isomorphic to the dual space $X^*$ of $X$. The general form of a linear functional on a separable rearrangement-invariant space $X$ is given by $G(f)=(f,g)$, where $g\in X'$ and $\|G\|_{X^*}=\|g\|_{X'}$. {\rm (c)} A rearrangement-invariant space $X$ is separable if and only if the space $C$ of all continuous functions on $\T$ is dense in $X$. \end{lemma} Due to the separability of the measure $dm$ (see, e.g., \cite[Ch.~1, Section~6.10]{KA}), Parts (a) and (b) follow from Corollaries 4.3, 4.4, and 5.6 \cite[Ch.~1]{BS88}. Part (c) is proved in \cite[Lemma~1.4]{K98} (see also \cite[Ch.~2, Section~5]{KPS}). Define the function $\chi_n\ (n\in\Z)$ by $\chi_n(t):=t^n\ (t\in\T)$. A function of the form $\sum_{n=-N}^N \lambda_n \chi_n \quad (\lambda_n\in\C)$ is called a trigonometric polynomial of order $N\ge 0$ and the linear set of all trigonometric polynomials is denoted by $\mathcal{P}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{corollary}\label{co:density} If $X$ is a reflexive rearrangement-invariant space, then $\mathcal{P}$ is dense in $X$ and in $X'$. \end{corollary} Since $\mathcal{P}$ is dense in $C$ and $C$ is imbedded into $X$ and $X'$ (see (\ref{eq:imbedding})), this statement follows from Lemma~\ref{le:separability}(a) and (c). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Norm convergence of Fourier series} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The Fourier coefficients of a function $f\in X\subset L^1$ are well defined by $f_j:=(f,\chi_j)$ for $j\in\Z$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Fourier-norm} {\rm (see \cite[Ch.~3, Corollary~6.11]{BS88}).} Let $X$ be a separable rearrange\-ment-invariant space. The Fourier series of every function $f \in X$ converges in $X$ if and only if $X$ has nontrivial Boyd indices. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Multiplication operators} For Banach spaces $E$ and $F$, the set of all bounded linear operators $A:E\to F$ is denoted by $\cL(E,F)$. If $E=F$, then $\cL(E):=\cL(E,E)$. If $a\in L^\infty$ and $X$ is a rearrangement-invariant space, then, by (\ref{eq:module}), the operator \[ M_a:X\to X, \quad f\mapsto af \] is bounded on $X$ and $\|M_a\|_{\cL(X)}\le\|a\|_\infty$. It is clear that $(M_a \chi_j,\chi_k)$ is equal to the $(k-j)$-th Fourier coefficient of $a$. The following lemma shows that every bounded operator with such a property is a multiplication operator. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:BS2.2} Let $X$ be a separable rearrangement-invariant space with nontrivial Boyd indices. Suppose $A\in\cL(X)$ and there is a sequence $\{a_n\}_{n\in\Z}$ of complex numbers such that \[ (A\chi_j,\chi_k)=a_{k-j} \quad\mbox{for all}\quad j,k\in\Z. \] Then there is an $a\in L^\infty$ such that $A=M_a$ and $\{a_n\}_{n\in\Z}$ is the Fourier coefficient sequence of $a$. Moreover, $\|M_a\|_{\cL(X)}=\|a\|_\infty$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} This statement is proved by analogy with \cite[Proposition~2.2]{BS90}. Put $a:=A\chi_0$. Then $a\in X$ and the $n$-th Fourier coefficient of $a$ is \[ (a,\chi_n)=(A\chi_0,\chi_n)=a_n. \] If $f\in L^\infty$, then both $Af$ and $af$ are in $X$. We claim that %%% \begin{equation}\label{eq:BS2.2-1} Af=af \quad\mbox{for all}\quad f\in L^\infty. \end{equation} %%% Let $\{f_n\}_{n\in\Z}$ denote the Fourier coefficient sequence of $f$. Then the $j$-th Fourier coefficient of $af$ is $\sum\limits_{k\in\Z}a_{j-k}f_k$. On the other hand, since $X$ is separable and has nontrivial Boyd indices, by Theorem~\ref{th:Fourier-norm}, the series $\sum\limits_{k\in\Z}f_k\chi_k$ converges to $f$ in $X$. Therefore, for every $\eps>0$ there exists $N\in\N$ such that for every $n\ge N$, %%% \begin{equation}\label{eq:BS2.2-2} \left\|f-\sum_{k=-n}^n f_k\chi_k\right\|_X< \frac{\eps}{\|A\|_{\cL(X)}\|1\|_{X'}}. \end{equation} %%% For $j\in\Z$ and $n\ge N$, we obtain %%% \begin{equation}\label{eq:BS2.2-3} (Af,\chi_j)-\sum_{k=-n}^n f_k(A\chi_k,\chi_j) = \left(A\left[ f-\sum_{k=-n}^n f_k\chi_k \right],\chi_j\right). \end{equation} %%% From Lemma~\ref{le:Hoelder}, from the boundedness of $A$ on $X$, and from (\ref{eq:BS2.2-2}) we have %%% \begin{equation}\label{eq:BS2.2-4} \begin{split} & \left|\left(A\left[ f-\sum_{k=-n}^n f_k\chi_k \right],\chi_j\right)\right| \le \left\|A\left[ f-\sum_{k=-n}^n f_k\chi_k \right]\right\|_X\|\chi_j\|_{X'} \\ & \le \|A\|_{\cL(X)} \|1\|_{X'} \left\|f-\sum_{k=-n}^n f_k\chi_k\right\|_X <\eps. \end{split} \end{equation} %%% Combining (\ref{eq:BS2.2-3})--(\ref{eq:BS2.2-4}) and taking into account that $\eps$ is arbitrary, we get \[ (Af,\chi_j)=\sum_{k\in\Z} f_k(A\chi_k,\chi_j) \quad \mbox{for all}\quad j\in\Z. \] This shows that the $j$-th Fourier coefficient of $Af$ equals $\sum\limits_{k\in\Z} a_{j-k}f_k$ too. Since all Fourier coefficients of $Af$ and $af$ coincide and $Af,af\in X\subset L^1$ (see (\ref{eq:imbedding})), we arrive at (\ref{eq:BS2.2-1}), in view of the uniqueness theorem for Fourier series (see, e.g., \cite[Theorem~2.7]{Kat76}). We now prove that $a\in L^\infty$. Assume the contrary. Let $E$ be a measurable subset of $\T$ with positive measure such that $|a(\tau)|>\|A\|_{\cL(X)}$ for $\tau\in E$. Then, by (\ref{eq:BS2.2-1}) and by the lattice property, \[ \|A\chi_E\|_X=\|a\chi_E\|_X > \|A\|_{\cL(X)} \|\chi_E\|_X. \] This contradicts to the boundedness of $A$ on $X$ and so $|a(\tau)|\le\|A\|_{\cL(X)}$ a.e. on $\T$. Hence $a\in L^\infty$ and $\|a\|_\infty\le\|A\|_{\cL(X)}$. By Lemma~\ref{le:separability}(c), $C$ is dense in $X$. Since, in view of (\ref{eq:BS2.2-1}), $A$ and $M_a$ coincide on the dense subset $C$ of $X$ and both operators are bounded, it follows that $A=M_a$. The norm equalities are now obvious. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Toeplitz operators} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Hardy type subspaces} \begin{theorem}\label{th:boundedness-S} The Cauchy singular integral operator $S$ is bounded on a rearran\-gement-invariant space $X$ if and only if $X$ 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]{BS88}). The proof for the unit circle is contained in \cite[Ch.~2, Section~8.6]{KPS}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma} If $X$ is a reflexive rearrangement-invariant space with nontrivial Boyd indices, then the operators $P_+$ and $P_-$ given by {\rm (\ref{eq:Riesz})} are bounded projections on $X$ and on $X'$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} From Theorem~\ref{th:boundedness-S} and (\ref{eq:duality}) it follows that $S$ is bounded on $X$ and on $X'$. It is not difficult to check that $S^2\varphi=\varphi$ for every $\varphi\in\mathcal{P}$. From the latter equality and Corollary~\ref{co:density} we derive that $S^2=I$ on both $X$ and $X'$. This immediately implies $P_\pm^2=P_\pm$ on both $X$ and $X'$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For $1\le p\le\infty$, consider the usual Hardy spaces (see, e.g., \cite{Duren70,Garnett,Hoffman}) \[ H^p := \Big\{ f\in L^p:\ f_j=0\ \mbox{for all}\ j<0 \Big\}, \quad H^p_-:= \Big\{ f\in L^p:\ f_j=0\ \mbox{for all}\ j\ge 0 \Big\}. \] %%% It is well known that $H^p=P_+L^p$ and $H^p_-=P_-L^p$ for $p\in(1,\infty)$. By analogy with $H^p$ and $H^p_-$, we define the subspaces of Hardy type of a reflexive rearrangement invariant space $X$ with nontrivial Boyd indices and of its associate space $X'$ by the formulas \[ X_+:= P_+X,\quad X_-:= P_-X, \quad\quad\quad X_+':=P_+X',\quad X_-':=P_-X'. \] We identify functions in $X_+, X_+', H^p$ (respectively, in $X_-,X_-', H^p_-$) defined on $\T$ with their analytic continuations into the unit disk $\D:=\{z\in\C: |z|<1\}$ (respectively, into the domain $\overline{\C}\setminus(\D\cup\T)$). It is easy to see that \[ X_+=H^1\cap X, \quad X_+'=H^1\cap X', \quad\quad X_-=H^1_-\cap X, \quad X_-'=H^1_-\cap X'. \] Let $\mathcal{Q}$ be a set of analytic functions in the open unit disk $\D$. We denote by $\mathcal{Q}^0$ the set of all $f\in\mathcal{Q}$ such that $f(0)=0$. Nevertheless, we will use the standard notation $H^1_0$ for $(H^1)^0=\{h\in H^1: h(0)=0\}$. Consider the set of (analytic) polynomials \[ \mathcal{P}_A:=\Big\{ \sum_{i=0}^n\alpha_i\chi_i: \ \alpha_i\in\C, \ n\ge 0 \Big\}. \] For a Banach space $E$ its closed unit ball is denoted by $B(E)$. For $\mathcal{F}\subset\cM$, put $\overline{\mathcal{F}}:=\{\overline{f}: \ f\in\mathcal{F}\}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:density-subspace} If $X$ is a reflexive rearrangement-invariant space with nontrivial Boyd indices, then {\rm (a)} $\cP_A\cap B(E)$ is dense in $B(E)$ for $E=X_+,X_+', (X_+)^0, (X_+')^0$. {\rm (b)} $\overline{\cP_A^0}\cap B(E)$ is dense in $B(E)$ for $E=X_-,X_-'$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} This statement follows from the obvious equalities $P_+\mathcal{P}=\mathcal{P}_A,P_-\mathcal{P}=\overline{\mathcal{P}_A^0}$ and Corollary~\ref{co:density}. \end{proof} %%% Consider the {\it flip operator} $J$ defined by $(Jf)(t):=\overline{t}f(\overline{t})$, where $t\in\T$. Clearly, $J$ is an isometry on $L^p,1\le p\le\infty$. By the Calder\'on-Mitjagin interpolation theorem (see, e.g., \cite[Theorem~2.a.10]{LT}), $\|J\|_{\cL(X)}\le\max\{\|J\|_{\cL(L^1)},\|J\|_{\cL(L^\infty)}\}=~1$. On the other hand, clearly, $\|J1\|_X=\|1\|_X$. Hence, $J$ is an isometry on every rearrangement-invariant space. It is easy to see that for reflexive rearrangement-invariant spaces with nontrivial Boyd indices, %%% \begin{equation}\label{eq:flip} J^2=I, \quad JP_\pm J=P_\mp, \quad JM_a J=M_{\widetilde{a}}, \end{equation} %%% where $a\in L^\infty$ and $\widetilde{a}(t):=a(1/t)$ for $t\in\T$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:normP} If $X$ is a reflexive rearrangement-invariant space with nontrivial Boyd indices, then %%% \begin{equation}\label{eq:normP-1} \|P_+\|_{\cL(X)}=\|P_-\|_{\cL(X)}=\|P_+\|_{\cL(X')}=\|P_-\|_{\cL(X')}=:\gamma. \end{equation} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} Since $J$ is an isometry, from (\ref{eq:flip}) we get %%% \begin{equation}\label{eq:normP-2} \|P_\pm\|_{\cL(X)}=\|JP_\pm J\|_{\cL(X)}=\|P_\mp\|_{\cL(X)}. \end{equation} %%% On the other hand, by \cite[Lemma~6.2]{K98}, $S=S^*$. Therefore, taking into account Lemma~\ref{le:separability}(a)--(b), we obtain %%% \begin{equation}\label{eq:normP-3} \|P_\pm\|_{\cL(X)}=\|(P_\pm)^*\|_{\cL(X')}=\|P_\pm\|_{\cL(X')}. \end{equation} %%% Combining (\ref{eq:normP-2}) and (\ref{eq:normP-3}), we arrive at (\ref{eq:normP-1}). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A lower estimate of $\gamma$ for an arbitrary reflexive rearrangement-invariant space with nontrivial Boyd indices was obtained in \cite[Theorem~4.5]{K00}. The exact value of this constant is unknown even for reflexive Orlicz spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Analogue of the Brown-Halmos theorem} The operator $T(a)$ defined for a function $a\in L^\infty$ by \[ T(a): X_+\to X_+, \quad f\mapsto P_+(af) \] is obviously bounded and $\|T(a)\|_{\cL(X_+)}\le \gamma \|a\|_\infty$. The operator $T(a)$ is called the {\it Toeplitz operator} generated by the {\it symbol} $a$. It is clear that %%% \begin{equation}\label{eq:T+} (T(a)\chi_j,\chi_k)=a_{k-j} \quad\mbox{for all}\quad j,k\ge 0, \end{equation} The following theorem states that every bounded operator on $X_+$ with this property is a Toeplitz operator and, moreover, relates the norm of a Toeplitz operator with the norm of the multiplication operator generated by the same function. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Brown-Halmos} Suppose $X$ is a reflexive rearrangement-invariant space with nontrivial Boyd indices. Let $A\in\cL(X_+)$ and suppose there is a sequence $\{a_n\}_{n\in\Z}$ of complex numbers such that %%% \begin{equation}\label{eq:BH-1} (A\chi_j,\chi_k)=a_{k-j} \quad\mbox{for all}\quad k,j\ge 0. \end{equation} %%% Then there exists an $a\in L^\infty$ such that $A=T(a)$ and $\{a_n\}_{n\in\Z}$ is the Fourier coefficient sequence of $a$. Moreover, %%% \begin{equation}\label{eq:BH-2} \|a\|_\infty\le\|T(a)\|_{\cL(X_+)}\le\gamma\|a\|_\infty. \end{equation} \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} We argue by analogy with the proof of \cite[Theorem~2.7]{BS90}. For $n\ge 0$, define $b_n\in X$ as $b_n:=\chi_{-n}A\chi_n$. Then, taking into account (\ref{eq:module}) and $A\in\cL(X_+)$, we get %%% \begin{equation}\label{eq:BH-5} \begin{split} \|b_n\|_{X} &= \|\chi_{-n}A\chi_n\|_X \le \|\chi_{-n}\|_\infty\|A\chi_n\|_X=\|A\chi_n\|_{X_+} \\ &\le \|A\|_{\cL(X_+)}\|\chi_n\|_X = \|A\|_{\cL(X_+)}\|1\|_X. \end{split} \end{equation} %%% Put \[ V:=\left\{ y\in X'\ : \|y\|_{X'}<\frac{1}{\|A\|_{\cL(X_+)}\|1\|_X} \right\}. \] Then from Lemma~\ref{le:Hoelder} and (\ref{eq:BH-5}) we get \[ |(b_n,y)|\le\|b_n\|_X\|y\|_{X'}< 1 \quad\mbox{for all}\quad y\in V,\quad n\ge 0. \] Since the space $X$ is reflexive, by Lemma~\ref{le:separability}(a)--(b), we infer that $X,X'$ are separable and $(X')^*=X^{**}=X$. Applying the Banach-Alaoglu theorem (see, e.g., \cite[Theorem~3.17]{Rudin73}) to $V$, $X'$, and $\{b_n\}_{n=0}^{+\infty} \subset X=(X')^*$, we deduce that there exists a $b\in X$ such that some subsequence $\{b_{n_k}\}_{k=1}^{+\infty}$ of $\{b_n\}_{n=0}^{+\infty}$ converges to $b$ in the weak topology on $X$. In particular, %%% \begin{equation}\label{eq:BH-6} \lim_{k\to+\infty}(b_{n_k},\chi_j)= (b,\chi_j) \quad\mbox{for all}\quad j\in\Z. \end{equation} %%% On the other hand, from the definition of $b_n$ and from (\ref{eq:BH-1}) we get %%% \begin{equation}\label{eq:BH-7} (b_{n_k},\chi_j)=(A\chi_{n_k},\chi_{n_k+j})=a_j \quad\mbox{whenever}\quad n_k+j\ge 0. \end{equation} %%% From (\ref{eq:BH-6}) and (\ref{eq:BH-7}) it follows that %%% \begin{equation}\label{eq:BH-8} (b,\chi_j)=a_j \quad\mbox{for all}\quad j\in\Z. \end{equation} %%% Now define the mapping $B$ by %%% \begin{equation}\label{eq:BH-9} B:\mathcal{P}\to X, \quad f\mapsto bf. \end{equation} Assume that $f$ and $g$ are trigonometric polynomials of order $m$ and $r$, respectively. Using (\ref{eq:BH-1}) and (\ref{eq:BH-8}) one can show that %%% \begin{equation}\label{eq:BH-13} (Bf,g)=(M_{\chi_{-n}}AM_{\chi_n}f,g), \quad n\ge\max\{m,r\}. \end{equation} %%% For $n\ge\max\{m,r\}$, obviously, $\chi_nf\in X_+$. Therefore, since $A\in\cL(X_+)$, %%% \begin{equation}\label{eq:BH-14} \begin{split} \|AM_{\chi_n}f\|_X & = \|A\chi_n f\|_{X_+}\le\|A\|_{\cL(X_+)}\|\chi_n\|_{X_+} \\ & \le \|A\|_{\cL(X_+)}\|\chi_n\|_\infty\|f\|_X = \|A\|_{\cL(X_+)}\|f\|_X. \end{split} \end{equation} %%% From Lemma~\ref{le:Hoelder}, (\ref{eq:module}), and (\ref{eq:BH-14}) we obtain for $n\ge\max\{m,r\}$, %%% \begin{equation}\label{eq:BH-15} \Big|(M_{\chi_{-n}}AM_{\chi_n}f,g)\Big| \le \|M_{\chi_{-n}}AM_{\chi_n}f\|_X\|g\|_{X'} \le \|A\|_{\cL(X_+)}\|f\|_X\|g\|_{X'}. \end{equation} %%% From (\ref{eq:BH-13}) and (\ref{eq:BH-15}) we deduce that \[ |(Bf,g)|\le\limsup_{n\to+\infty}\Big| (M_{\chi_{-n}}AM_{\chi_n}f,g) \Big| \le\|A\|_{\cL(X_+)}\|f\|_X\|g\|_{X'}. \] Thus, taking into account that $X'=X^*$ (see Lemma~\ref{le:separability}(a)--(b)), from the latter inequality we obtain \[ \|Bf\|_X=\sup\Big\{|(Bf,g)|\ : \ g\in\mathcal{P}, \ \|g\|_{X'}\le 1\Big\} \le\|A\|_{\cL(X_+)}\|f\|_X \] for all $f\in\mathcal{P}$. Since $\mathcal{P}$ is dense in $X$ (see Corollary~\ref{co:density}), this shows that the linear mapping (\ref{eq:BH-9}) extends to an operator $B\in\cL(X)$ with %%% \begin{equation}\label{eq:BH-16} \|B\|_{\cL(X)}\le\|A\|_{\cL(X_+)}. \end{equation} %%% Again from (\ref{eq:BH-8}) we deduce that \[ (B\chi_j,\chi_k)=(b,\chi_{k-j})=a_{k-j} \quad\mbox{for all}\quad j,k\in\Z. \] Now Lemma~\ref{le:BS2.2} gives the existence of an $a\in L^\infty$ such that $B=M_a$ and $\{a_n\}_{n\in\Z}$ is the Fourier coefficient sequence of $a$. Moreover, %%% \begin{equation}\label{eq:BH-17} \|B\|_{\cL(X)}=\|M_a\|_{\cL(X)}=\|a\|_\infty. \end{equation} %%% Combining (\ref{eq:T+}) and (\ref{eq:BH-1}), we obtain %%% \begin{equation}\label{eq:BH-18} (T(a)\chi_j,\chi_k)=(A\chi_j,\chi_k)=a_{k-j}, \quad j,k\ge 0. \end{equation} %%% Since $T(a)\chi_j,A\chi_j\in X_+\subset H^1$, from the uniqueness theorem for Fourier series (see, e.g., \cite[Theorem~2.7]{Kat76}) and (\ref{eq:BH-18}) it follows that %%% \begin{equation}\label{eq:BH-19} T(a)\chi_j=A\chi_j, \quad j\ge 0. \end{equation} %%% On the other hand, the set $\{\chi_j, \ j\ge 0\}$ is dense in $X_+$. Therefore, from (\ref{eq:BH-19}) we obtain $T(a)f=Af$ for all $f\in X_+$. Thus, $T(a)=A$ and %%% \begin{equation}\label{eq:BH-20} \|T(a)\|_{\cL(X_+)}=\|A\|_{\cL(X_+)}. \end{equation} %%% Combining (\ref{eq:BH-16}), (\ref{eq:BH-17}), and (\ref{eq:BH-20}), we arrive at the first inequality in (\ref{eq:BH-2}). The second inequality in (\ref{eq:BH-2}) is obvious. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For a function $a\in L^\infty$, together with the Toeplitz operator $T(a)$ given on $X_+$, we consider also the Toeplitz operators defined on the whole space $X$ by the formulas %%% \begin{equation}\label{eq:Tpm-def} T_a^+:=P_+ M_a P_+, \quad T_a^-:=P_- M_a P_-. \end{equation} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{corollary}\label{co:Brown-Halmos} Let $X$ be a reflexive rearrangement-invariant space with nontrivial Boyd indices. If $a\in L^\infty$, then \[ \|a\|_\infty\le\|T_a^\pm\|_{\cL(X)}\le\gamma^2\|a\|_\infty. \] \end{corollary} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} If $\varphi\in X_+\setminus\{0\}$, then $\varphi=P_+\varphi$. Therefore, \[ \frac{\|P_+a\varphi\|_X}{\|\varphi\|_X} = \frac{\|P_+aP_+\varphi\|_X}{\|\varphi\|_X} \le \sup_{f\in X\setminus\{0\}} \frac{\|P_+aP_+f\|_X}{\|f\|_X}=\|T_a^+\|_{\cL(X)}. \] Taking the supremum on the left hand side over all $\varphi\in X_+\setminus\{0\}$, we get %%% \begin{equation}\label{eq:Brown-Halmos-1} \|T(a)\|_{\cL(X_+)}\le\|T_a^+\|_{\cL(X)}. \end{equation} %%% On the other hand, since $T_a^+=T(a)P_+$, we deduce that %%% \begin{equation}\label{eq:Brown-Halmos-2} \|T_a^+\|_{\cL(X)}\le\|T(a)\|_{\cL(X_+)}\|P_+\|_{\cL(X)}=\gamma\|T(a)\|_{\cL(X_+)}. \end{equation} %%% From the inequalities (\ref{eq:Brown-Halmos-1}), (\ref{eq:Brown-Halmos-2}), and (\ref{eq:BH-2}) we obtain %%% \begin{equation}\label{eq:Brown-Halmos-3} \|a\|_\infty\le\|T_a^+\|_{\cL(X)}\le\gamma^2\|a\|_\infty. \end{equation} %%% Relations (\ref{eq:flip}) and (\ref{eq:Tpm-def}) imply $JT_a^-J= %(JP_-J)(JM_aJ)(JP_-J)=P_+M_aP_+= T_{\widetilde{a}}^+$. Since $J$ is an isometry on $X$, the latter equality gives $\|T_a^-\|_{\cL(X)}=\|T_{\widetilde{a}}^+\|_{\cL(X)}$. On the other hand, $\|\widetilde{a}\|_\infty=\|a\|_\infty$. Combining these equalities with (\ref{eq:Brown-Halmos-3}), we arrive at \[ \|a\|_\infty\le\|T_a^-\|_{\cL(X)}\le\gamma^2\|a\|_\infty, \] which finishes the proof. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Hankel operators} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Factorization of the unit ball of $H^1$} In this subsection we prove that every function in the unit ball of $H^1$ can be represented as a product of two functions which lie in $B(X_+)$ and in $B(X_+')$, respectively. Put $\log_+x:=\max\{0,\log x\}$ for $x\in[0,\infty)$. First, we need the following auxiliary proposition. \begin{proposition}\label{pr:log-L1} If $\log_+|f|,\log_+|g|\in L^1$ and $fg\in H^1\setminus\{0\}$, then \[ \log|f|\in L^1, \quad \log|g|\in L^1. \] \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} Since $fg\in H^1\setminus\{0\}$, by \cite[Ch.~2, Corollary~4.2]{Garnett}, $|f(\tau)g(\tau)|>0$ for almost every $\tau\in\T$. Combining this property with $\log_+|f|,\log_+|g|\in L^1$, we deduce that %%% \begin{equation}\label{eq:log-L1-1} 0<|f(\tau)|<\infty, \quad 0<|g(\tau)|<\infty \quad\mbox{for almost all}\quad \tau\in\T. \end{equation} %%% Since $fg\in H^1$, by \cite[Theorems~5.3 and 5.4]{Garnett}, $\log |fg|\in L^1$. Taking into account (\ref{eq:log-L1-1}), we get %%% \begin{equation}\label{eq:log-L1-3} \log|f|=\log_+|f|-\log_+(1/|f|), \quad \log|g|=\log_+|g|-\log_+(1/|g|). \end{equation} %%% From (\ref{eq:log-L1-3}) and $h=fg$ we get %%% \begin{equation}\label{eq:log-L1-4} \log_+(1/|f|)+\log_+(1/|g|) = \log_+|f|+\log_+|g|-\log|h|\in L^1. \end{equation} %%% Since the functions $\log_+(1/|f|)$ and $\log_+(1/|g|)$ are nonnegative, (\ref{eq:log-L1-4}) implies that %%% \begin{equation}\label{eq:log-L1-5} \log_+(1/|f|)\in L^1, \quad \log_+(1/|g|)\in L^1. \end{equation} %%% Combining (\ref{eq:log-L1-5}) with (\ref{eq:log-L1-3}) and taking into account that $\log_+|f|\in L^1$, we infer that $\log|f|\in L^1$. Analogously we derive that $\log|g|\in L^1$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% An {\it inner function} is a function $f\in H^\infty$ such that $|f(\tau)|=1$ almost everywhere on $\T$. Given a nonnegative function $\varphi$ on $\T$ with $\log\varphi\in L^1$, the corresponding {\it outer function} $\Phi$ is defined by \[ \Phi(z):=\exp\left( \int_\T\frac{\tau+z}{\tau-z}\log\varphi(\tau)\,dm \right), \quad z\in\D, \] and has the crucial property that $|\Phi|=\varphi$ a.e. on $\T$ (see, e.g., \cite[Ch.~5]{Hoffman}). Now we are in a position to prove the main result of this subsection. The idea of its proof is due to N.~Kalton (personal communication), who pointed {out the} importance of the Lozanovskii factorization theorem (Theorem~\ref{th:Lozanovskii}) in this connection. Note that the corresponding result for the classical Hardy spaces $H^p, 10$, there exist $p\in\mathcal{P}_A\cap B(X_+)$ and $q\in\mathcal{P}_A\cap B(X_+')$ such that %%% \begin{equation}\label{eq:H1-2} \|f-p\|_X<\frac{\eps}{2}, \quad \|g-q\|_{X'}<\frac{\eps}{2}. \end{equation} %%% Applying Lemma~\ref{le:Hoelder} and (\ref{eq:H1-2}), we obtain %%% \begin{equation}\label{eq:H1-3} \begin{split} \|h-pq\|_{L^1} &\le \|fg-fq\|_{L^1}+\|fq-pq\|_{L^1} \\ &\le \|f\|_X\|g-q\|_{X'}+\|f-p\|_X\|q\|_{X'}<\frac{\eps}{2}+\frac{\eps}{2}=\eps. \end{split} \end{equation} %%% Since $h(0)=f(0)g(0)=0$, either $f\in(X_+)^0$ or $g\in(X_+')^0$. Then, in view of Proposition~\ref{pr:density-subspace}(a) and (\ref{eq:H1-2}), either $p(0)=0$ or $q(0)=0$. If $q(0)\ne 0$, then set $\widetilde{p}:=p\chi_{-1}, \widetilde{q}:=q\chi_1$. In that case $\widetilde{q}(0)=\chi_1(0)=0$ and $pq=\widetilde{p}\widetilde{q}$. Since $|\chi_n(t)|=1$ on $\T$ for $n\in\Z$, we have $|p(t)|=|\widetilde{p}(t)|, |q(t)|=|\widetilde{q}(t)|$ on $\T$. Therefore, \[ \|\widetilde{p}\|_X=\|p\|_X\le 1, \quad \|\widetilde{q}\|_X=\|q\|_X\le 1. \] Clearly, $\widetilde{p}\in\mathcal{P}_A$ and $\widetilde{q}\in\mathcal{P}_A^0$. Hence, $\widetilde{p}\widetilde{q}\in\Pi$ and, in view of (\ref{eq:H1-3}), $\|h-\widetilde{p}\widetilde{q}\|_{L^1}<\eps$. Thus, $\Pi$ is dense in $B(H_0^1)$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The dual space for $X_-$} In this subsection we describe the dual space for $X_-$ using the scheme of \cite[Section~7.2]{Duren70}. Let $E$ be a Banach space and let $S$ be a subspace of $E$. The {\it annihilator} of the subspace $S$ is the set $S^\perp$ of all linear functionals $\varphi\in E^*$ such that $\varphi(f)=0$ for all $f\in S$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:annihilator} Let $X$ be a reflexive rearrangement-invariant space with nontrivial Boyd indices. Then $(X_-)^\perp=X_+'$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} By Lemma~\ref{le:separability}(a)--(b), $X^*=X'$. If $g\in X'$ annihilates every function in $X_-$, then surely $0=(\chi_i,g)=(\overline{g},\chi_{-i})$ for all $i<0$. Hence, $\overline{g}\in\overline{H^1}$. Therefore, $g\in H^1\cap X'=X'_+$. Thus, $(X_-)^\perp\subset X_+'$. Conversely, if $g\in X_+'$, then for every $f\in X_-$, one can show that %%% \begin{equation}\label{eq:annihilator-1} (f,g)=0. \end{equation} %%% Indeed, since $f\in X_-$ and $g\in X_+'$, their Fourier series are given by \[ \sum_{i=-\infty}^{-1}f_i\chi_i, \quad \sum_{j=0}^\infty g_j\chi_j, \] respectively. From Lemma~\ref{le:separability}(a), relations (\ref{eq:duality}), and Theorem~\ref{th:Fourier-norm} it follows that for every $\eps>0$ there exist numbers $M,N>0$ such that for all $n\ge N, m\ge M$, %%% \begin{equation}\label{eq:annihilator-2} \|f-p_n\|_X<\eps, \quad \|g-q_m\|_{X'}<\eps, \end{equation} %%% where \[ p_n:=\sum_{i=-n}^{-1}f_i\chi_i, \quad q_m:=\sum_{j=0}^m g_j\chi_j. \] Clearly, %%% \begin{equation}\label{eq:annihilator-3} |(f,g)| \le |(p_n,q_m)|+|(f,g-q_m)|+|(f-p_n,q_m)|. \end{equation} %%% In view of Lemma~\ref{le:Hoelder} and (\ref{eq:annihilator-2}), for $n\ge N,m\ge M$ we get %%% \begin{equation}\label{eq:annihilator-4} \begin{split} & |(f,g-q_m)|\le \|f\|_X \|g-q_m\|_{X'}<\eps\|f\|_X, \\ & |(f-p_n,q_m)|\le \|f-p_n\|_X \|q_m\|_{X'} <\eps \|q_m\|_{X'}<\eps(\|g\|_{X'}+\eps). \end{split} \end{equation} %%% Obviously, for $n\ge 1, m\ge 1$, %%% \begin{equation}\label{eq:annihilator-5} (p_n,q_m)=\sum_{i=-n}^{-1}\sum_{j=0}^m f_i\overline{g_j}(\chi_i,\chi_j) = \sum_{i=-n}^{-1}\sum_{j=0}^m f_i\overline{g_j}\delta_{ij}=0. \end{equation} %%% Combining (\ref{eq:annihilator-3}) -- (\ref{eq:annihilator-5}), we obtain \[ |(f,g)|<\eps(\|f\|_X+\|g\|_{X'}+\eps). \] Since $\eps$ is arbitrary, the latter inequality immediately gives (\ref{eq:annihilator-1}). Therefore, we have proved that $X_+'\subset (X_-)^\perp$. Thus, $(X_-)^\perp=X_+'$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:Xminus-dual} Let $X$ be a reflexive rearrangement-invariant space with nontrivial Boyd indices and let $X'$ be its associate space. Then the dual space of $X_-$ is isometrically isomorphic to $X'/X_+'$. The general form of a functional $G\in(X_-)^*$ is given by $G(f)= (f,g)$, where $g\in X_-'$ is uniquely determined by $G$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} From Lemma~\ref{le:separability}(a)--(b) and Lemma~\ref{le:annihilator} we obtain \[ X^*/(X_-)^\perp=X^*/X_+'=X'/X_+'. \] Further we can almost literally repeat the proof of \cite[Theorem~7.3]{Duren70}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:1} Let $X$ be a reflexive rearrangement-invariant space with nontrivial Boyd indices. For $G\in(X_-)^*$, we have %%% \begin{equation}\label{eq:1-1} \frac{1}{\gamma}\|g\|_{X_-'}\le\|G\|_{(X_-)^*}\le\|g\|_{X_-'}. \end{equation} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} Lemma~\ref{le:Xminus-dual} states that $(X_-)^*=X'/X_+'$. By the definition of the norm in a quotient space, %%% \begin{equation}\label{eq:functional} \|G\|_{(X_-)^*}= \inf\Big\{\|g+h\|_{X'}:\quad h\in{X_+'}\Big\}. \end{equation} Suppose $g\in X_-'$ and $h\in X_+'$. Then $g= P_-g$, $0= P_-h$. Therefore, %%% \begin{equation}\label{eq:1-2} \|g\|_{X_-'} = \|P_-g+P_-h\|_{X'}\le\|P_-\|_{\cL(X')}\|g+h\|_{X'} = \gamma\|g+h\|_{X'}. \end{equation} %%% Taking the infimum over all $h\in X_+'$, we derive from (\ref{eq:functional}) and (\ref{eq:1-2}) the first inequality in (\ref{eq:1-1}). The second inequality in (\ref{eq:1-1}) is obvious. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Estimates for the norm of $A\in\cL(X_+,X_-)$} In this subsection we formulate general estimates for the norm of an arbitrary bounded linear operator $A:X_+\to X_-$. To this aim we define the quantity %%% \begin{equation}\label{eq:Phi} \Phi(A) = \sup\Big\{ |(Af,g)|: \quad f\in B(X_+), \quad g\in B(X_-') \Big\}. \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:2} Let $X$ be a reflexive rearrangement-invariant space with nontrivial Boyd indices. For $A\in\cL(X_+,X_-)$, we have %%% \begin{equation}\label{eq:2-1} \frac{1}{\gamma}\|A\|_{\cL(X_+,X_-)}\le\Phi(A)\le\|A\|_{\cL(X_+,X_-)}. \end{equation} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} It is well known (see, e.g., \cite[p.~93]{Rudin73}) that the norm of a bounded linear operator $A:X_+\to X_-$ can be computed by the formula %%% \begin{equation}\label{eq:2-3} \|A\|_{\cL(X_+,X_-)} = \sup\Big\{ |G(Af)|: \quad f\in B(X_+), \quad G\in B((X_-)^*) \Big\}. \end{equation} %%% Since \[ \|G\|_{(X_-)^*}= \inf_{h\in X_+'}\|g+h\|_{X'}\le \|g\|_{X'}, \] we infer that the inequality $\|g\|_{X'}\le 1$ implies $\|G\|_{(X_-)^*}\le 1$. Therefore, taking into account $(Af,g)= G(Af)$, we deduce that the set \[ S_1:= \Big\{ |(Af,g)|: \quad f\in B(X_+), \quad g\in B(X_-') \Big\} \] is contained in the set \[ S_2:= \Big\{ |G(Af)|: \quad f\in B(X_+), \quad G\in B((X_-)^*) \Big\}. \] Hence, in view of (\ref{eq:Phi}), (\ref{eq:2-3}), and the latter observation, we obtain %%% \begin{equation}\label{eq:2-4} \Phi(A)= \sup S_1 \le \sup S_2 = \|A\|_{\cL(X_+,X_-)}. \end{equation} %%% On the other hand, by Proposition~\ref{pr:1}, if $\|G\|_{(X_-)^*}\le 1$, then $\left\|\frac{g}{\gamma}\right\|_{X'}\le 1$. Therefore, the set $S_2$ is contained in the set %%% \begin{eqnarray*} && \left\{ \left| \gamma\left(Af,\frac{g}{\gamma}\right) \right|: \quad f\in B(X_+), \quad g/\gamma\in B(X_-') \right\} \\ &&= \Big\{ \gamma |(Af,h)|: \quad f\in B(X_+), \quad h\in B(X_-') \Big\} = :\gamma S_1. \end{eqnarray*} %%% In view of (\ref{eq:Phi}), (\ref{eq:2-3}), and the latter observation, we have %%% \begin{equation}\label{eq:2-5} \|A\|_{\cL(X_+,X_-)} = \sup S_2 \le \sup(\gamma S_1) = \gamma\sup S_1 = \gamma\Phi(A). \end{equation} %%% From (\ref{eq:2-4}) and (\ref{eq:2-5}) we immediately get (\ref{eq:2-1}). \end{proof} The next proposition allows us to recalculate $\Phi(A)$ by using only polynomials. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:3} Let $X$ be a reflexive rearrangement-invariant space with nontrivial Boyd indices. For $A\in\cL(X_+,X_-)$, we have %%% \begin{equation}\label{eq:3-1} \Phi(A)= \sup\Big\{ |(Ap,\overline{q})|:\quad p\in\mathcal{P}_A, \quad q\in\mathcal{P}_A^0, \quad \|p\|_X\le 1, \quad \|q\|_{X'}\le 1 \Big\}. \end{equation} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} If $q\in\mathcal{P}_A^0$, then, obviously, $\overline{q}\in X_-'$. Therefore, the set \[ \Big\{ |(Ap,\overline{q})|: \quad p\in\mathcal{P}_A, \quad q\in\mathcal{P}_A^0, \quad \|p\|_X\le 1, \quad \|q\|_{X'}\le 1 \Big\} \] is contained in the set \[ \Big\{ |(Af,g)|:\quad f\in B(X_+), \quad g\in B(X_-') \Big\}. \] Thus, %%% \begin{equation}\label{eq:3-3} \sup\Big\{ |(Ap,\overline{q})|:\quad p\in\mathcal{P}_A, \quad q\in\mathcal{P}_A^0, \quad \|p\|_X\le 1, \quad \|q\|_{X'}\le 1 \Big\} \le\Phi(A). \end{equation} %%% On the other hand, by the definition of $\Phi(A)$, for every $\eps>0$ there exist $f\in B(X_+)$ and $g\in B(X_-')$ such that %%% \begin{equation}\label{eq:3-6} |(Af,g)|>\Phi(A)-\frac{\eps}{2}. \end{equation} %%% In view of Proposition~\ref{pr:density-subspace}(a)--(b), there exist $p,q\in\mathcal{P}_A$ such that $q(0)=0$ and %%% \begin{equation}\label{eq:3-7} \begin{array}{ll} \|p\|_X\le 1, & \displaystyle \|f-p\|_X<\frac{\eps}{4\|A\|_{\cL(X_+,X_-)}}, \\[3ex] \|q\|_{X'}\le 1, & \displaystyle \|g-\overline{q}\|_{X'}<\frac{\eps}{4\|A\|_{\cL(X_+,X_-)}}. \end{array} \end{equation} %%% Further, %%% \begin{equation}\label{eq:3-8} |(Af,g)| \le |(Ap,\overline{q})|+|(Af,g-\overline{q})|+|(A(f-p),\overline{q})|. \end{equation} %%% Taking into account Lemma~\ref{le:Hoelder}, $A\in\cL(X_+,X_-)$, and (\ref{eq:3-7}), we get %%% \begin{eqnarray} \nonumber |(Af,g-\overline{q})| &\le & \|Af\|_X\|g-\overline{q}\|_{X'} \\ \label{eq:3-9} &\le& \|A\|_{\cL(X_+,X_-)}\|f\|_X\|g-\overline{q}\|_{X'}<\frac{\eps}{4}, \\ \nonumber |(A(f-p),\overline{q})| &\le & \|A(f-p)\|_X\|\overline{q}\|_{X'} \\ \label{eq:3-10} &\le& \|A\|_{\cL(X_+,X_-)}\|f-p\|_X\|\overline{q}\|_{X'}<\frac{\eps}{4}. \end{eqnarray} %%% From (\ref{eq:3-6}) and (\ref{eq:3-8})--(\ref{eq:3-10}) we deduce that \[ \Phi(A)-\frac{\eps}{2}<|(Af,g)|<|(Ap,\overline{q})|+\frac{\eps}{2}. \] Thus, for every $\eps>0$ there exist $p,q\in\mathcal{P}_A$ such that \[ q(0)=0,\quad \|p\|_X\le 1, \quad \|q\|_{X'}\le 1, \quad \Phi(A)-\eps<|(Ap,\overline{q})|. \] Combining this with (\ref{eq:3-3}), we arrive at (\ref{eq:3-1}). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Analogue of the Nehari theorem} The operator $H(a)$ defined for $a\in L^\infty$ by \[ H(a): X_+\to X_-, \quad f\mapsto P_-(af) \] is obviously bounded. It is called the {\it Hankel operator} generated by the {\it symbol} $a$. It is clear that %%% \begin{equation}\label{eq:H+} (H(a)\chi_{-j},\chi_k) =a_{k+j}, \quad j\le 0, \quad k<0. \end{equation} %%% The following theorem describes the bounded operators from $X_+$ to $X_-$ with this property and provides important norm estimates for Hankel operators. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Nehari} Suppose $X$ is a reflexive rearrangement-invariant space with nontrivial Boyd indices. Let $A\in\cL(X_+,X_-)$ and suppose there is a sequence $\{a_n\}_{n=-\infty}^{-1}$ such that %%% \begin{equation}\label{eq:N-1} (A\chi_{-j},\chi_k) =a_{k+j}, \quad j\le 0, \quad k<0. \end{equation} %%% Then there exists an $a\in L^\infty$ such that $A=H(a)$ and the $n$-th Fourier coefficient of $a$ is equal to $a_n$ for all $n\le -1$. Moreover, %%% \begin{equation}\label{eq:N-1*} \inf_{\psi\in H^\infty}\|a-\psi\|_\infty \le \|H(a)\|_{\cL(X_+,X_-)} \le \gamma\inf_{\psi\in H^\infty}\|a-\psi\|_\infty. \end{equation} \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} Here we follow Sarason's idea for the Hilbert space $L^2$ \cite[p.~100]{Sarason78} (see also \cite[Lecture~8]{Nik86}) and its modification for the case of the Banach spaces $L^p$, $p\in(1,\infty)$, in \cite[Theorem~2.11]{BS90}. From (\ref{eq:N-1}) it follows that %%% \begin{equation}\label{eq:N-3} (A\chi_{-j},\chi_k)=(A\chi_0,\chi_{k+j}), \quad j\le 0, \quad k<0. \end{equation} %%% Using this relation, one can show that %%% \begin{equation}\label{eq:N-4} (Ap,\overline{q})=\int_\T (A\chi_0)pq\,dm, \quad p\in\mathcal{P}_A, \quad q\in\mathcal{P}_A^0. \end{equation} From Proposition~\ref{pr:3}, (\ref{eq:N-4}), and Lemma~\ref{co:H1} we get %%% \begin{equation}\label{eq:N-5} \begin{split} \Phi(A) &= \sup\left\{ \left| \int_\T(A\chi_0)pq\,dm \right|:\quad \begin{array}{ll} p\in\mathcal{P}_A, & \|p\|_X\le 1, \\[1ex] q\in\mathcal{P}_A^0, & \|q\|_{X'}\le 1 \end{array} \right\} \\[2ex] &= \sup\left\{ \left| \int_\T(A\chi_0)g\,dm \right|:\quad g\in\mathcal{P}_A^0,\quad\|g\|_{L^1}\le 1 \right\}. \end{split} \end{equation} %%% On the other hand, by Proposition~\ref{pr:2}, %%% \begin{equation}\label{eq:N-6} \frac{1}{\gamma}\|A\|_{\cL(X_+,X_-)} \le \Phi(A) \le \|A\|_{\cL(X_+,X_-)}. \end{equation} %%% Hence, the functional %%% \begin{equation}\label{eq:N-7} g\mapsto\int_\T(A\chi_0)g\,dm, \quad g\in\mathcal{P}_A^0, \end{equation} %%% is bounded. Since $\mathcal{P}_A^0$ is dense in $H^1_0$, the functional (\ref{eq:N-7}) extends to a continuous functional $C\in(H^1_0)^*$ and $\|C\|_{(H^1_0)^*}=\Phi(A)$. In that case there is a function $a\in L^\infty$ such that %%% \begin{equation}\label{eq:N-8} \int_\T(A\chi_0)h\,dm = \int_\T ah\,dm, \quad h\in H^1_0. \end{equation} %%% Letting $h=\chi_{-n}, n\le -1$, from (\ref{eq:N-1}) and (\ref{eq:N-8}) we infer that the $n$-th Fourier coefficient of $a$ coincides with $a_n$. From (\ref{eq:N-1}) and (\ref{eq:N-3}) we get %%% \begin{equation}\label{eq:N-9} (H(a)\chi_{-j},\chi_k)=(A\chi_{-j},\chi_k)=a_{k+j}, \quad j\le 0, \quad k<0. \end{equation} %%% Since $H(a)\chi_{-j},A\chi_{-j}\in X_-\subset H^1_-$ for $j\le 0$, from the uniqueness theorem for Fourier series (see, e.g., \cite[Theorem~2.7]{Kat76}) and (\ref{eq:N-9}) it follows that %%% \begin{equation}\label{eq:N-10} H(a)\chi_i=A\chi_i, \quad i\ge 0. \end{equation} %%% On the other hand, the set $\{\chi_i, i\ge 0\}$ is dense in $X_+$. Therefore, from (\ref{eq:N-10}) we obtain $H(a)f=Af$ for all $f\in X_+$. Thus, $H(a)=A$. We are left with the norm estimate (\ref{eq:N-1*}). From (\ref{eq:N-5}), (\ref{eq:N-7}), (\ref{eq:N-8}), and \cite[Ch.~4, Theorem~1.3]{Garnett} it follows that %%% \begin{equation}\label{eq:N-11} \begin{split} \Phi(A) &= \|C\|_{(H^1_0)^*} =\sup\left\{ \left|\int_\T ah\,dm \right|:\quad h\in H^1_0, \quad \|h\|_{L^1}\le 1 \right\} \\ &= \inf_{\psi\in H^\infty}\|a-\psi\|_\infty. \end{split} \end{equation} %%% Combining (\ref{eq:N-6}) and (\ref{eq:N-11}), we arrive at (\ref{eq:N-1*}). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A.~Hartmann proved an analogue of the Nehari theorem for Banach lattices (i.e., for spaces in which axioms (a) and (b) in the definition of Banach function spaces hold) satisfying the so-called {\it approximate factorization property (AFP)} \cite[Theorem~3.2]{H98}. He also proved a lower estimate for the norm of $H(a)$. We are not able to verify the AFP for reflexive rearrangement-invariant spaces because we cannot guarantee continuity (or even ``higher integrability'') of $f$ and $g$ in Theorem~\ref{th:unit-ball} (as is necessary in the AFP). Instead of this we require density of analytic polynomials in $X'_+$ and apply the Lozanovskii factorization theorem (Theorem~\ref{th:Lozanovskii}). Thanks to this we can prove the two-sided estimate for the norm of $H(a)$. For a function $a\in L^\infty$, together with the Hankel operator $H(a):X_+\to X_-$, we consider also the Hankel operators defined on the whole space $X$ by the formulas %%% \begin{equation}\label{eq:Hpm-def} H_a^+:=P_-M_a P_+, \quad H_a^-:= P_+M_a P_-. \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{corollary}\label{co:Nehari} Let $X$ be a reflexive rearrangement-invariant space with nontrivial Boyd indices. If $a\in L^\infty$, then \begin{eqnarray} \label{eq:Nehari-1} && \inf_{\psi\in H^\infty}\|a-\psi\|_\infty \le \|H_a^+\|_{\cL(X)} \le\gamma^2 \inf_{\psi\in H^\infty}\|a-\psi\|_\infty, \\ \label{eq:Nehari-2} && \inf_{\psi\in\overline{H^\infty}}\|a-\psi\|_\infty \le \|H_a^-\|_{\cL(X)} \le\gamma^2 \inf_{\psi\in\overline{H^\infty}}\|a-\psi\|_\infty. \end{eqnarray} \end{corollary} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} As in the proof of Corollary~\ref{co:Brown-Halmos}, one can show that %%% \begin{equation}\label{eq:Nehari-3} \|H(a)\|_{\cL(X_+,X_-)} \le \|H_a^+\|_{\cL(X)} \le \gamma\|H(a)\|_{\cL(X_+,X_-)}. \end{equation} %%% Combining these inequalities with (\ref{eq:N-1*}), we arrive at (\ref{eq:Nehari-1}). Relations (\ref{eq:flip}) and (\ref{eq:Hpm-def}) imply $JH_a^-J=H_{\widetilde{a}}^+$. Since $J$ is an isometry on $X$, the latter equality gives %%% \begin{equation}\label{eq:Nehari-4} \|H_a^-\|_{\cL(X)}=\|H_{\widetilde{a}}^+\|_{\cL(X)}. \end{equation} %%% On the other hand, obviously, for $\widetilde{\psi}(t)=\psi(1/t)$, we obtain $\widetilde{\psi}_j=-\psi_{-j}\ (j\in\Z)$. Therefore, by the definition of $\overline{H^\infty}$, we have $\widetilde{\psi}\in\overline{H^\infty}$ if and only if $\psi\in H^\infty$. Hence, %%% \begin{equation}\label{eq:Nehari-5} \inf_{\psi\in H^\infty}\|\widetilde{a}-\psi\|_\infty = \inf_{\psi\in\overline{H^\infty}}\|a-\psi\|_\infty. \end{equation} Combining (\ref{eq:Nehari-3}) -- (\ref{eq:Nehari-5}), we arrive at (\ref{eq:Nehari-2}). \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Estimates for the norms of the Toeplitz operators $T_a^+, T_a^-$ and the Hankel operators $H_a^+, H_a^-$ play an important role in the theory of Banach algebras of one-dimensional singular integral operators (see, e.g., \cite{BS90,GD99}). Using the results of Corollaries~\ref{co:Brown-Halmos} and~\ref{co:Nehari}, we are able to extend all the results of \cite{GD99}, where singular integral operators are studied in Lebesgue spaces $L^p,1