\documentclass[12pt]{article} \usepackage{amsfonts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Karlovich's and Santos' commands \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\I}{\mathbb{I}} \newcommand{\N}{\mathbb{N}} \newcommand{\T}{\mathbb{T}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cW}{\mathcal{W}} \newcommand{\ind}{\mathrm{ind}\,} \newcommand{\im}{\mathrm{Im}\,} \newcommand{\slim}{{\rm s-}\!\!\!\lim_{n\to\infty}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{claim}[theorem]{Claim} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{definition}[theorem]{Definition} \newtheorem{problem}[theorem]{Problem} \renewcommand{\thesubsection}{\arabic{section}.\arabic{subsection}} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{On asymptotics of Toeplitz determinants\\ with symbols of nonstandard smoothness} \author{Alexei Yu. Karlovich \and Pedro A. Santos} \date{~} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We prove Szeg\H{o}'s strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary $N$-functions both satisfying the $\Delta_2^0$-condition and by weight sequences satisfying some regularity and compatibility conditions. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction and the main results} \setcounter{equation}{0} Let $\T$ be the unit circle. For a complex-valued function $a\in L^1(\T)$, let $\{a_k\}_{k=-\infty}^\infty$ be the sequence of the Fourier coefficients of $a$, \[ a_k:=\frac{1}{2\pi}\int_0^{2\pi}a(e^{i\theta})e^{-ik\theta}d\theta. \] Let $W$ be the Wiener algebra of all functions $a$ on $\T$ of the form \[ a(t)=\sum_{k=-\infty}^\infty a_k t^k \quad (t\in\T) \] for which \[ \|a\|_W:=\sum_{k=-\infty}^\infty |a_k|<\infty. \] It is well known that $W$ is a Banach algebra under the norm $\|\cdot\|_W$ and that $W$ is continuously imbedded into $C(\T)$, the Banach algebra of all complex-valued continuous functions with the maximum norm. Let $F\ell^{p,r}_{\alpha,\beta}$, where $1\le p,r<\infty$ and $-\infty<\alpha,\beta<\infty$, denote the set of all functions $a\in L^1(\T)$ for which \[ \sum_{k=1}^\infty |a_{-k}|^p(k+1)^{\alpha p} + \sum_{k=0}^\infty |a_k|^r(k+1)^{\beta r}<\infty. \] For a function $a\in L^1(\T)$, consider the determinants $D_n(a)$ of the finite Toeplitz matrices $T_n(a)$, \[ D_n(a)=\det T_n(a)=\det (a_{j-k})_{j,k=0}^n \quad (n\in\Z_+) \] where, as usual, $\Z_+:=\N\cup\{0\}$ and $\N:=\{1,2,\dots\}$. We shall denote the Cauchy index of a continuous function $a$ by $\ind a$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:BoetSilb} {\rm (see \cite[Corollary~10.42]{BS90})}. Suppose $p\in(1,\infty)$ and $\alpha\in[0,1]$. If $a\in W\cap F\ell^{p,p/(p-1)}_{\alpha,1-\alpha}$, $a(t)\ne 0$ for $t\in\T$, and $\ind a=0$, then %%% \begin{equation}\label{eq:Szego} \lim_{n\to\infty}\frac{D_n(a)}{\big(G(a)\big)^{n+1}}=E(a)\ne 0, \end{equation} where %%% \begin{equation}\label{eq:G-def} G(a):= e^{(\log a)_0} = \exp\left(\frac{1}{2\pi}\int_0^{2\pi}\log a(e^{i\theta})d\theta\right) \end{equation} %%% and %%% \begin{equation}\label{eq:E-def} E(a):= \exp\left( \sum_{k=1}^\infty k(\log a)_k(\log a)_{-k} \right). \end{equation} \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In 1952, G.~Szeg\H{o} showed that if the function $a\in C^{1+\varepsilon}$ is positive, then (\ref{eq:Szego}) holds. The smoothness conditions needed by Szeg\H{o} was subsequently relaxed by many authors including M.~Kac ($a\in F\ell^{1,1}_{1,1}$, 1954), G.~Baxter ($a\in F\ell^{1,1}_{1/2,1/2}$, 1961), I.~I.~Hirschman ($a\in W\cap F\ell^{2,2}_{1/2,1/2}$, 1966), M.~G.~Krein ($a\in C(\T)\cap F\ell^{2,2}_{1/2,1/2}$, 1966), A.~Devinatz ($a\in L^\infty(\T)\cap F\ell^{2,2}_{1/2,1/2}$, 1967), B.~L.~Golinski and I.~A.~Ibragimov finally proved (1971) that (\ref{eq:Szego}) is true if it makes sense and $a$ is positive. G. Baxter (1963) and I.~I.~Hirschman (1966) were the first to replace the positivity of $a$ by the condition that $a(t)\ne 0$ for $t\in\T$ and $\ind a=0$. H.~Widom \cite{Widom76} then gave easy proofs for symbols in the Krein algebra $K^{2,2}_{1/2,1/2}:=L^\infty(\T)\cap F\ell^{2,2}_{1/2,1/2}$. A more detailed history with precise references can be found in \cite[Ch.~6]{BS83}, \cite[Ch.~10]{BS90}, and also in \cite[Ch.~5]{BS99}. Nowadays the result of this type is frequently referred to as Szeg\H{o}'s (or Szeg\H{o}-Widom's) strong limit theorem for Toeplitz determinants. Theorem~\ref{th:BoetSilb} was established by I.~I.~Hirschman for $p=2$ and $\alpha=1/2$ and by A.~B\"ottcher and B.~Silbermann \cite[Section~6.32]{BS83} under the assumption that $W\cap F\ell^{p,p/(p-1)}_{\alpha,1-\alpha},10$ for $t>0$, and $\lim\limits_{t\to\infty} p(t)=\infty$. Then the function $q(s)=\sup\{t :\ p(t)\le s\}$ (defined for $s\ge 0$) has the same properties as the function $p$. The convex functions $\Phi$ and $\Psi$ defined by the equalities \[ \Phi(x):=\int_0^x p(t) dt, \quad \Psi(x):=\int_0^x q(s)ds \quad (x\ge 0) \] are called \textit{complementary $N$-functions} (see, e.g., \cite[Section~1.3]{KR58}, \cite[Ch.~8]{Maligranda89}, \cite[Section~13]{Musielak83}). An $N$-function $\Phi$ is said to satisfy the $\Delta_2^0$-condition if \[ \limsup_{x\to 0}\frac{\Phi(2x)}{\Phi(x)}<\infty. \] Any sequence $\{\nu_k\}_{k=0}^\infty$ of positive numbers is called a weight sequence. We denote by $\cW$ the set of all weight sequences $\{\nu_k\}_{k=0}^\infty$ such that \begin{enumerate} \item[(a)] $\nu_0=1$; \item[(b)] $\nu_{k-1}\le\nu_k$ for $k\in\N$; \item[(c)] $\{\nu_k\}_{k=0}^\infty$ satisfies the $\Delta_2^\N$-condition, that is, there is a constant $C_\nu\in(0,\infty)$ such that $\nu_{2k}\le C_\nu\nu_k$ for $k\in\N$. \end{enumerate} It is easy to see that $C_\nu\ge 1$. The main result of the paper reads as follows. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:main} Suppose $\Phi,\Psi$ are complementary $N$-functions both satisfying the $\Delta_2^0$-condition, $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ are weight sequences in $\cW$, and there exists a constant $M\in(0,\infty)$ such that %%% \begin{equation}\label{eq:main-1} k\le M\varphi_k\psi_k \quad\mbox{for all }\quad k\in\Z_+. \end{equation} %%% If $a\in W$, $a(t)\ne 0$ for $t\in\T$, $\ind a=0$, and %%% \begin{equation}\label{eq:main-2} \sum_{k=1}^\infty \Phi(|a_{-k}|\varphi_k) + \sum_{k=0}^\infty \Psi(|a_k|\psi_k)<\infty, \end{equation} %%% then {\rm (\ref{eq:Szego})} holds. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Since $x^p/p$ and $x^{p'}/p'$, where $1/p+1/p'=1$, are complementary $N$-functions both satisfying the $\Delta_2^0$-condition, the weight sequence $\{(k+1)^\alpha\}_{k=0}^\infty$ belongs to $\cW$ for $\alpha\in[0,1]$, and $(k+1)^\alpha(k+1)^{1-\alpha}\ge k$ for all $k\in\Z_+$; Theorem~\ref{th:BoetSilb} can be considered as a partial case of Theorem~\ref{th:main}. The next lemma shows that Theorem~\ref{th:main} is applicable to strictly wider class of functions $a\in W$ than Theorem~\ref{th:BoetSilb}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:example} There exist weight sequences $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ in $\cW$, a constant $M\in(0,\infty)$, and a complex-valued function $a\in W$ such that $a(t)\ne 0$ for $t\in\T, \ind a=0$, \[ a\notin F\ell^{p,p'}_{\alpha,1-\alpha} \quad\mbox{for every}\quad p\in (1,\infty),\quad \alpha\in[0,1], \] and \[ k\le M\varphi_k\psi_k\quad\mbox{for all}\quad k\in\Z_+, \quad \sum_{k=1}^\infty(|a_{-k}|\varphi_k)^2+\sum_{k=0}^\infty(|a_k|\psi_k)^2<\infty. \] \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This lemma will be proved constructively in Section~\ref{sec:6.3}. For Banach spaces $X,Y$ we denote by $\cB(X,Y)$ the set of all bounded linear operators from $X$ into $Y$ and by $\cK(X,Y)$ the set of all compact operators from $X$ into $Y$. We simply write $\cB(X)$ (resp. $\cK(X)$) instead of $\cB(X,X)$ (resp. $\cK(X,X)$). The group of all bounded invertible operators on $X$ is denoted by $G\cB(X)$. The rest of the paper is devoted to the proof of Theorem~\ref{th:main} and Lemma~\ref{le:example}. We follow an approach of A.~B\"ottcher and B.~Silbermann (see \cite[Ch.~6]{BS83} or \cite[Ch.~10]{BS90}) and give a self-contained and detailed proof of Theorem~\ref{th:main}. The paper is organized as follows. In Section~\ref{sec:2} we collect necessary facts from the theory of weighted Orlicz sequence spaces and weighted Orlicz sequence classes. Let $\Phi$ and $\Psi$ be (not necessarily complementary) $N$-functions. The set of all functions in $L^1(\T)$, whose sequence of Fourier coefficients with negative (resp. nonnegative) indices belongs to the weighted Orlicz sequence space $\ell^\Phi_\varphi(\N)$ (resp. to $\ell^\Psi_\psi(\Z_+)$), is denoted by $F\ell^{\Phi,\Psi}_{\varphi,\psi}$. One of the authors \cite{Karlovich03} has recently proved that if the weight sequences $\{\varphi_k\}_{k=0}^\infty$ and $\{\psi_k\}_{k=0}^\infty$ are in $\cW$, then the set $W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$ is a Banach algebra with the factorization property. We state this result precisely in the end of Section~\ref{sec:2} because we will need it in the present paper. In Section~\ref{sec:3} we study Toeplitz operators $T(a)$ and Hankel operators $H(a), H(\widetilde{a})$, where $\widetilde{a}(t):=a(1/t)$, in two different situations: %%% \begin{eqnarray*} \mbox{(a)}\ X &:=&\ell^\Psi(\Z_+),\quad a\in W;\\ \mbox{(b)}\ X &:=&\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+),\ Y :=\ell^1(\Z_+)\cap\ell^\Phi_\varphi(\Z_+),\ a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}. \end{eqnarray*} %%% In case (a) we verify that $H(a),H(\widetilde{a})\in\cK(X)$, but in case (b) we prove that $H(a)\in\cB(Y,X), H(\widetilde{a})\in\cB(X,Y)$. The operators $H(a)$ and $H(\widetilde{a})$ are compact in case (b) if, in addition, $\Phi$ and $\Psi$ satisfy the $\Delta_2^0$-condition. Notice that if $\Phi$ and $\Psi$ satisfy the $\Delta_2^0$-condition, then $F\ell^{\Phi,\Psi}_{\varphi,\psi}$ coincides with the class of functions $a\in L^1(\T)$ satisfying (\ref{eq:main-2}). In Section~\ref{sec:4} we verify the applicability of the finite section method to the Toeplitz operator $T(a)$ in both cases (a) and (b). In Section~\ref{sec:5} we prove a representation formula for the zeroth finite section $P_0T_n^{-1}(a)P_0$ on the space $\ell^\Psi(\Z_+)$ under the assumption $a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, $a(t)\ne 0$ for $t\in\T$, and $\ind a=0$. That representation formula implies an asymptotic formula for $D_{n-1}(a)/D_n(a)$. The verification of the above mentioned asymptotic formula is essentially based on duality arguments, so the assumption that $\Phi$ and $\Psi$ are complementary $N$-functions (both satisfying the $\Delta_2^0$-condition) is important in this part of the proof. On the basis of the results of Sections~\ref{sec:4}--\ref{sec:5} and Hirschman's formula we finish the proof of Theorem~\ref{th:main} in Section~\ref{sec:6}. Finally, we give a constructive proof of Lemma~\ref{le:example}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Weighted Orlicz sequence spaces\\ and related algebras of functions} \label{sec:2} \subsection{Weighted Orlicz sequence spaces} \setcounter{equation}{0} Let $\I$ be either $\N:=\{1,2,\dots\}$ or $\Z_+:=\N\cup\{0\}$. Suppose $\Phi$ is an $N$-function and $\{\varphi_k\}_{k=0}^\infty$ is a weight sequence. The set $\ell^\Phi_\varphi(\I)$ of all sequences $c=\{c_k\}_{k\in\I}$ of complex numbers such that %%% \begin{equation}\label{eq:Orlicz-def} \sum_{k\in\I}\Phi\left(\frac{|c_k|\varphi_k}{\lambda}\right)<\infty \end{equation} %%% for some $\lambda=\lambda(c)>0$ is a Banach space when equipped with the norm \[ \|c\|_{\ell^\Phi_\varphi(\I)} = \inf\left\{\lambda>0\ :\ \sum_{k\in\I}\Phi\left(\frac{|c_k|\varphi_k}{\lambda}\right)\le 1 \right\}. \] The space $\ell^\Phi_\varphi(\I)$ is called a \textit{weighted Orlicz sequence space}. If $\varphi_k=1$ for all $k\in\Z_+$, then we will simply write $\ell^\Phi(\I)$ instead of $\ell^\Phi_\varphi(\I)$ and we will say that $\ell^\Phi(\I)$ is an \textit{Orlicz sequence space}. The set $\widetilde{\ell}^\Phi_\varphi(\I)$ of all sequences $c=\{c_k\}_{k\in\I}$ of complex numbers for which (\ref{eq:Orlicz-def}) holds with $\lambda=1$ is called a \textit{weighted Orlicz sequence class}. Of particular interest is the subspace $h^\Phi_\varphi(\I)$ of $\ell^\Phi_\varphi(\I)$ consisting of those sequences $c=\{c_k\}_{k\in\I}\in\ell^\Phi_\varphi(\I)$ for which (\ref{eq:Orlicz-def}) holds for every $\lambda>0$. The subspace $h^\Phi_\varphi(\I)$ is called the \textit{subspace of finite elements} of $\ell^\Phi_\varphi(\I)$. Weighted Orlicz sequence spaces are the partial case of so-called Musielak-Orlicz sequence spaces (= modular sequence spaces). Good sources for the theory of Musielak-Orlicz sequence spaces are \cite[Section~4.d]{LT77} and \cite{Musielak83}; for Orlicz sequence spaces see also \cite{Maligranda89}. For properties of $N$-functions we refer to \cite[Ch.~1]{KR58} and also to \cite{Maligranda89}, \cite[Section~13]{Musielak83}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:AC} If a sequence $c=\{c_k\}_{k\in\I}$ belongs to $h^\Phi_\varphi(\I)$, then %%% \begin{equation}\label{eq:AC-1} \lim_{n\to\infty}\|c-c^{(n)}\|_{\ell^\Phi_\varphi(\I)}=0, \end{equation} %%% where $c^{(n)}_k=c_n$ for $k\le n$ and $c^{(n)}_k=0$ for $k>n$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} If $c\in h^\Phi_\varphi(\I)$, then (\ref{eq:Orlicz-def}) holds for every $\lambda>0$. Therefore, %%% \begin{equation}\label{eq:AC-2} \lim_{n\to\infty}\sum_{k\in\I} \Phi\left(\frac{|c_k-c_k^{(n)}|\varphi_k}{\lambda}\right) = \lim_{n\to\infty}\sum_{k=n+1}^\infty \Phi\left(\frac{|c_k|\varphi_k}{\lambda}\right)=0 \end{equation} %%% for every $\lambda>0$. Note that $\rho(x):=\sum_{k\in\I}\Phi(x\varphi_k)$ is a convex modular (see, e.g., \cite[Section~7.1]{Musielak83}). Then, by \cite[Theorem~1.6]{Musielak83}, (\ref{eq:AC-1}) holds if and only if for every $\lambda>0$ (\ref{eq:AC-2}) is fulfilled. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Applying the results of \cite[Theorem~8.14(b)]{Musielak83} (see also \cite[Proposition~4.d.3]{LT77}) to the sequence of $N$-functions $\Phi_k(x)=\Phi(x\varphi_k)$, we get the following. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Orlicz-class} If $\Phi$ is an $N$-function satisfying the $\Delta_2^0$-condition and $\{\varphi_k\}_{k=0}^\infty$ is a weight sequence, then $\ell^\Phi_\varphi(\I)= \widetilde{\ell}^\Phi_\varphi(\I)= h^\Phi_\varphi(\I)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Now we formulate an analog of H\"older's inequality for weighted Orlicz sequence spaces. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:Hoelder} Let $\Phi$ and $\Psi$ be complementary $N$-functions and let $\{\varphi_k\}_{k=0}^\infty$ and $\{\psi_k\}_{k=0}^\infty$ be weight sequences satisfying {\rm (\ref{eq:main-1})}. Suppose $c=\{c_k\}_{k=1}^\infty$ belongs to $\ell^\Phi_\varphi(\N)$ and $d=\{d_k\}_{k=1}^\infty$ belongs to $\ell^\Psi_\psi(\N)$. Then \[ \sum_{k=1}^\infty k|c_kd_k|\le 2M \|c\|_{\ell^\Phi_\varphi(\N)} \|d\|_{\ell^\Psi_\psi(\N)}. \] \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This lemma follows from (\ref{eq:main-1}) and H\"older's inequality for non-weighted Orlicz sequence spaces (see \cite[Theorems~13.11 and 13.13]{Musielak83} and also \cite[Section~9.7]{KR58}, \cite[Section~4.b]{LT77}, \cite[Ch.~8]{Maligranda89}). Below we formulate the result about the general form of a linear functional on Orlicz sequence spaces. The next theorem follows from \cite[Theorems~13.11 and 13.18]{Musielak83} (see also \cite[Section~4.b]{LT77}, \cite[Ch.~8]{Maligranda89}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:duality} Let $\Phi$ and $\Psi$ be complementary $N$-functions and let $\Psi$ satisfy the $\Delta_2^0$-condition. The general form of a linear functional on $\ell^\Psi(\Z_+)$ is given by \[ f(c)=\sum_{k=0}^\infty c_k\overline{d_k} \quad \Big(c=\{c_k\}_{k=0}^\infty\in\ell^\Psi(\Z_+)\Big), \] where $d=\{d_k\}_{k=0}^\infty$ is an arbitrary element of $\ell^\Phi(\Z_+)$ and \[ \|d\|_{\ell^\Phi(\Z_+)} \le \|f\|_{(\ell^\Psi(\Z_+))^*} \le 2\|d\|_{\ell^\Phi(\Z_+)}. \] \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The space $\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$} Let $\Phi$ and $\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ be weight sequences. We denote by $\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$ the Banach space of all sequences $\{c_k\}_{k=-\infty}^\infty$ of complex numbers for which $\{c_{-k}\}_{k=1}^\infty\in\ell^\Phi_\varphi(\N)$ and $\{c_k\}_{k=0}^\infty\in\ell^\Psi_\psi(\Z_+)$. The norm in $\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$ is defined by %%% \begin{equation}\label{eq:normPhiPsi} \|c\|_{\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)} := \Big\|\{c_{-k}\}_{k\in\N}\Big\|_{\ell^\Phi_\varphi(\N)} + \Big\|\{c_k\}_{k\in\Z_+}\Big\|_{\ell^\Psi_\psi(\Z_+)}. \end{equation} %%% We will identify $\ell^\Psi_\psi(\Z_+)$ with the closed subspace of $\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$ of the form \[ \Big\{\{c_k\}_{k=-\infty}^\infty\in\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)\ :\quad c_k=0\quad\mbox{for}\quad k<0\Big\}. \] It is easy to see that $\ell^{\Psi,\Psi}(\Z)$ coincides (as a set) with the Orlicz space $\ell^\Psi(\Z)$ equipped with the norm \[ \|c\|_{\ell^\Psi(\Z)}= \inf\left\{\lambda>0 \ : \ \sum_{k=-\infty}^\infty\Psi\left(\frac{|c_k|}{\lambda}\right)\le 1 \right\} \] and \[ \frac{1}{2}\|c\|_{\ell^{\Psi,\Psi}(\Z)} \le \|c\|_{\ell^\Psi(\Z)} \le 2\|c\|_{\ell^{\Psi,\Psi}(\Z)}. \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The algebra $W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$} Let $\Phi$ and $\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty$ and $\{\psi_k\}_{k=0}^\infty$ be weight sequences. We denote by $F\ell^{\Phi,\Psi}_{\varphi,\psi}$ the set of all functions $a\in L^1(\T)$ such that the sequence $\{a_k\}_{k=-\infty}^\infty$ of its Fourier coefficients belongs to $\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$. The set $F\ell^{\Phi,\Psi}_{\varphi,\psi}$ is a Banach space with respect to the norm \[ \|a\|_{F\ell^{\Phi,\Psi}_{\varphi,\psi}} := \|a\|_-+\|a\|_+, \] where %%% \begin{eqnarray*} \|a\|_- &:=& \Big\|\{a_{-k}\}_{k\in\N}\Big\|_{\ell^\Phi_\varphi(\N)} = \inf\left\{\lambda>0:\quad \sum_{k=1}^\infty \Phi\left(\frac{|a_{-k}|\varphi_k}{\lambda}\right) \le 1\right\}, \\ \|a\|_+ &:=& \Big\|\{a_k\}_{k\in\Z_+}\Big\|_{\ell^\Psi_\psi(\Z_+)} = \inf\left\{\mu>0:\quad \sum_{k=0}^\infty \Psi\left(\frac{|a_k|\psi_k}{\mu}\right) \le 1\right\}. \end{eqnarray*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:bar} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences. If $a$ belongs to $F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then its complex conjugate $\overline{a}$ belongs to $F\ell^{\Psi,\Phi}_{\psi,\varphi}$ and \[ \|\overline{a}\|_{F\ell^{\Psi,\Phi}_{\psi,\varphi}} \le K\|a\|_{F\ell^{\Phi,\Psi}_{\varphi,\psi}}, \] where %%% \begin{equation}\label{eq:bar-1} K:=1+\max\left\{1,\frac{\varphi_0\Psi^{-1}(1)}{\psi_0\Phi^{-1}(1/2)}\right\}. \end{equation} \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} It is easy to see that for a function $a\in L^1(\T)$, the Fourier coefficients of its complex conjugate $\overline{a}$ are calculated by the formula \[ (\overline{a})_k=\overline{a_{-k}}\quad (k\in\Z). \] Then %%% \begin{equation}\label{eq:bar-2} \|\overline{a}\|_{F\ell^{\Psi,\Phi}_{\psi,\varphi}} = \Big\|\{\overline{a_k}\}_{k\in\N}\Big\|_{\ell^\Psi_\psi(\N)} + \Big\|\{\overline{a_{-k}}\}_{k\in\Z_+}\Big\|_{\ell^\Phi_\varphi(\Z_+)}. \end{equation} %%% Denote %%% \begin{eqnarray*} A_-^0 &:=& \Big\|\{a_{-k}\}_{k\in\Z_+}\Big\|_{\ell^\Phi_\varphi(\Z_+)}, \quad A_- := \Big\|\{a_{-k}\}_{k\in\N}\Big\|_{\ell^\Phi_\varphi(\N)}, \\ A_+^0 &:=& \Big\|\{a_k\}_{k\in\Z_+}\Big\|_{\ell^\Psi_\psi(\Z_+)}, \quad\ A_+ := \Big\|\{a_k\}_{k\in\N}\Big\|_{\ell^\Psi_\psi(\N)}. \end{eqnarray*} %%% Taking into account (\ref{eq:bar-2}), we get %%% \begin{equation}\label{eq:bar-3} \|a\|_{F\ell^{\Phi,\Psi}_{\varphi,\psi}}=A_-+A_+^0, \quad \|\overline{a}\|_{F\ell^{\Psi,\Phi}_{\psi,\varphi}}=A_-^0+A_+. \end{equation} %%% It is easy to see that %%% \begin{equation}\label{eq:bar-4} A_+\le A_+^0. \end{equation} %%% If $A_+^0=0$, then $a_0=0$ and $A_+=0, A_-^0=A_-$. Then from (\ref{eq:bar-3}) we get %%% \begin{equation}\label{eq:bar-5} \|\overline{a}\|_{F\ell^{\Psi,\Phi}_{\psi,\varphi}} = \|a\|_{F\ell^{\Phi,\Psi}_{\varphi,\psi}}. \end{equation} If $A_+^0>0$, then \[ \Psi\left(\frac{|a_0|\psi_0}{A_+^0}\right) \le \sum_{k=0}^\infty\Psi\left(\frac{|a_k|\psi_k}{A_+^0}\right) \le 1. \] Therefore $|a_0|\le \Psi^{-1}(1)A_+^0/\psi_0$ and %%% \begin{equation}\label{eq:bar-6} \Phi\left(|a_0|\cdot\frac{\psi_0\Phi^{-1}(1/2)}{\Psi^{-1}(1)A_+^0}\right) \le \Phi\Big(\Phi^{-1}(1/2)\Big)=\frac{1}{2}. \end{equation} If $A_+^0>0$ and $A_-=0$, then from (\ref{eq:bar-6}) we get \[ \sum_{k=0}^\infty\Phi\left( |a_{-k}|\varphi_k\cdot \frac{\psi_0\Phi^{-1}(1/2)}{\varphi_0\Psi^{-1}(1)A_+^0} \right) = \Phi\left( \frac{|a_0|\varphi_0}{\varphi_0} \cdot \frac{\psi_0\Phi^{-1}(1/2)}{\Psi^{-1}(1)A_+^0} \right) \le \frac{1}{2}. \] Hence, %%% \begin{equation}\label{eq:bar-7} A_-^0\le \frac{\varphi_0\Psi^{-1}(1)}{\psi_0\Phi^{-1}(1/2)}A_+^0. \end{equation} If $A_+^0>0$ and $A_->0$, then from (\ref{eq:bar-6}) we deduce that %%% \begin{eqnarray} && \sum_{k=0}^\infty\Phi\left( \frac{|a_{-k}|\varphi_k} {\displaystyle 2A_-+\frac{\varphi_0\Psi^{-1}(1)}{\psi_0\Phi^{-1}(1/2)}A_+^0} \right) \le \sum_{k=1}^\infty\Phi\left(\frac{|a_{-k}|\varphi_k}{2A_-}\right) \nonumber\\ && + \Phi\left(\frac{|a_0|\varphi_0}{\varphi_0}\cdot \frac{\psi_0\Phi^{-1}(1/2)}{\Phi^{-1}(1)A_+^0}\right) \le \sum_{k=1}^\infty\Phi\left(\frac{|a_{-k}|\varphi_k}{2A_-}\right) +\frac{1}{2}. \label{eq:bar-8} \end{eqnarray} %%% Since $\Phi(x/2)\le\Phi(x)/2$ for $x\in[0,\infty)$ (see, e.g., \cite[p.~139]{LT77}), we have %%% \begin{equation}\label{eq:bar-9} \sum_{k=1}^\infty\Phi\left(\frac{|a_{-k}|\varphi_k}{2A_-}\right) \le \frac{1}{2} \sum_{k=1}^\infty\Phi\left(\frac{|a_{-k}|\varphi_k}{A_-}\right) \le \frac{1}{2}. \end{equation} %%% From (\ref{eq:bar-8}) and (\ref{eq:bar-9}) it follows that %%% \begin{equation}\label{eq:bar-10} A_-^0\le 2A_-+\frac{\varphi_0\Psi^{-1}(1)}{\psi_0\Phi^{-1}(1/2)}A_+^0. \end{equation} %%% Combining (\ref{eq:bar-3})--(\ref{eq:bar-5}), (\ref{eq:bar-7}), and (\ref{eq:bar-10}), we get %%% \begin{eqnarray*} \|\overline{a}\|_{F\ell^{\Psi,\Phi}_{\psi,\varphi}} &=& A_-^0+A_+ \le 2A_-+\frac{\varphi_0\Psi^{-1}(1)}{\psi_0\Phi^{-1}(1/2)}A_+^0+A_+^0 \\ &\le& K(A_-+A_+^0)=K\|a\|_{F\ell^{\Phi,\Psi}_{\varphi,\psi}}, \end{eqnarray*} %%% where $K$ is given by (\ref{eq:bar-1}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\{a_k\}_{k=-\infty}^\infty$ be the sequence of Fourier coefficients of a function $a\in L^1(\T)$. For $n\in\N$, put %%% \begin{equation}\label{eq:a-truncation} a^{(n)}(t):=\sum_{k=-n}^n a_kt^k \quad (t\in\T). \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:approximation} {\rm (a)} If $a\in W$, then \[ \lim_{n\to\infty} \|a-a^{(n)}\|_W=0. \] {\rm (b)} Let $\Phi,\Psi$ be $N$-functions both satisfying the $\Delta_2^0$-condition and let \\ $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ be weight sequences. If $a\in F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then \[ \lim_{n\to\infty}\|a-a^{(n)}\|_\pm=0, \quad \lim_{n\to\infty} \|a-a^{(n)}\|_{F\ell^{\Phi,\Psi}_{\varphi,\psi}}=0. \] \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part (a) is obvious. Part (b) follows from Proposition~\ref{pr:AC} and Theorem~\ref{th:Orlicz-class}. We equip the set $W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$ with the norm \[ \|a\|_{W\cap F}:=\|a\|_W+\|a\|_{F\ell^{\Phi,\Psi}_{\varphi,\psi}}. \] The next theorem generalizes Horbach's theorem \cite[Theorem~6.54]{BS90}. It was recently proved by one of the authors (see \cite[Theorem~2.2 and Corollary~2.3]{Karlovich03}) in a slightly more general form. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Karlovich} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. {\rm (a)} For every $a,b\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, \[ \|ab\|_{W\cap F}\le C\|a\|_{W\cap F}\|b\|_{W\cap F}, \] where %%% \begin{equation}\label{eq:Karlovich-1} C:=1+2C_\varphi+2C_\psi. \end{equation} {\rm (b)} If %%% \begin{equation}\label{eq:Karlovich-2} a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}, \quad a(t)\ne 0 \mbox{ for }t\in\T, \quad \ind a=0, \end{equation} %%% then $a$ has a logarithm in $W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. If we let for $t\in\T$, %%% \begin{eqnarray} a_-(t) &:=& \exp\left(\sum_{k=1}^\infty(\log a)_{-k}t^{-k}\right), \label{eq:WH-} \\ a_+(t) &:=& \exp\left(\sum_{k=0}^\infty(\log a)_kt^k\right), \label{eq:WH+} \end{eqnarray} %%% then $a=a_-a_+$ and $a_\pm^{\pm 1}\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:constant-Ea} Suppose $\Phi$ and $\Psi$ are complementary $N$-functions and $\{\varphi_k\}_{k=0}^\infty$ and $\{\psi_k\}_{k=0}^\infty$ are weight sequences in $\cW$. If {\rm (\ref{eq:main-1})} and {\rm (\ref{eq:Karlovich-2})} are valid, then \[ \sum_{k=1}^\infty k\Big|(\log a)_k(\log a)_{-k}\Big|<\infty. \] \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} By Theorem~\ref{th:Karlovich}(b), $\log a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. Hence, $\{(\log a)_{-k}\}_{k=1}^\infty$ belongs to $\ell^\Phi_\varphi(\N)$ and $\{(\log a)_k\}_{k=1}^\infty$ belongs to $\ell^\Psi_\psi(\N)$. Combining these facts with Lemma~\ref{le:Hoelder}, we arrive at the concluison. \rule{2mm}{2mm} Lemma~\ref{le:constant-Ea} immediately implies that the constant $E(a)$ is well defined by (\ref{eq:E-def}) and $E(a)\ne 0$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Laurent, Toeplitz, flip,\\ and Hankel operators}\label{sec:3} \subsection{Boundedness of the Laurent operator} \setcounter{equation}{0} For Banach spaces $X$ and $Y$ of one- or two-sided sequences of complex numbers we will frequently consider the set $X\cap Y$ of sequences $c$ such that $c\in X$ and $c\in Y$. This is a Banach space with respect to the norm \[ \|c\|_{X\cap Y}:=\|c\|_X+\|c\|_Y. \] Let $a$ be a function in $L^1(\T)$ with the Fourier coefficients $\{a_k\}_{k=-\infty}^\infty$. Let $c=\{c_k\}_{k=-\infty}^\infty$ be a sequence of complex numbers. We formally define the \textit{Laurent operator} with the symbol $a$ by \[ L(a)\ : \quad \{c_k\}_{k=-\infty}^\infty \mapsto \Big\{\sum_{k=-\infty}^\infty a_{j-k}c_k \Big\}_{j=-\infty}^\infty. \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:Laurent} {\rm (a)} If $\Psi$ is an $N$-function and $a\in W$, then \[ \|L(a)\|_{\cB(\ell^\Psi(\Z))}\le \|a\|_W. \] {\rm (b)} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If $a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then \[ \|L(a)\|_{\cB(\ell^1(\Z)\cap\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z))} \le C\|a\|_{W\cap F}, \] where the constant $C$ is given by {\rm (\ref{eq:Karlovich-1})}. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Part (a) follows from the proof of \cite[Lemma~3.1]{Luxemburg68}. Regarding part (b), let $c=\{c_k\}_{k=-\infty}^\infty\in\ell^1(\Z)\cap \ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$. Then \[ b(t):=\sum_{k=-\infty}^\infty c_kt^k \quad (t\in\T) \] belongs to $W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. Clearly, \[ (L(a)c)_j=\sum_{k=-\infty}^\infty a_{j-k}c_k=(ab)_j \quad (j\in\Z). \] Therefore, by Theorem~\ref{th:Karlovich}(a), %%% \begin{eqnarray*} \|L(a)c\|_{\ell^1(\Z)\cap \ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)} &=& \|ab\|_{W\cap F} \le C\|a\|_{W\cap F}\|b\|_{W\cap F} \\ &=& C\|a\|_{W\cap F}\|c\|_{\ell^1(\Z)\cap\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)}, \end{eqnarray*} %%% where the constant $C$ is given by (\ref{eq:Karlovich-1}). Part (b) follows from the latter inequality. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Note that the spectrum of the Laurent operator $L(a)$ with $a\in W$ on Orlicz spaces $\ell^\Psi(\Z)$ was studied by W.~A.~J.~Luxemburg \cite{Luxemburg68}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Boundedness of Toeplitz operators} Let $c=\{c_k\}_{k=-\infty}^\infty$ be a two-sided sequence of complex numbers. Define the operators $P$ and $Q$ as follows \[ (Pc)_k:=\left\{ \begin{array}{ccc} c_k &\mbox{for} & k\ge 0,\\ 0 &\mbox{for} & k<0, \end{array} \right. \quad \quad (Qc)_k:=\left\{ \begin{array}{ccc} 0 &\mbox{for} & k\ge 0,\\ c_k &\mbox{for} & k<0. \end{array} \right. \] Obviously, the operators $P$ and $Q$ are bounded projections on $\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$ and on $\ell^1(\Z)$. Moreover, $P+Q=I$. The operator \[ T(a):=PL(a)P|\im P \] is said to be the \textit{Toeplitz operator} with the symbol $a$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:Toeplitz} {\rm (a)} If $\Psi$ is an $N$-function and $a\in W$, then the Toeplitz operator $T(a)$ is bounded on the space $\ell^\Psi(\Z_+)$. {\rm (b)} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If $a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then the Toeplitz operator $T(a)$ is bounded on the space $\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+)$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This statement immediately follows from Proposition~\ref{pr:Laurent}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Boundedness of the flip operator} We define the \textit{flip operator} $J$ by the formula \[ (Jc)_k=c_{-k-1},\quad k\in\Z. \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:flip} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. Then \[ J\in\cB\Big( \ell^{\Phi,\Psi}_{\varphi,\psi}(\Z), \ell^{\Psi,\Phi}_{\psi,\varphi}(\Z) \Big), \quad J\in\cB\Big( \ell^{\Psi,\Phi}_{\psi,\varphi}(\Z), \ell^{\Phi,\Psi}_{\varphi,\psi}(\Z) \Big), \] and $J^2=I$ on both spaces $\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$ and $\ell^{\Psi,\Phi}_{\psi,\varphi}(\Z)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Let $c=\{c_k\}_{k=-\infty}^\infty\in\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$. Put \[ C_-:=\Big\|\{c_{-k}\}_{k\in\N}\Big\|_{\ell^\Phi_\varphi(\N)}, \quad C_+:=\Big\|\{c_k\}_{k\in\Z_+}\Big\|_{\ell^\Psi_\psi(\Z_+)}. \] Without loss of generality, assume that $C_\pm>0$. Then, taking into account that the weight sequence $\{\varphi_k\}_{k=0}^\infty$ is non-decreasing, we have %%% \begin{eqnarray*} && \sum_{k=0}^\infty\Phi\left(\frac{(Jc)_k\varphi_k}{C_-}\right) = \sum_{k=0}^\infty\Phi\left(\frac{c_{-k-1}\varphi_k}{C_-}\right) \\ && \le \sum_{k=0}^\infty\Phi\left(\frac{c_{-(k+1)}\varphi_{k+1}}{C_-}\right) = \sum_{k=1}^\infty\Phi\left(\frac{c_{-k}\varphi_k}{C_-}\right) \le 1. \end{eqnarray*} %%% Therefore, %%% \begin{equation}\label{eq:flip-1} \Big\|\{(Jc)_k\}_{k\in\Z_+}\Big\|_{\ell^\Phi_\varphi(\Z_+)} \le \Big\|\{c_{-k}\}_{k\in\N}\Big\|_{\ell^\Phi_\varphi(\N)}. \end{equation} %%% On the other hand, since $C_\psi\ge 1$ and $\psi_0=1$, we have %%% \begin{equation}\label{eq:flip-2} \psi_1=\psi_1\psi_0\le C_\psi\psi_1\psi_0. \end{equation} %%% If $k>1$, then, in view of the definition of the class $\cW$, %%% \begin{eqnarray} \label{eq:flip-3} \psi_k &\le& \psi_{2(k-1)}\le C_\psi\psi_{k-1}\le C_\psi\psi_1\psi_{k-1}. \end{eqnarray} %%% Then, taking into account (\ref{eq:flip-2})--(\ref{eq:flip-3}), we obtain %%% \begin{eqnarray*} && \sum_{k=1}^\infty \Psi\left(\frac{(Jc)_{-k}\psi_k}{C_+C_\psi\psi_1}\right) = \sum_{k=1}^\infty \Psi\left(\frac{c_{k-1}\psi_k}{C_+C_\psi\psi_1}\right) \nonumber\\ && \le \sum_{k=1}^\infty \Psi\left(\frac{c_{k-1}\psi_{k-1}}{C_+}\right) = \sum_{k=0}^\infty \Psi\left(\frac{c_k\psi_k}{C_+}\right) \le 1. \end{eqnarray*} %%% Thus, %%% \begin{equation}\label{eq:flip-6} \Big\|\{(Jc)_{-k}\}_{k\in\N}\Big\|_{\ell^\Psi_\psi(\N)} \le C_\psi\psi_1 \Big\|\{c_k\}_{k\in\Z_+}\Big\|_{\ell^\Psi_\psi(\Z_+)}. \end{equation} %%% From (\ref{eq:normPhiPsi}), (\ref{eq:flip-1}), and (\ref{eq:flip-6}) it follows that \[ \|Jc\|_{\ell^{\Psi,\Phi}_{\psi,\varphi}(\Z)} \le C_\psi\psi_1 \|c\|_{\ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)}. \] Analogously one can check that $J\in\cB\Big( \ell^{\Psi,\Phi}_{\psi,\varphi}(\Z), \ell^{\Phi,\Psi}_{\varphi,\psi}(\Z) \Big)$. The equality $J^2=I$ is obvious. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Boundedness and compactness\\ of Hankel operators} Define the \textit{Hankel operator} with the \textit{symbol} $a$ by \[ H(a):=PL(a)QJ|\im P. \] Put $\widetilde{a}(t):=a(1/t)$ for $t\in\T$, then it is easy to see that \[ H(\widetilde{a})=JQL(a)P|\im P. \] Let %%% \begin{eqnarray*} H^\infty &:=& \Big\{a\in L^\infty(\T)\ :\ a_k=0\quad\mbox{for}\quad k<0\Big\}, \\ \overline{H^\infty} &:=& \Big\{a\in L^\infty(\T)\ :\ a_k=0\quad\mbox{for}\quad k>0\Big\}. \end{eqnarray*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:Hankel-Orlicz} Let $\Psi$ be an $N$-function. \begin{enumerate} \item[{\rm (a)}] If $a\in W$, then $H(a),H(\widetilde{a})\in\cK(\ell^\Psi(\Z_+))$; \item[{\rm (b)}] If $a\in W\cap H^\infty$, then $H(\widetilde{a})=0$. \item[{\rm (c)}] If $a\in W\cap\overline{H^\infty}$, then $H(a)=0$. \end{enumerate} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} (a) Since $J$ is an isometry on the Orlicz space $\ell^\Psi(\Z)$ and \[ \|P\|_{\cB(\ell^\Psi(\Z))}\le 1, \quad \|Q\|_{\cB(\ell^\Psi(\Z))}\le 1, \] from Proposition~\ref{pr:Laurent}(a) it follows that %%% \begin{eqnarray} \|H(a)-H(a^{(n)})\|_{\cB(\ell^\Psi(\Z_+))} &\le& \|a-a^{(n)}\|_W, \label{eq:Hankel-Orlicz-1} \\ \|H(\widetilde{a})-H(\widetilde{a}^{(n)})\|_{\cB(\ell^\Psi(\Z_+))} &\le& %\|\widetilde{a}-\widetilde{a}^{(n)}\|_W %= \|a-a^{(n)}\|_W, \label{eq:Hankel-Orlicz-2} \end{eqnarray} where the function $a^{(n)}$ is given by (\ref{eq:a-truncation}). Clearly, both operators $H(a^{(n)})$ and $H(\widetilde{a}^{(n)})$ are finite-dimensional and therefore compact. From Proposition~\ref{pr:approximation}(a) and (\ref{eq:Hankel-Orlicz-1})--(\ref{eq:Hankel-Orlicz-2}) it follows that %%% \begin{eqnarray*} \lim_{n\to\infty}\|H(a)-H(a^{(n)})\|_{\cB(\ell^\Psi(\Z_+))} &=&0, \\ \lim_{n\to\infty} \|H(\widetilde{a})-H(\widetilde{a}^{(n)})\|_{\cB(\ell^\Psi(\Z_+))} &=&0. \end{eqnarray*} %%% Thus the operators $H(a)$ and $H(\widetilde{a})$ are compact on $\ell^\Psi(\Z_+)$. Part (a) is proved. Parts (b) and (c) are obvious. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Put \[ X :=\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+), \quad Y :=\ell^1(\Z_+)\cap\ell^\Phi_\varphi(\Z_+). \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:Hankel-weighted-Orlicz} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. \begin{enumerate} \item[{\rm (a)}] If $a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then there exists a constant $L>0$ such that \[ \|H(a)\|_{\cB(Y,X)}\le L\|a\|_{W\cap F}, \quad \|H(\widetilde{a})\|_{\cB(X,Y)}\le L\|a\|_{W\cap F}. \] \item[{\rm (b)}] If $a\in (W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi})\cap H^\infty$, then $H(\widetilde{a})=0$. \item[{\rm (c)}] If $a\in (W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi})\cap\overline{H^\infty}$, then $H(a)=0$. \end{enumerate} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Part (a) follows from Proposition~\ref{pr:Laurent}(b) and Lemma~\ref{le:flip}. Parts (b) and (c) are trivial. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:Hankel-compactness} Suppose $\Phi,\Psi$ are $N$-functions both satisfying the $\Delta_2^0$-condi\-tion and $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ are weight sequences in $\cW$. If $a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then $H(a)\in\cK(Y,X), H(\widetilde{a})\in\cK(X,Y)$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Let $a^{(n)}$ be the function given by (\ref{eq:a-truncation}). It is not difficult to see that the Hankel operators $H(a^{(n)})\in \cB(Y,X)$ and $H(\widetilde{a}^{(n)})\in\cB(X,Y)$ are finite-dimensional and therefore compact. Proposition~\ref{pr:Hankel-weighted-Orlicz}(a) and Proposition~\ref{pr:approximation} imply %%% \begin{eqnarray*} \lim_{n\to\infty}\|H(a)-H(a^{(n)})\|_{\cB(Y,X)} &\le& L\lim_{k\to\infty}\|a-a^{(n)}\|_{W\cap F}=0, \\ \lim_{n\to\infty} \|H(\widetilde{a})-H(\widetilde{a}^{(n)})\|_{\cB(X,Y)} &\le& L\lim_{n\to\infty}\|a-a^{(n)}\|_{W\cap F}=0. \end{eqnarray*} %%% Thus, $H(a)\in\cK(Y,X)$ and $H(\widetilde{a})\in\cK(X,Y)$. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Widom's formulas} Now we are in a position to formulate an analog of Widom's formulas (see, e.g., \cite[Proposition~2.7]{BS83}, \cite[Proposition~2.14]{BS90}, \cite[Proposition~1.12]{BS99}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:Widom} {\rm (a)} If $\Psi$ is an $N$-function and $a,b\in W$, then %%% \begin{eqnarray} \label{eq:Widom-1} T(ab) &=& T(a)T(b)+H(a)H(\widetilde{b}), \\ \label{eq:Widom-2} H(ab) &=& T(a)H(b)+H(a)T(\widetilde{b}) \end{eqnarray} on the space $\ell^\Psi(\Z_+)$. {\rm (b)} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If $a,b\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then {\rm (\ref{eq:Widom-1})} is valid on the space $\ell^1(\Z_+)\cap \ell^\Psi_\psi(\Z_+)$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} In the case (a), if $a,b\in W$, then $ab\in W$. By Proposition~\ref{pr:Laurent}(a), the operators $L(a), L(b)$, and $L(ab)$ are bounded on $\ell^\Psi(\Z)$. Similarly, in the case (b), if $a,b\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then, in view of Theorem~\ref{th:Karlovich}(a), $ab\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. By Proposition~\ref{pr:Laurent}(b), the operators $L(a), L(b)$, and $L(ab)$ are bounded on $\ell^1(\Z)\cap \ell^{\Phi,\Psi}_{\varphi,\psi}(\Z)$. The result now follows by writing the operators explicitly as defined above. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Invertibility of Toeplitz operators} Arguing as in \cite[p.~15]{BS99}, with the help of Propositions \ref{pr:Hankel-Orlicz}(b)-(c), \ref{pr:Hankel-weighted-Orlicz}(b)-(c), and \ref{pr:Widom}, one can prove the following. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:Toeplitz-inverse} Suppose $a_\pm$ are defined by {\rm (\ref{eq:WH-})--(\ref{eq:WH+})}, respectively. {\rm (a)} Let $\Psi$ be an $N$-function. If $a_\pm^{\pm 1}\in W$, then the operators $T(a_-), T(a_+)$, $T(a_-a_+)$, and $T(a_-^{-1}a_+^{-1})$ are invertible on $\ell^\Psi(\Z_+)$ and %%% \begin{equation}\label{eq:Toeplitz-inverse} T^{-1}(a_-a_+)=T(a_+^{-1})T(a_-^{-1}), \ T^{-1}(a_-^{-1}a_+^{-1})=T(a_+)T(a_-). \end{equation} %%% {\rm (b)} Let $\Phi,\Psi$ be arbitrary $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If $a_\pm^{\pm 1}\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then the operators $T(a_-), T(a_+), T(a_-a_+)$, and $T(a_-^{-1}a_+^{-1})$ are invertible on the space $\ell^1(\Z_+)\cap \ell^\Psi_\psi(\Z_+)$ and the inverses of $T(a_-a_+)$ and $T(a_-^{-1}a_+^{-1})$ are given by {\rm (\ref{eq:Toeplitz-inverse})}. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The finite section method}\label{sec:4} \subsection{Projection methods} \setcounter{equation}{0} Let $X$ be a Banach space and $A\in\cB(X)$. A projection method is a method for the approximate solution of the equation %%% \begin{equation}\label{eq:proj-1} Ax=y \end{equation} %%% which can be described as follows. Let $\{\cP_n\}_{n=0}^\infty$ be the sequence of projections $\cP_n\in\cB(X)$ such that \[ \slim\cP_n=I, \] where ${\rm s}-\lim\limits_{n\to\infty}$ represents the strong limit, and let $A_n:=\cP_nA\cP_n|\cP_nX$. We shall identify $A_n$ with $A_n\cP_n$ and may therefore regard $A_n$ as an element of $\cB(X)$. It is easy to see that \[ \slim A_n\cP_n = \slim\cP_nA_n\cP_n=A. \] Now we consider the equation %%% \begin{equation}\label{eq:proj-2} A_nx_n=\cP_ny \quad (x_n\in\cP_nX). \end{equation} %%% We write $A\in\Pi(X,\cP_n)$ and say that the projection method is applicable to $A$ if %%% \begin{enumerate} \item[{\rm (i)}] there exists an $n_0\in\N$ such that for each $y\in X$ the equation (\ref{eq:proj-2}) has a unique solution $x_n\in\cP_nX$ for all $n\ge n_0$; \item[{\rm (ii)}] $x_n$ converges in the norm of $X$ to a solution $x\in X$ of (\ref{eq:proj-1}). \end{enumerate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:proj-method} {\rm (see \cite[Proposition~7.3]{BS90}).} The following conditions are equivalent: \begin{enumerate} \item[{\rm (i)}] $A\in\Pi(X,\cP_n)$; \item[{\rm (ii)}] $A\in G\cB(X)$, $A_n\in G\cB(\cP_n X)$ for all sufficiently large $n$ ($n\ge n_0$, say), and \[ \sup_{n\ge n_0}\|A_n^{-1}\cP_n\|_{\cB(X)}<\infty; \] \item[{\rm (iii)}] $A\in G\cB(X)$, $A_n\in G\cB(\cP_n X)$ for all sufficiently large $n$, and \[ \slim A^{-1}\cP_n=A^{-1}. \] \end{enumerate} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:GF} {\rm (see, \cite[Ch.~2, Theorem~3.1]{GF74}).} Let $A\in\Pi(X,\cP_n)$ and\\ $K\in\cK(X)$. If $A+K\in G\cB(X)$, then $A+K\in\Pi(X,\cP_n)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Applicability of the finite section method} Let $\Psi$ be an $N$-function and let $\{\psi_k\}_{k=0}^\infty$ be a weight sequence. Let $n\in\Z_+$ and $P_n$ be the operator defined by \[ P_n:\{c_0,c_1,c_2,\dots\}\mapsto\{c_0,c_1,\dots, c_n,0,0,\dots\} \] and $Q_n:=I-P_n$. Clearly, $P_n,Q_n\in\cB(X)$ and $P_n^2=P_n, Q_n^2=Q_n$ on the space $X$, where $X$ is one of the spaces $\ell^\Psi(\Z_+), \ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:projections} If $\Psi$ is an $N$-function satisfying the $\Delta_2^0$-condition and $\{\psi_k\}_{k=0}^\infty$ is a weight sequence, then \[ \slim P_n=I \] on both spaces $\ell^\Psi(\Z_+)$ and $\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+)$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This statement follows from Theorem~\ref{th:Orlicz-class} and Proposition~\ref{pr:AC}. The \textit{finite section method} is a projection method defined by the projections $\cP_n=P_n$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:zero-trunc} {\rm (a)} Let $\Psi$ be an $N$-function. If $a_+\in W\cap H^\infty$ and $a_-\in W\cap \overline{H^\infty}$, then %%% \begin{equation}\label{eq:zero-trunc} P_n T(a_+)Q_n=Q_nT(a_-)P_n=0 \end{equation} %%% on the space $\ell^\Psi(\Z_+)$. {\rm (b)} Let $\Phi,\Psi$ be $N$-functions and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If \[ a_+\in (W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}) \cap H^\infty, \quad a_-\in (W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}) \cap \overline{H^\infty}, \] then {\rm (\ref{eq:zero-trunc})} holds on the space $\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+)$. \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This statement is proved by direct and elementary computations. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:applicability} Suppose $a_\pm$ are defined by {\rm (\ref{eq:WH-})--(\ref{eq:WH+})}, respectively. {\rm (a)} Let $\Psi$ be an $N$-function satisfying the $\Delta_2^0$-condition. Suppose \\ $a_\pm^{\pm 1}\in W$. Then $T(a_-a_+)\in\Pi(\ell^\Psi(\Z_+),P_n)$. {\rm (b)} Let $\Phi,\Psi$ be $N$-functions both satisfying the $\Delta_2^0$-condition and let \\ $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If $a_\pm^{\pm 1}\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then $T(a_-a_+)\in\Pi(\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+),P_n)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Let us prove Part (b). The idea of the proof goes back to Gohberg and Fel'dman (see \cite{GF74}). Here we adopt arguments of \cite[Section~2.4, p.~45]{BS99} to our case. Put $X:=\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+)$. By Proposition~\ref{pr:Toeplitz-inverse}(b), %%% \begin{equation}\label{eq:applicability-1} A:=T(a_+)T(a_-)\in G\cB(X). \end{equation} %%% We have %%% \begin{equation}\label{eq:applicability-2} P_nAP_n = P_nT(a_+)P_nT(a_-)P_n+P_nT(a_+)Q_nT(a_-)P_n. \end{equation} %%% It is easy to see that $a_+\in H^\infty$ and $a_-\in\overline{H^\infty}$. From (\ref{eq:applicability-2}) and Proposition~\ref{pr:zero-trunc}(b) we immediately get %%% \begin{equation}\label{eq:applicability-4} P_nAP_n = T_n(a_+)T_n(a_-). \end{equation} %%% Since $a_-$ and $a_-^{-1}$ belong to $\overline{H^\infty}$, the finite Toeplitz matrices $T_n(a_-)$ and $T_n(a_-^{-1})$ are upper-triangular. Hence, \[ T_n(a_-)T_n(a_-^{-1})=I=T_n(a_-^{-1})T_n(a_-). \] Thus, for every $n\in\Z_+$, $T_n(a_-)$ is invertible on $P_nX$ and %%% \begin{equation}\label{eq:applicability-5} T_n^{-1}(a_-)=T_n(a_-^{-1}). \end{equation} %%% Analogously, for every $n\in\Z_+$, $T_n(a_+)$ is invertible on $P_nX$ and %%% \begin{equation}\label{eq:applicability-6} T_n^{-1}(a_+)=T_n(a_+^{-1}). \end{equation} %%% Combining (\ref{eq:applicability-4})--(\ref{eq:applicability-6}), we see that %%% \begin{equation}\label{eq:applicability-7} P_nAP_n\in G\cB(P_nX) \quad (n\in\Z_+) \end{equation} %%% and \[ (P_nAP_n)^{-1}P_n=T_n(a_-^{-1})T_n(a_+^{-1})P_n= P_nT(a_-^{-1})P_n\cdot P_nT(a_+^{-1})P_n. \] Hence, for every $n\in\Z_+$, %%% \begin{equation}\label{eq:applicability-8} \|(P_nAP_n)^{-1}P_n\|_{\cB(X)} \le \|T(a_-^{-1})\|_{\cB(X)}\|T(a_+^{-1})\|_{\cB(X)}. \end{equation} %%% From (\ref{eq:applicability-1}), (\ref{eq:applicability-7}), (\ref{eq:applicability-8}) and Propositions~\ref{pr:proj-method} and~\ref{pr:projections} it follows that %%% \begin{equation}\label{eq:applicability-9} T(a_+)T(a_-)\in\Pi(X,P_n). \end{equation} %%% On the other hand, by Proposition~\ref{pr:Widom}(b), %%% \begin{equation}\label{eq:applicability-10} T(a_-a_+)=T(a_+a_-)=T(a_+)T(a_-)+H(a_+)H(\widetilde{a_-}). \end{equation} %%% From Proposition~\ref{pr:Hankel-compactness} and Proposition~\ref{pr:Toeplitz-inverse}(b) we deduce that %%% \begin{equation}\label{eq:applicability-11} H(a_+)H(\widetilde{a_-})\in\cK(X), \quad\quad T(a_-a_+)\in G\cB(X), \end{equation} %%% respectively. Combining (\ref{eq:applicability-9})--(\ref{eq:applicability-11}) and taking into account Theorem~\ref{th:GF}, we conclude that $T(a_-a_+)\in\Pi(\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+),P_n)$. Part (b) is proved. The proof of Part (a) is the same. We need only to replace the space $\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+)$ by the space $\ell^\Psi(\Z_+)$ and apply Propositions \ref{pr:Hankel-Orlicz}(a), \ref{pr:Widom}(a), \ref{pr:Toeplitz-inverse}(a), \ref{pr:zero-trunc}(a) instead of Propositions~\ref{pr:Hankel-compactness}, \ref{pr:Widom}(b), \ref{pr:Toeplitz-inverse}(b), \ref{pr:zero-trunc}(b) respectively. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Asymptotics of finite sections of Toeplitz operators} \label{sec:5} \subsection{Representation of $P_0T_n^{-1}(a)P_0$} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:Kozak} Suppose $X$ is a linear space, $\cP$ and $\cQ$ are complementary projections on $X$ (i.e., $\cP^2=\cP, \cQ^2=\cQ, \cP+\cQ=I$), and $A$ is an invertible operator on $X$. Then the compression $\cP A\cP|\im\cP$ is invertible on $\im\cP$ if and only if the compression $\cQ A^{-1}\cQ|\im\cQ$ is invertible on $\im\cQ$. In that case \[ (\cP A\cP)^{-1}\cP = \cP A^{-1}\cP - \cP A^{-1}\cQ(\cQ A^{-1}\cQ)^{-1}\cQ A^{-1}\cP. \] \end{lemma} The above statement can be found in \cite[Proposition~3.8]{BS83}, or in \cite[Proposition~7.15]{BS90}, or in \cite[Lemma~2.9]{BS99}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:representation} Suppose $\Phi,\Psi$ are $N$-functions both satisfying the $\Delta_2^0$-condition and $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ are weight sequences in $\cW$. Suppose the functions $a_-$ and $a_+$ are defined by {\rm (\ref{eq:WH-})} and {\rm (\ref{eq:WH+})}, respectively, and $a_\pm^{\pm 1}\in W$. Then, for all sufficiently large $n$, we have $D_n(a)\ne 0$ and %%% \begin{equation}\label{eq:representation-1} P_0T_n^{-1}(a)P_0 = P_0T(a_+^{-1})P_0\left\{I-\sum_{k=0}^\infty G_{n,k}\right\}P_0T(a_-^{-1})P_0 \end{equation} %%% on the space $\ell^\Psi(\Z_+)$, where $a:=a_-a_+, b:=a_-a_+^{-1}$, and %%% \begin{eqnarray} \label{eq:representation-2} G_{n,0} &:=& P_0T(b^{-1})Q_nT(b)P_0,\\ \label{eq:representation-3} G_{n,k} &:=& P_0T(b^{-1})Q_n\Big(Q_nH(b)H(\widetilde{b^{-1}})Q_n\Big)^kQ_nT(b)P_0 \end{eqnarray} %%% for $k\ge 1$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} This statement is proved by analogy with \cite[Section~6.15]{BS83} or \cite[Section~10.32]{BS90}. In view of Proposition~\ref{pr:Toeplitz-inverse}(a), $T(a), T(a^{-1})\in G\cB(\ell^\Psi(\Z_+))$ and %%% \begin{equation}\label{eq:representation-4} T^{-1}(a)=T(a_+^{-1})T(a_-^{-1}), \quad T^{-1}(a^{-1})=T(a_+)T(a_-). \end{equation} %%% By Lemma~\ref{le:applicability}(a), $T(a)\in\Pi(\ell^\Psi(\Z_+),P_n)$. In view of Proposition~\ref{pr:proj-method}, there exists an $n_1\in\N$ such that the operators $T_n(a)=P_nT(a)P_n$ are invertible on $P_n\ell^\Psi(\Z_+)$ for all $n\ge n_1$. Hence, $D_n(a)\ne 0$ for $n\ge n_1$. Lemma~\ref{le:Kozak} with $\cP=P_n, \cQ=Q_n, A=T(a)$, and $X=\ell^\Psi(\Z_+)$ gives that the operators $Q_nT^{-1}(a)Q_n$ are invertible on $Q_n\ell^\Psi(\Z_+)$ and %%% \begin{eqnarray*} && T_n^{-1}(a) = \Big(P_nT(a)P_n\Big)^{-1}P_n \\ &&= P_nT^{-1}(a)P_n-P_nT^{-1}(a)Q_n\Big(Q_nT^{-1}(a)Q_n\Big)^{-1}Q_nT^{-1}(a)P_n. \end{eqnarray*} %%% Multiplying this formula from the right and from the left by $P_0$ and taking into account that $P_0=P_0P_n=P_nP_0$, we obtain %%% \begin{eqnarray} && P_0T_n^{-1}(a)P_0 = P_0T^{-1}(a)P_0 \nonumber\\ &&- P_0T^{-1}(a)Q_n\Big(Q_nT^{-1}(a)Q_n\Big)^{-1}Q_nT^{-1}(a)P_0. \label{eq:representation-5} \end{eqnarray} %%% Since $a_+^{-1}\in H^\infty$ and $a_-^{-1}\in\overline{H^\infty}$, from Proposition~\ref{pr:zero-trunc}(a) we deduce that %%% \begin{equation}\label{eq:representation-6} P_0T(a_+^{-1})Q_0=Q_0T(a_-^{-1})P_0=0. \end{equation} %%% Then from (\ref{eq:representation-4})--(\ref{eq:representation-6}) and $I=P_0+Q_0$ we obtain %%% \begin{eqnarray} && P_0T_n^{-1}(a)P_0 = P_0T(a_+^{-1})T(a_-^{-1})P_0 \nonumber\\ &&- P_0T(a_+^{-1})T(a_-^{-1})Q_n \Big(Q_nT^{-1}(a)Q_n\Big)^{-1} Q_nT(a_+^{-1})T(a_-^{-1})P_0 \nonumber\\ &&= P_0T(a_+^{-1})P_0T(a_-^{-1})P_0 \nonumber\\ &&- P_0T(a_+^{-1})P_0T(a_-^{-1})Q_n \Big(Q_nT^{-1}(a)Q_n\Big)^{-1} Q_nT(a_+^{-1})P_0T(a_-^{-1})P_0 \nonumber\\ &&= P_0T(a_+^{-1})P_0 \nonumber\\ &&\times \Big\{I-P_0T(a_-^{-1})Q_n \Big(Q_nT^{-1}(a)Q_n\Big)^{-1} Q_nT(a_+^{-1})P_0\Big\} \nonumber\\ &&\times P_0T(a_-^{-1})P_0. \label{eq:representation-7} \end{eqnarray} %%% We now consider $\Big(Q_nT^{-1}(a)Q_n\Big)^{-1}Q_n$. From the first formula in (\ref{eq:representation-4}) and (\ref{eq:Widom-1}) it follows that \[ T^{-1}(a)=T(a_+^{-1})T(a_-^{-1}) = T(a_+^{-1}a_-^{-1}) - H(a_+^{-1})H(\widetilde{a_-^{-1}}). \] Since $a_+^{-1}\in W$, by Proposition~\ref{pr:Hankel-Orlicz}(a), we have %%% \begin{equation}\label{eq:representation-8} H(a_+^{-1})H(\widetilde{a_-^{-1}})=:K\in\cK(\ell^\Psi(\Z_+)). \end{equation} %%% Hence, %%% \begin{equation}\label{eq:representation-9} T^{-1}(a)=T(a^{-1})-K. \end{equation} %%% From the second formula in (\ref{eq:representation-4}) and Proposition~\ref{pr:zero-trunc}(a) we can easily see that %%% \begin{equation}\label{eq:representation-12} B_n:=Q_nT(a^{-1})Q_n|\im Q_n \end{equation} %%% is invertible and %%% \begin{equation}\label{eq:representation-13} B_n^{-1}Q_n = Q_nT(a_+)Q_nT(a_-)Q_n. \end{equation} %%% Furthemore, for every $n\in\Z_+$, %%% \begin{equation}\label{eq:representation-14} \|B_n^{-1}Q_n\|_{\cB(\ell^\Psi(\Z_+))} \le \|T(a_+)\|_{\cB(\ell^\Psi(\Z_+))}\|T(a_-)\|_{\cB(\ell^\Psi(\Z_+))}. \end{equation} %%% From (\ref{eq:representation-9}), taking into account that $B_n$ given by (\ref{eq:representation-12}) is invertible on $Q_n\ell^\Psi(\Z_+)$, we obtain %%% \begin{eqnarray} && \Big(Q_nT^{-1}(a)Q_n\Big)^{-1}Q_n = \Big(Q_nT(a^{-1})Q_n-Q_nKQ_n\Big)^{-1}Q_n \nonumber\\ &&= (B_n-K_n)^{-1}Q_n=(I-B_n^{-1}K_n)^{-1}B_n^{-1}Q_n, \label{eq:representation-15} \end{eqnarray} %%% where %%% \begin{equation}\label{eq:representation-16} K_n=Q_nKQ_n. \end{equation} %%% From Proposition~\ref{pr:projections} it follows that \[ \slim Q_n=0 \] on the space $\ell^\Psi(\Z_+)$. From the latter equality and (\ref{eq:representation-8}), by \cite[Lem\-ma~2.8]{BS99}, it follows that %%% \begin{equation}\label{eq:representation-17} \lim_{n\to\infty}\|Q_nKQ_n\|_{\cB(\ell^\Psi(\Z_+))}=0. \end{equation} %%% From (\ref{eq:representation-14}), (\ref{eq:representation-16}), and (\ref{eq:representation-17}) we deduce that \[ \lim_{n\to\infty}\|B_n^{-1}K_n\|_{\cB(\ell^\Psi(\Z_+))}=0. \] Hence, there exists an $n_2\ge n_1$ such that $\|B_n^{-1}K_n\|_{\cB(\ell^\Psi(\Z_+))}<1$ for $n\ge n_2$. We therefore can use Neumann's series expansion for (\ref{eq:representation-15}), which gives for $n\ge n_2$, %%% \begin{equation}\label{eq:representation-18} \Big(Q_nT^{-1}(a)Q_n\Big)^{-1}Q_n=\sum_{k=0}^\infty(B_n^{-1}K_n)^kB_n^{-1}Q_n. \end{equation} %%% Combining (\ref{eq:representation-7}) and (\ref{eq:representation-18}), we arrive at (\ref{eq:representation-1}) for $n\ge n_2$, where %%% \begin{eqnarray} \label{eq:representation-19} G_{n,0} &:=& P_0T(a_-^{-1})Q_nB_n^{-1}Q_nT(a_+^{-1})P_0, \\ \label{eq:representation-20} G_{n,k} &:=& P_0T(a_-^{-1})Q_n(B_n^{-1}K_n)^kB_n^{-1}Q_nT(a_+^{-1})P_0 \end{eqnarray} %%% for $k\ge 1$. Let us prove that $G_{n,k} (k\ge 0)$ can be rewritten in the form (\ref{eq:representation-2}) for $k=0$ and (\ref{eq:representation-3}) for $k\ge 1$. From (\ref{eq:representation-19}), (\ref{eq:representation-13}), and Proposition~\ref{pr:zero-trunc}(a) it follows that %%% \begin{eqnarray} G_{n,0} &=& P_0T(a_-^{-1})Q_n\cdot Q_nT(a_+)Q_nT(a_-)Q_n\cdot Q_nT(a_+^{-1})P_0 \nonumber\\ &=& P_0T(a_-^{-1})T(a_+)Q_nT(a_-)T(a_+^{-1})P_0. \label{eq:representation-21} \end{eqnarray} %%% Since $a_-\in\overline{H^\infty}$ and $a_+\in H^\infty$, we have $H(a_-)=H(\widetilde{a_+})=0$. Then, by (\ref{eq:Widom-1}), %%% \begin{equation}\label{eq:representation-22} T(a_-^{-1})T(a_+)=T(b^{-1}), \quad T(a_-)T(a_+^{-1})=T(b), \end{equation} %%% where $b=a_-a_+^{-1}$. Combining (\ref{eq:representation-21}) and (\ref{eq:representation-22}), we arrive at (\ref{eq:representation-2}). Taking into account (\ref{eq:representation-16}) and (\ref{eq:representation-20}), we represent $G_{n,k} (k\ge 1)$ in the form %%% \begin{eqnarray} G_{n,k} &=& P_0T(a_-^{-1})Q_nB_n^{-1}Q_n \cdot KQ_n\Big(B_n^{-1}Q_nKQ_n\Big)^{k-1} \nonumber\\ &&\times B_n^{-1}Q_nT(a_+^{-1})P_0. \label{eq:representation-23} \end{eqnarray} %%% From (\ref{eq:representation-13}), (\ref{eq:representation-22}), and Proposition~\ref{pr:zero-trunc}(a) it follows that %%% \begin{eqnarray} P_0T(a_-^{-1})Q_nB_n^{-1}Q_n &=& P_0T(a_-^{-1})Q_nT(a_+)Q_nT(a_-)Q_n \nonumber\\ &=& P_0T(b^{-1})Q_nT(a_-)Q_n, \label{eq:representation-24} \\[2mm] B_n^{-1}Q_nT(a_+^{-1})P_0 &=& Q_nT(a_+)Q_nT(a_-)Q_nT(a_+^{-1})P_0 \nonumber\\ &=& Q_nT(a_+)Q_nT(b)P_0. \label{eq:representation-25} \end{eqnarray} %%% From (\ref{eq:representation-23})--(\ref{eq:representation-25}) we get %%% \begin{equation}\label{eq:representation-26} G_{n,k}=P_0T(b^{-1})Q_n\cdot F_{n,k}\cdot Q_nT(b)P_0, \end{equation} %%% where (recall (\ref{eq:representation-13}) and (\ref{eq:representation-16})) %%% \begin{eqnarray} F_{n,k} &:=& Q_nT(a_-)Q_nKQ_n\Big(B_n^{-1}Q_nKQ_n\Big)^{k-1}Q_nT(a_+)Q_n \nonumber\\ &=& Q_nT(a_-)Q_nH(a_+^{-1})H(\widetilde{a_-^{-1}})Q_n \nonumber\\ &&\times \Big(Q_nT(a_+)Q_nT(a_-)Q_nH(a_+^{-1})H(\widetilde{a_-^{-1}})Q_n\Big)^{k-1} \nonumber\\ &&\times Q_nT(a_+)Q_n. \label{eq:representation-27} \end{eqnarray} %%% Since $a_-\in\overline{H^\infty}$, we have $H(a_-)=0$. Then from (\ref{eq:Widom-2}) it follows that \[ T(a_-)H(a_+^{-1})=H(b). \] From the latter formula and Proposition~\ref{pr:zero-trunc}(a) we obtain %%% \begin{equation}\label{eq:representation-28} Q_nT(a_-)Q_nH(a_+^{-1}) = Q_nH(b). \end{equation} %%% Combining (\ref{eq:representation-27}) and (\ref{eq:representation-28}), we obtain for $k\ge 1$, %%% \begin{eqnarray} F_{n,k} &=& Q_nH(b)H(\widetilde{a_-^{-1}})Q_n \Big(Q_nT(a_+)Q_nH(b)H(\widetilde{a_-^{-1}})Q_n\Big)^{k-1} \nonumber\\ &&\times Q_nT(a_+)Q_n. \label{eq:representation-29} \end{eqnarray} %%% In particular, %%% \begin{equation}\label{eq:representation-30} F_{n,1}=Q_nH(b)H(\widetilde{a_-^{-1}})Q_nT(a_+)Q_n. \end{equation} %%% Since $a_+\in H^\infty$, we have $H(\widetilde{a_+})=0$. Then from (\ref{eq:Widom-2}) it follows that \[ H(\widetilde{a_-^{-1}})T(a_+) = H(\widetilde{a_-^{-1}})T\Big((\widetilde{a_+})^{\widetilde{~}}\Big) =H(\widetilde{b^{-1}}). \] From the latter formula and Proposition~\ref{pr:zero-trunc}(a) we obtain %%% \begin{equation}\label{eq:representation-31} H(\widetilde{a_-^{-1}})Q_nT(a_+)Q_n = H(\widetilde{b^{-1}})Q_n. \end{equation} %%% From (\ref{eq:representation-30}) and (\ref{eq:representation-31}) we get %%% \begin{equation}\label{eq:representation-32} F_{n,1}=Q_nH(b)H(\widetilde{b^{-1}})Q_n. \end{equation} %%% If $k>1$, then from (\ref{eq:representation-29}) and (\ref{eq:representation-31}) we deduce that %%% \begin{eqnarray} F_{n,k} &=& Q_nH(b)H(\widetilde{a_-^{-1}})Q_nT(a_+)Q_nH(b)H(\widetilde{a_-^{-1}})Q_n \nonumber\\ &&\times \Big(Q_nT(a_+)Q_nH(b)H(\widetilde{a_-^{-1}})Q_n\Big)^{k-2} Q_nT(a_+)Q_n \nonumber\\ &=& Q_nH(b)H(\widetilde{b^{-1}})Q_n\cdot Q_nH(b)H(\widetilde{a_-^{-1}})Q_n \nonumber\\ &&\times \Big(Q_nT(a_+)Q_nH(b)H(\widetilde{a_-^{-1}})Q_n\Big)^{k-2} Q_nT(a_+)Q_n \nonumber\\ &=& Q_nH(b)H(\widetilde{b^{-1}})Q_n\cdot F_{n,k-1}. \label{eq:representation-33} \end{eqnarray} %%% Combining (\ref{eq:representation-32}) and (\ref{eq:representation-33}), by induction one can show that %%% \begin{equation}\label{eq:representation-34} F_{n,k}=\Big(Q_nH(b)H(\widetilde{b^{-1}})Q_n\Big)^k, \quad k\ge 1. \end{equation} %%% Finally, from (\ref{eq:representation-26}) and (\ref{eq:representation-34}) we get (\ref{eq:representation-3}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The norm of an one-dimensional operator} \begin{proposition}\label{pr:onedimensional} Let $\Psi$ be an $N$-function, $b\in\C$, and $M(b)$ be the operator on $\ell^\Psi(\Z_+)$ given by the infinite matrix \[ \left( \begin{array}{cccc} b & 0 & 0 & \cdots\\ 0 & 0 & 0 & \cdots\\ \vdots & \vdots & \ddots &\ddots \end{array} \right). \] Then %%% \begin{equation}\label{eq:onedimensional-1} \|M(b)\|_{\cB(\ell^\Psi(\Z_+))}=|b|. \end{equation} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Without loss of generality assume that $b\ne 0$. We have \[ \Big\|\{b,0,0,\dots\}\Big\|_{\ell^\Psi(\Z_+)} = \frac{|b|}{\Psi^{-1}(1)}. \] Therefore, %%% \begin{equation}\label{eq:onedimensional-2} |b| = \frac{\|M(b)e_0\|_{\ell^\Psi(\Z_+)}} {\|e_0\|_{\ell^\Psi(\Z_+)}} \le \|M(b)\|_{\cB(\ell^\Psi(\Z_+))}, \end{equation} %%% where $e_0=\{1,0,0,\dots\}$. Now suppose that $c=\{c_k\}_{k=0}^\infty$ is not identically zero. Then %%% \begin{eqnarray*} && \sum_{k=0}^\infty\Psi\left( \frac{|(M(b)c)_k|}{|b|\cdot\|c\|_{\ell^\Psi(\Z_+)}} \right) = \Psi\left( \frac{|bc_0|}{|b|\cdot\|c\|_{\ell^\Psi(\Z_+)}} \right) \\[2mm] &&= \Psi\left( \frac{|c_0|}{\|c\|_{\ell^\Psi(\Z_+)}} \right) \le \sum_{k=0}^\infty \Psi\left( \frac{|c_k|}{\|c\|_{\ell^\Psi(\Z_+)}} \right) \le 1. \end{eqnarray*} %%% Therefore $\|M(b)c\|_{\ell^\Psi(\Z_+)} \le |b|\cdot\|c\|_{\ell^\Psi(\Z_+)}$. Thus, %%% \begin{equation}\label{eq:onedimensional-3} \|M(b)\|_{\cB(\ell^\Psi(\Z_+))} \le |b|. \end{equation} %%% From (\ref{eq:onedimensional-2}) and (\ref{eq:onedimensional-3}) we immediately get (\ref{eq:onedimensional-1}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Asymptotics of the norms of truncated Toeplitz operators} \begin{lemma}\label{le:Toeplitz-trunc} Let $\Phi,\Psi$ be complementary $N$-functions both satisfying the $\Delta_2^0$-condition and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If $a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$, then %%% \begin{eqnarray} \label{eq:Toeplitz-trunc-1} \|Q_nT(a)P_0\|_{\cB(\ell^\Psi(\Z_+))} &=& o(1/\psi_{n+1}), \\ \label{eq:Toeplitz-trunc-2} \|P_0T(a)Q_n\|_{\cB(\ell^\Psi(\Z_+))} &=& o(1/\varphi_{n+1}) \end{eqnarray} %%% as $n\to\infty$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} Let $c=\{c_k\}_{k=0}^\infty\in\ell^\Psi(\Z_+)\setminus\{0\}$. Clearly, \[ \Psi\left(\frac{|c_0|}{\|c\|_{\ell^\Psi(\Z_+)}}\right) \le \sum_{k=0}^\infty\Psi\left(\frac{|c_k|}{\|c\|_{\ell^\Psi(\Z_+)}}\right) \le 1. \] Therefore, %%% \begin{equation}\label{eq:Toeplitz-trunc-3} |c_0|\le\Psi^{-1}(1)\|c\|_{\ell^\Psi(\Z_+)}. \end{equation} %%% It is easy to check that \[ (Q_nT(a)P_0c)_k=\left\{ \begin{array}{ll} 0, & 0\le k\le n, \\ a_kc_0, & k> n. \end{array}\right. \] Recall the definition (\ref{eq:a-truncation}). In view of Proposition~\ref{pr:approximation}(b), %%% \begin{equation}\label{eq:Toeplitz-trunc-6} \|a-a^{(n)}\|_+=o(1) \quad\mbox{as}\quad n\to\infty. \end{equation} %%% Without loss of generality we assume that $\|a-a^{(n)}\|_+>0$. Then %%% \begin{eqnarray} S(n) &:=& \sum_{k=0}^\infty\Psi\left( \frac{|(Q_nT(a)P_0c)_k|\psi_{n+1}} {\Psi^{-1}(1)\|a-a^{(n)}\|_+\|c\|_{\ell^\Psi(\Z_+)}} \right) \nonumber\\ &=& \sum_{k=n+1}^\infty \Psi\left( \frac{|a_k|\psi_{n+1}} {\|a-a^{(n)}\|_+}\cdot\frac{|c_0|}{\Psi^{-1}(1)\|c\|_{\ell^\Psi(\Z_+)}} \right). \label{eq:Toeplitz-trunc-4} \end{eqnarray} %%% Since $\Psi$ is a non-decreasing function and $\{\psi_k\}_{k=0}^\infty$ is a non-decreasing sequence, from (\ref{eq:Toeplitz-trunc-3}) and (\ref{eq:Toeplitz-trunc-4}) it follows that %%% \begin{eqnarray*} S(n) &\le& \sum_{k=n+1}^\infty\Psi\left( \frac{|a_k|\psi_{n+1}}{\|a-a^{(n)}\|_+} \right) \le \sum_{k=n+1}^\infty\Psi\left( \frac{|a_k|\psi_k}{\|a-a^{(n)}\|_+} \right) \\ &=& \sum_{k=0}^\infty \Psi\left( \frac{|a_k-a_k^{(n)}|\psi_k}{\|a-a^{(n)}\|_+} \right) \le 1. \end{eqnarray*} %%% Therefore, %%% \begin{equation}\label{eq:Toeplitz-trunc-5} \|Q_nT(a)P_0c\|_{\ell^\Psi(\Z_+)} \le \frac{\Psi^{-1}(1)}{\psi_{n+1}}\|a-a^{(n)}\|_+\|c\|_{\ell^\Psi(\Z_+)}. \end{equation} %%% Combining (\ref{eq:Toeplitz-trunc-5}) and (\ref{eq:Toeplitz-trunc-6}), we arrive at (\ref{eq:Toeplitz-trunc-1}). Since the function $\Psi$ satisfies the $\Delta_2^0$-condition and $\Phi,\Psi$ are complementary $N$-functions, by using Theorem~\ref{th:duality}, we can easily get $(T(a))^*=T(\overline{a})$, $P_0^*=P_0, Q_n^*=Q_n$ on the space $\ell^\Phi(\Z_+)$ and %%% \begin{eqnarray} \|P_0T(a)Q_n\|_{\cB(\ell^\Psi(\Z_+))} &=& \|(P_0T(a)Q_n)^*\|_{\cB\big((\ell^{\Psi}(\Z_+))^*\big)} \nonumber\\ &\le& 2\|Q_nT(\overline{a})P_0\|_{\cB(\ell^{\Phi}(\Z_+))}. \label{eq:Toeplitz-trunc-7} \end{eqnarray} %%% By Lemma~\ref{le:bar}, $\overline{a}\in W\cap F\ell^{\Psi,\Phi}_{\psi,\varphi}$. Then, applying (\ref{eq:Toeplitz-trunc-1}) to $\overline{a}$ and $\ell^\Phi(\Z_+)$, we obtain %%% \begin{equation}\label{eq:Toeplitz-trunc-8} \|Q_nT(\overline{a})P_0\|_{\cB(\ell^{\Phi}(\Z_+))}=o(1/\varphi_{n+1}) \quad\mbox{as}\quad n\to\infty. \end{equation} %%% From (\ref{eq:Toeplitz-trunc-7}) and (\ref{eq:Toeplitz-trunc-8}) we immediately get (\ref{eq:Toeplitz-trunc-2}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:Dn/Dn-1} Let $\Phi,\Psi$ be complementary $N$-functions both satisfying the $\Delta_2^0$-condition and let $\{\varphi_k\}_{k=0}^\infty,\{\psi_k\}_{k=0}^\infty$ be weight sequences in $\cW$. If for a function $a\in L^1(\T)$ the conditions {\rm (\ref{eq:Karlovich-2})} are fulfilled, then $D_n(a)\ne 0$ for all sufficiently large $n$ and \[ \frac{D_{n-1}(a)}{D_n(a)}=\frac{1}{G(a)} \left(1-o\left(\frac{1}{\varphi_{n+1}\psi_{n+1}}\right)\right) \quad\mbox{as}\quad n\to\infty, \] where $G(a)$ is given by {\rm (\ref{eq:G-def})}. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\it Proof.} In view of Theorem~\ref{th:Karlovich}(b), the function $a$ admits the factorization $a=a_-a_+$, where $a_-$ and $a_+$ are given by (\ref{eq:WH-}) and (\ref{eq:WH+}), respectively, and $a_\pm^{\pm 1}\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. Define $b:=a_-a_+^{-1}$. By Theorem~\ref{th:Karlovich}(a), %%% \begin{equation}\label{eq:Dn-1} b\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}, \quad b^{-1}=a_-^{-1}a_+\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}. \end{equation} %%% By Proposition~\ref{pr:Hankel-Orlicz}(a), $H(b)H(\widetilde{b^{-1}})\in\cK(\ell^\Psi(\Z_+))$. Then from Proposition~\ref{pr:projections} and \cite[Lemma~2.8]{BS99} we deduce that \[ \lim_{n\to\infty}\|Q_nH(b)H(\widetilde{b^{-1}})Q_n\|_{\cB(\ell^\Psi(\Z_+))} =0. \] Then, for sufficiently large $n$, $\|Q_n H(b)H(\widetilde{b^{-1}})Q_n\|_{\cB(\ell^\Psi(\Z_+))}\le 1/2$. Hence, %%% \begin{equation}\label{eq:Dn-2} \left\|\sum_{k=0}^\infty \Big(Q_nH(b)H(\widetilde{b^{-1}})Q_n\Big)^k \right\|_{\cB(\ell^\Psi(\Z_+))} \le \sum_{k=0}^\infty 2^{-k}<\infty. \end{equation} %%% From (\ref{eq:Dn-1}) and Lemma~\ref{le:Toeplitz-trunc} it follows that %%% \begin{eqnarray} && \|P_0T(b^{-1})Q_n\|_{\cB(\ell^\Psi(\Z_+))} = o(1/\varphi_{n+1}), \label{eq:Dn-3} \\ && \|Q_nT(b)P_0\|_{\cB(\ell^\Psi(\Z_+))} = o(1/\psi_{n+1}) \label{eq:Dn-4} \end{eqnarray} %%% as $n\to\infty$. Combining (\ref{eq:Dn-2})--(\ref{eq:Dn-4}), we deduce that %%% \begin{equation}\label{eq:Dn-5} \left\|\sum_{k=0}^\infty G_{n,k}\right\|_{\cB(\ell^\Psi(\Z_+))} =o\left(\frac{1}{\varphi_{n+1}\psi_{n+1}}\right) \quad\mbox{as}\quad n\to\infty, \end{equation} %%% where $G_{n,k}$ are given by (\ref{eq:representation-2})--(\ref{eq:representation-3}). From (\ref{eq:Dn-5}) and Proposition~\ref{pr:onedimensional} it follows that %%% \begin{equation}\label{eq:Dn-6} P_0\left\{I-\sum_{k=0}^\infty G_{n,k}\right\}P_0 = 1-o\left(\frac{1}{\varphi_{n+1}\psi_{n+1}}\right) \quad\mbox{as}\quad n\to\infty. \end{equation} %%% Clearly, %%% \begin{equation}\label{eq:Dn-7} P_0T(a_+^{-1})P_0=(a_+^{-1})_0, \quad P_0T(a_-^{-1})P_0=(a_-^{-1})_0. \end{equation} %%% Since $a_+^{-1}\in H^\infty$ and $a_-^{-1}\in\overline{H^\infty}$, we have %%% \begin{equation}\label{eq:Dn-8} \frac{1}{G(a)}=(a^{-1})_0 = (a_+^{-1}a_-^{-1})_0 = (a_+^{-1})_0(a_-^{-1})_0. \end{equation} %%% From Lemma~\ref{le:representation} we know that $D_n(a)\ne 0$ for all sufficiently large $n$. Hence, by Cramer's rule, %%% \begin{equation}\label{eq:Dn-9} P_0T_n^{-1}(a)P_0=\frac{D_{n-1}(a)}{D_n(a)}. \end{equation} %%% Combining (\ref{eq:Dn-6})--(\ref{eq:Dn-9}) with (\ref{eq:representation-1}), we obtain %%% \begin{eqnarray*} \frac{D_{n-1}(a)}{D_n(a)} &=& P_0T_n^{-1}(a)P_0 = P_0T(a_+^{-1})P_0\left\{I-\sum_{k=0}^\infty G_{n,k}\right\}P_0T(a_-^{-1})P_0 \\ &=& \frac{1}{G(a)} \left(1-o\left(\frac{1}{\varphi_{n+1}\psi_{n+1}}\right)\right) \end{eqnarray*} %%% as $n\to\infty$. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proof of the main results}\label{sec:6} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Hirschman's formula} The next theorem is due to I.~I.~Hirschman. Its proof can be found in \cite[Theorem~6.30]{BS83} or in \cite[Corollary~10.41]{BS90}. Let $e_0:=\{1,0,0,\dots\}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:Hirschman} Let $n\in\N, a\in L^\infty(\T)$, and $D_n(a)\ne 0$. Suppose the polynomials $p_n$ and $q_n$, defined in the complex plane by %%% \begin{equation}\label{eq:Hirschman-1} p_n(z):=\sum_{k=0}^n\Big(T^{-1}_n(a)e_0\Big)_kz^k, \quad q_n(z):=\sum_{k=0}^n\Big(T^{-1}_n(\overline{a})e_0\Big)_kz^k, \end{equation} %%% have no zeros on the closed unit disk $\overline{\D}:=\{z\in\C: |z|\le 1\}$. Then $D_{n-1}(a)\ne 0$ and \[ D_n(a)=\left(\frac{D_n(a)}{D_{n-1}(a)}\right)^{n+1} E_n(a), \] where \begin{equation}\label{eq:Hirschman-2} E_n(a):= \exp\left( \sum_{k=1}^\infty k(\log p_n)_k(\log\overline{q_n})_{-k} \right). \end{equation} \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Proof of Theorem~\ref{th:main}} {\it Proof.} In view of Theorem~\ref{th:Orlicz-class}, since the $N$-functions $\Phi$ and $\Psi$ satisfy the $\Delta_2^0$-condition, the weighted Orlicz sequence classes $\widetilde{\ell}^\Phi_\varphi(\N)$ and $\widetilde{\ell}^\Psi_\psi(\Z_+)$ coincide with the corresponding Orlicz sequence spaces $\ell^\Phi_\varphi(\N)$ and $\ell^\Psi_\psi(\Z_+)$, respectively. Therefore (\ref{eq:main-2}) implies $a\in W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. By Lemma~\ref{le:bar}, $\overline{a}\in W\cap F\ell^{\Psi,\Phi}_{\psi,\varphi}$. Clearly, $\overline{a(t)}\ne 0$ for all $t\in\T$ and $\ind\overline{a}=\ind a=0$. By Theorem~\ref{th:Karlovich}(b), the function $a$ (resp. $\overline{a}$) has a logarithm in $W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$ (resp. in $W\cap F\ell^{\Psi,\Phi}_{\psi,\varphi}$) and it admits the factorization $a=a_-a_+$ (resp. $\overline{a}=\overline{a}_-\overline{a}_+$), where $a_-$ and $a_+$ are given by (\ref{eq:WH-}) and (\ref{eq:WH+}), respectively, and %%% \begin{eqnarray*} \overline{a}_-(t) &:=& \exp\left(\sum_{k=1}^\infty(\log\overline{a})_{-k}t^{-k}\right) = \frac{\overline{a_+(t)}}{\overline{G(a)}}, \\[2mm] \overline{a}_+(t) &:=& \exp\left(\sum_{k=0}^\infty(\log\overline{a})_kt^k\right) = \overline{a_-(t)G(a)} \end{eqnarray*} %%% with $G(a)$ is given by (\ref{eq:G-def}). Hence, %%% \begin{equation}\label{eq:main-3} \overline{a}_-^{-1}(t)=\overline{a_+^{-1}(t)G(a)}, \quad \overline{a}_+^{-1}(t)=\overline{a_-^{-1}(t)(G(a))^{-1}}. \end{equation} %%% Put $X:=\ell^1(\Z_+)\cap\ell^\Psi_\psi(\Z_+)$ and $Y:=\ell^1(\Z_+)\cap\ell^\Phi_\varphi(\Z_+)$. In view of Lemma~\ref{le:applicability}(b), we have $T(a)\in\Pi(X,P_n)$ and $T(\overline{a})\in\Pi(Y,P_n)$. Due to Proposition~\ref{pr:proj-method}(iii), $T_n(a)\in G\cB(P_nX), T_n(\overline{a})\in G\cB(P_nY)$ for all sufficiently large $n$ and %%% \begin{equation}\label{eq:main-4} \slim T_n^{-1}(a)=T^{-1}(a), \quad \slim T_n^{-1}(\overline{a})=T^{-1}(\overline{a}) \end{equation} on $X$ and $Y$, respectively. Clearly, $e_0$ belongs to $X$ and to $Y$. By Proposition~\ref{pr:Toeplitz-inverse}(b), we have %%% \begin{equation}\label{eq:main-5} T^{-1}(a)e_0=T(a_+^{-1})T(a_-^{-1})e_0, \quad T^{-1}(\overline{a})e_0=T(\overline{a}_+^{-1})T(\overline{a}_-^{-1})e_0. \end{equation} %%% Since $a_-^{-1}\in\overline{H^\infty}$ and $a_+^{-1}\in H^\infty$, from the first formula in (\ref{eq:main-5}) it follows that %%% \begin{equation}\label{eq:main-7} (T^{-1}(a)e_0)_j =(a_+^{-1})_j(a_-^{-1})_0, \quad j\in\Z_+. \end{equation} %%% Analogously, taking into account (\ref{eq:main-3}), we get %%% \begin{eqnarray} (T^{-1}(\overline{a})e_0)_j &=& (\overline{a}_+^{-1})_j(\overline{a}_-^{-1})_0 = \Big(\overline{a_-^{-1}(G(a))^{-1}}\Big)_j \Big(\overline{a_+^{-1}G(a)}\Big)_0 \nonumber\\ &=& \Big(\overline{a_-^{-1}}\Big)_j\Big(\overline{a_+^{-1}}\Big)_0, \quad j\in\Z_+. \label{eq:main-8} \end{eqnarray} %%% Let $p_n$ and $q_n$ be polynomials defined by (\ref{eq:Hirschman-1}). From (\ref{eq:main-7}), (\ref{eq:main-8}), and the definition of $p_n$ and $q_n$ it follows that %%% \begin{eqnarray} \|T_n^{-1}(a)e_0-T^{-1}(a)e_0\|_X &=& \| p_n-(a_-^{-1})_0a_+^{-1} \|_{W\cap F}, \label{eq:main-9} \\ \|T_n^{-1}(\overline{a})e_0-T^{-1}(\overline{a})e_0\|_Y &=& \Big\| q_n-\Big(\overline{a_+^{-1}}\Big)_0\overline{a_-^{-1}} \Big\|_{W\cap \overline{F}}, \label{eq:main-10} \end{eqnarray} %%% where $F$ stands for $F\ell^{\Phi,\Psi}_{\varphi,\psi}$ and $\overline{F}$ stands for $F\ell^{\Psi,\Phi}_{\psi,\varphi}$. From Lemma~\ref{le:bar} we obtain %%% \begin{eqnarray} && \|\overline{q_n}-(a_+^{-1})_0a_-^{-1}\|_{W\cap F} = \left\| \overline{q_n-\Big(\overline{a_+^{-1}}\Big)_0\overline{a_-^{-1}}} \right\|_{W\cap F} \nonumber\\ &&\le K\Big\| q_n-\Big(\overline{a_+^{-1}}\Big)_0\overline{a_-^{-1}} \Big\|_{W\cap \overline{F}} \label{eq:main-11} \end{eqnarray} %%% with $K$ given by (\ref{eq:bar-1}). From (\ref{eq:main-4}) and (\ref{eq:main-9})--(\ref{eq:main-11}) we get %%% \begin{eqnarray} \lim_{n\to\infty}\|p_n-(a_-^{-1})_0a_+^{-1}\|_{W\cap F} &=&0, \label{eq:main-12} \\ \lim_{n\to\infty}\|\overline{q_n}-(a_+^{-1})_0a_-^{-1}\|_{W\cap F} &=&0. \label{eq:main-13} \end{eqnarray} %%% From (\ref{eq:main-7}) and invertibility of $T^{-1}(a)$ it follows that $(a_-^{-1})_0\ne 0$. Hence, $\log(a_-^{-1})_0$ is well defined. Analogously, from (\ref{eq:main-8}) and invertibility of $T^{-1}(\overline{a})$ we deduce that $\log (a_+^{-1})_0$ is well defined. On the other hand, \[ \log a_-^{-1}(t)= -\sum_{k=1}^\infty (\log a)_{-k}t^{-k}, \quad \log a_+^{-1}(t)= -\sum_{k=0}^\infty (\log a)_kt^k \] belong to $W\cap F\ell^{\Phi,\Psi}_{\varphi,\psi}$. Hence from (\ref{eq:main-12}) and (\ref{eq:main-13}) we infer that %%% \begin{eqnarray} \lim_{n\to\infty}\|\log p_n-\log(a_-^{-1})_0+\log a_+\|_{W\cap F} &=&0, \label{eq:main-14} \\ \lim_{n\to\infty}\|\log\overline{q_n}-\log(a_+^{-1})_0+\log a_-\|_{W\cap F} &=&0. \label{eq:main-15} \end{eqnarray} %%% Let us show that \[ S_n(a):=\sum_{k=1}^\infty k(\log p_n)_k(\log\overline{q_n})_{-k} \to S(a):=\sum_{k=1}^\infty k(\log a)_k(\log a)_{-k} \] as $n\to\infty$. Since $\log (a_-^{-1})_0$ and $\log (a_+^{-1})_0$ are finite constants, we get for $k>0$, %%% \begin{eqnarray*} (\log p_n+\log a)_k &=& (\log p_n-\log(a_-^{-1})_0+\log a_+)_k, \\ (\log\overline{q_n}+\log a)_{-k} &=& (\log\overline{q_n}-\log(a_+^{-1})_0+\log a_-)_{-k}. \end{eqnarray*} %%% Then we have %%% \begin{eqnarray*} |S_n(a)-S(a)| &\le& \left| \sum_{k=1}^\infty k(\log p_n)_k(\log\overline{q_n})_{-k} + \sum_{k=1}^\infty k(\log a)_k(\log\overline{q_n})_{-k} \right| \\[2mm] &&+ \left| \sum_{k=1}^\infty k(\log a)_k(\log\overline{q_n})_{-k} + \sum_{k=1}^\infty k(\log a)_k(\log a)_{-k} \right| \\[2mm] &\le& \sum_{k=1}^\infty k\Big|(\log p_n+\log a)_k\Big| |(\log\overline{q_n})_{-k}| \\[2mm] &&+ \sum_{k=1}^\infty k|(\log a)_k| \Big|(\log \overline{q_n}+\log a)_{-k}\Big| \\[2mm] &=& \sum_{k=1}^\infty k|(\log\overline{q_n})_{-k}| \cdot \Big|(\log p_n -\log(a_-^{-1})_0 +\log a_+)_k\Big| \\[2mm] &&+ \sum_{k=1}^\infty k\Big|(\log\overline{q_n}-\log(a_+^{-1})_0+\log a_-)_{-k}\Big| \cdot |(\log a)_k|. \end{eqnarray*} %%% Applying Lemma~\ref{le:Hoelder} to each sum on the right-hand side of the above formula and taking into account that \[ \Big\|\{c_k\}_{k\in\N}\Big\|_{\ell^\Psi_\psi(\N)} \le \Big\|\{c_k\}_{k\in\Z_+}\Big\|_{\ell^\Psi_\psi(\Z_+)} \] for any sequence $\{c_k\}_{k=0}^\infty$ in $\ell^\Psi_\psi(\Z_+)$, we obtain %%% \begin{eqnarray} && |S_n(a) - S(a)| \le 2M\Big\| \Big\{(\log\overline{q_n})_{-k}\Big\}_{k\in\N} \Big\|_{\ell^\Phi_\varphi(\N)} \nonumber\\ &&\times \Big\| \Big\{(\log p_n-\log(a_-^{-1})_0+\log a_+)_k\Big\}_{k\in\Z_+} \Big\|_{\ell^\Psi_\psi(\Z_+)} \nonumber\\ &&+ 2M\Big\|\Big\{(\log a)_k\Big\}_{k\in\Z_+}\Big\|_{\ell^\Psi_\psi(\Z_+)} \nonumber\\ &&\times \Big\| \Big\{(\log\overline{q_n}-\log(a_+^{-1})_0+\log a_-)_{-k}\Big\}_{k\in\N} \Big\|_{\ell^\Phi_\varphi(\N)} \nonumber\\ &&\le 2M\|\log\overline{q_n}\|_{W\cap F} \|\log p_n-\log(a_-^{-1})_0+\log a_+\|_{W\cap F} \nonumber\\ &&+ 2M\|\log a\|_{W\cap F} \|\log\overline{q_n}-\log(a_+^{-1})_0+\log a_-\|_{W\cap F}. \label{eq:main-19} \end{eqnarray} %%% From (\ref{eq:main-14}), (\ref{eq:main-15}), and (\ref{eq:main-19}) it follows that \[ \lim_{n\to\infty}S_n(a)=S(a). \] Put $E_n(a):=\exp S_n(a)$ (see (\ref{eq:Hirschman-1})) and remind that $E(a)=\exp S(a)$ (see (\ref{eq:E-def})). Then %%% \begin{equation}\label{eq:main-20} \lim_{n\to\infty} E_n(a)=\lim_{n\to\infty}\exp S_n(a)=\exp S(a)=E(a). \end{equation} %%% By Theorem~\ref{th:Dn/Dn-1}, $D_n(a)\ne 0$ for all sufficiently large $n$ and %%% \begin{equation}\label{eq:main-21} \left(\frac{D_n(a)}{D_{n-1}(a)}\right)^{n+1} = (G(a))^{n+1}(1-v_{n+1})^{-(n+1)}, \end{equation} %%% where %%% \begin{equation}\label{eq:main-22} v_n:=o\left(\frac{1}{\varphi_n\psi_n}\right) \quad\mbox{as}\quad n\to\infty. \end{equation} %%% From (\ref{eq:main-22}) and (\ref{eq:main-1}) it follows that %%% \begin{eqnarray} \lim_{n\to\infty}(1-v_{n+1})^{-(n+1)} &=& \lim_{n\to\infty}\left(1-\frac{v_nn}{n}\right)^{-n} \nonumber\\ &=& \exp\left(\lim_{n\to\infty}(v_nn)\right)=1. \label{eq:main-23} \end{eqnarray} %%% Lemma~\ref{le:Hirschman} and (\ref{eq:main-21}) give %%% \begin{eqnarray} \frac{D_n(a)}{(G(a))^{n+1}} &=& \left(\frac{D_n(a)}{D_{n-1}(a)}\right)^{n+1}\cdot\frac{E_n(a)}{(G(a))^{n+1}} \nonumber\\ &=& E_n(a)(1-v_{n+1})^{-(n+1)}. \label{eq:main-24} \end{eqnarray} %%% From (\ref{eq:main-20}), (\ref{eq:main-23}), and (\ref{eq:main-24}) we immediately get (\ref{eq:Szego}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Proof of Lemma~\ref{le:example}}\label{sec:6.3} {\it Proof.} Define the weight sequences %%% \begin{equation}\label{eq:weight-1} \varphi_k:= \left\{\begin{array}{l} 1, \quad k\in\{0,\dots,8\},\\ \frac{k^{1/2}\log k}{3\log 9}, \quad k\ge 9, \end{array}\right. \quad \psi_k:= \left\{\begin{array}{l} 1, \quad k\in\{0,\dots,8\},\\ \frac{\log 9}{3}\cdot \frac{k^{1/2}}{\log k}, \quad k\ge 9. \end{array}\right. \end{equation} %%% It is not difficult to verify that $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ belong to $\cW$ and satisfy $k\le 9\varphi_k\psi_k$ for all $k\in\Z_+$. Now we define the function %%% \begin{equation}\label{eq:example-1} a(t):=a_0+\sum_{k=1}^\infty c_k t^{-k}+\sum_{k=1}^\infty d_k t^k, \quad t\in\T, \end{equation} %%% where $a_0\in\C$ and $c_k,d_k\ge 0$ will be chosen in such a way that all the conditions of Lemma~\ref{le:example} will be fulfilled. Fix $\gamma\in(3/4,1)$ and put %%% \begin{eqnarray} c_k &:=& \left\{ \begin{array}{ll} k^{-3/4}(\log k)^{-\gamma-1}, & k=m^2, \quad m=3,4,5,\dots,\\ 0, &\mbox{otherwise}, \end{array} \right. \label{eq:example-2} \\ d_k &:=& \left\{ \begin{array}{ll} k^{-3/4}(\log k)^{-\gamma+1}, & k=m^2, \quad m=3,4,5,\dots,\\ 0, &\mbox{otherwise}. \end{array} \right. \label{eq:example-3} \end{eqnarray} %%% Then %%% \begin{equation}\label{eq:example-4} \sum_{k=0}^\infty c_k = \sum_{m=3}^\infty m^{-3/2}(\log m^2)^{-\gamma-1} \le \sum_{m=3}^\infty m^{-3/2}<\infty. \end{equation} %%% Since $\gamma\in(3/4,1)$, we have $1-\gamma<1/4$ and \[ (\log m^2)^{1-\gamma} \le 2^{1-\gamma}m^{1-\gamma} \le 2m^{1/4} \quad\mbox{for}\quad m\ge 3. \] From the latter inequality and (\ref{eq:example-3}) it follows that %%% \begin{equation}\label{eq:example-5} \sum_{k=1}^\infty d_k = \sum_{m=3}^\infty m^{-3/2}(\log m^2)^{1-\gamma} \le 2\sum_{m=3}^\infty m^{-5/4}<\infty. \end{equation} %%% Combining (\ref{eq:example-1}), (\ref{eq:example-4}), and (\ref{eq:example-5}), we deduce that $a\in W$. Let $\{\varphi_k\}_{k=0}^\infty$ and $\{\psi_k\}_{k=0}^\infty$ be weight sequences given by (\ref{eq:weight-1}). Let us show that %%% \begin{equation}\label{eq:example-6} \sum_{k=1}^\infty(c_k\varphi_k)^2+\sum_{k=1}^\infty(d_k\psi_k)^2<\infty. \end{equation} %%% It is well known (see, e.g., \cite[Problem~3.2.29(a)]{KN00}) that %%% \begin{equation}\label{eq:example-7} \sum_{m=3}^\infty \frac{1}{m(\log m)^\delta}<\infty \quad\mbox{for}\quad\delta>1. \end{equation} %%% We have %%% \begin{eqnarray} \sum_{k=1}^\infty (c_k\varphi_k)^2 &=& (3\log 9)^{-2}\sum_{m=3}^\infty \left(\frac{m\log m^2}{m^{3/2}(\log m^2)^{\gamma+1}}\right)^2 \nonumber\\ &=& (3\log 9\cdot 2^\gamma)^{-2} \sum_{m=3}^\infty\frac{1}{m(\log m)^{2\gamma}}, \label{eq:example-8} \\ \sum_{k=1}^\infty (d_k\psi_k)^2 &=& \left(\frac{\log 9}{3}\right)^2 \sum_{m=3}^\infty\left( \frac{1}{m^{3/2}(\log m^2)^{\gamma-1}}\cdot\frac{m}{\log m^2} \right)^2 \nonumber\\ &=& \left(\frac{\log 9}{3\cdot 2^\gamma}\right)^2 \sum_{m=3}^\infty\frac{1}{m(\log m)^{2\gamma}}. \label{eq:example-9} \end{eqnarray} %%% From (\ref{eq:example-7})--(\ref{eq:example-9}), taking into account that $\gamma\in(3/4,1)$, we get (\ref{eq:example-6}). Now we show that the function $a$ given by (\ref{eq:example-1})--(\ref{eq:example-3}) does not belong to $F\ell^{p,p'}_{\alpha,1-\alpha}$, where $1/p+1/p'=1$, for any $p\in(1,\infty)$ and any $\alpha\in[0,1]$. To this end, put %%% \begin{eqnarray} S_1(\alpha,p) &:=& \sum_{k=1}^\infty (c_kk^\alpha)^p = \sum_{m=3}^\infty\left( \frac{m^{2\alpha}}{m^{3/2}(\log m^2)^{\gamma+1}} \right)^p \nonumber\\ &=& 2^{-p(\gamma+1)}\sum_{m=3}^\infty m^{(2\alpha-3/2)p}(\log m)^{-(\gamma+1)p}, \label{eq:example-10} \\ S_2(\alpha,p) &:=& \sum_{k=1}^\infty (d_kk^{1-\alpha})^{p'} = \sum_{m=3}^\infty\left( \frac{m^{2(1-\alpha)}}{m^{3/2}(\log m^2)^{\gamma-1}} \right)^{p'} \nonumber\\ &=& 2^{(1-\gamma)p'}\sum_{m=3}^\infty m^{(1/2-2\alpha)p'}(\log m)^{(1-\gamma)p'}. \label{eq:example-11} \end{eqnarray} %%% Consider two disjoint cases (a) $(3/2-2\alpha)p\ge 1$, (b) $(3/2-2\alpha)p<1$. In the case (a) we have \[ 1/2-2\alpha\ge 1/p-1=-1/p'. \] Then $(2\alpha-1/2)p'\le 1$. Taking into account that $(1-\gamma)p'>0$, we get \[ S_2(\alpha,p) = 2^{(1-\gamma)p'}\sum_{m=3}^\infty \frac{(\log m)^{(1-\gamma)p'}}{m^{(2\alpha-1/2)p'}} \ge 2^{(1-\gamma)p'}\sum_{m=3}^\infty m^{(1/2-2\alpha)p'}. \] The latter series diverges in view of $(2\alpha-1/2)p'\le 1$. Therefore $S_2(\alpha,p)=\infty$ whenever $(3/2-2\alpha)p\ge 1$. In the case (b), %%% \begin{equation}\label{eq:example-12} S_1(\alpha,p)=2^{-(\gamma+1)p}\sum_{m=3}^\infty m^{-\beta}(\log m)^{-\delta}, \end{equation} %%% where $\beta=(3/2-2\alpha)p<1$ and $\delta=(\gamma+1)p>1$. Since $m^{-\beta}(\log m)^{-\delta}$ is positive and monotonically decreasing sequence, we can apply the condensation test of Cauchy (see, e.g., \cite[Problem~3.2.28]{KN00}) to (\ref{eq:example-12}). In view of that test, the series (\ref{eq:example-12}) is convergent if and only if the series \[ \sum_{m=3}^\infty\frac{2^m}{2^{\beta m}(\log 2^m)^\delta} = (\log 2)^{-\delta} \sum_{m=3}^\infty \frac{2^{m(1-\beta)}}{m^\delta} \] is convergent. But the latter series diverges because $1-\beta>0$ and $\delta>1$. Hence $S_1(\alpha,p)=\infty$ whenever $(3/2-2\alpha)p<1$. From both cases (a) and (b) we deduce that \[ S_1(\alpha,p)+S_2(\alpha,p)=\infty \quad\quad\mbox{for all}\quad p\in(1,\infty),\quad\alpha\in[0,1]. \] This means that $a\notin F\ell^{p,p'}_{\alpha,1-\alpha}$ for any $p\in(1,\infty)$ and any $\alpha\in[0,1]$. The constant $a_0\in\C$ can be chosen such that $a(t)\ne 0$ for all $t\in\T$ and $\ind a=0$. Thus the function $a$ given by (\ref{eq:example-1})--(\ref{eq:example-3}) with the corresponding choice of $a_0$ and the weight sequences $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ given by (\ref{eq:weight-1}) satisfy all the conditions of Lemma~\ref{le:example}. This finishes the proof of Lemma~\ref{le:example}. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection*{Acknowledgements} This work was supported by F.C.T. (Portugal) through ``Centro de Matem\'atica e Aplica\c{c}\~oes'' (IST, UTL, Lisboa). A.~Yu.~Karlovich also was supported by F.C.T. (Portugal) grant SFRH/BPD/11619/2002. The authors wish to thank Albrecht B\"ottcher and Bernd Silbermann for useful remarks concerning a preliminary version of this paper. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{99} \bibitem{BE01} Basor,~E., Ehrhardt,~T.: Asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices. Math. Nachr. \textbf{228,} 5--45 (2001) \bibitem{BE02} Basor,~E., Ehrhardt,~T.: Asymptotic formulas for the determinants of symmetric Toeplitz plus Hankel matrices. In: Toeplitz matrices and singular integral equations (Pobershau, 2001). 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