Content-Type: multipart/mixed; boundary="-------------0603311240907" This is a multi-part message in MIME format. ---------------0603311240907 Content-Type: text/plain; name="06-102.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-102.comments" 2000 Mathematics Subject Classification: Primary 47B36; Secondary 47A10 Revised version of the manuscript originally posted as 05-371 ---------------0603311240907 Content-Type: text/plain; name="06-102.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-102.keywords" Unbounded Jacobi Operators Jacobi Matrices Discrete Spectrum ---------------0603311240907 Content-Type: application/x-tex; name="siltol1final_rev3.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="siltol1final_rev3.tex" \documentclass[12pt]{amsart} \usepackage{hyperref,latexsym} \usepackage{enumerate} \DeclareMathOperator{\re}{Re} \DeclareMathOperator{\im}{Im} \DeclareMathOperator{\dom}{Dom} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator*{\res}{Res} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\I}{{\rm i}} \newcounter{aux1} \newcounter{aux2} \newenvironment{hypothesis}[1] {\setcounter{aux1}{\value{equation}} \setcounter{equation}{\value{aux2}} \renewcommand{\theequation}{H.\arabic{equation}} \begin{equation} #1} {\end{equation} \setcounter{equation}{\value{aux1}} \renewcommand{\theequation}{\arabic{equation}} \stepcounter{aux2}\newline} \begin{document} \title[Jacobi Matrices with Rapidly Growing Weights] {Jacobi Matrices with Rapidly Growing Weights Having Only Discrete Spectrum} \date{} \author{Luis O. Silva} \address{Departamento de M{\'e}todos Matem{\'a}ticos y Num{\'e}ricos\\ IIMAS, Universidad Nacional Aut{\'o}noma de M{\'e}xico\\ Apdo. Postal 20-726 M{\'e}xico, D.F. 01000} \email{silva@leibniz.iimas.unam.mx} \thanks{Research partially supported by Universidad Nacional Aut\'onoma de M\'exico under Project PAPIIT-DGAPA IN 101902 and by CONACYT under Project P42553F} \author{Julio H. Toloza} \email{jtoloza@leibniz.iimas.unam.mx} \subjclass[2000]{Primary 47B36; Secondary 47A10} \keywords{Unbounded Jacobi Operators; Jacobi Matrices; Discrete Spectrum} \begin{abstract} We establish sufficient conditions for self-adjointness on a class of unbounded Jacobi operators defined by matrices with main diagonal sequence of very slow growth and rapidly growing off-diagonal entries. With some additional assumptions, we also prove that these operators have only discrete spectrum. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\{b_n\}_{n\in{\mathbb N}}$ be a real sequence whose elements are given by \begin{equation} \label{eq:weights} b_n = n^{\alpha} \left(1+\frac{c_n}{n}+h_n\right)\,, \quad \alpha>1\,, \quad 1+\frac{c_n}{n}+h_n\neq 0\,\quad \forall n\in\mathbb{N}\,, \end{equation} where $\{c_n\}_{n\in{\mathbb N}}$ is bounded with $\liminf\abs{c_n}>0$, and $\{h_n\}_{n\in{\mathbb N}}$ is such that $h_n=O(n^{-\gamma})$ when $n\to\infty$ for some $\gamma>1$. Let $\{q_n\}_{n\in{\mathbb N}}$ be a real sequence that grows not faster than logarithmically, viz., \begin{equation} \label{eq:diagonal} q_n=O(\log n)\quad\text{as }n\to\infty\,. \end{equation} In the Hilbert space $l_2(\mathbb{N})$, we define the Jacobi operator $J$ as the minimal closed operator such that \begin{equation} \label{eq:matrix-rep} (Je_n,e_n)=q_n\,,\quad (Je_n,e_{n+1})=(Je_{n+1},e_n) =b_n\,,\quad\forall n\in\mathbb{N}\,, \end{equation} where $\{e_n\}_{n\in{\mathbb N}}$ is the canonical basis in $l_2(\mathbb{N})$. Clearly, there are closed operators satisfying (\ref{eq:matrix-rep}), thus $J$ is correctly defined. This operator is unbounded and we have defined it in such a way that the semi-infinite Jacobi matrix \begin{equation} \label{eq:jm} \begin{pmatrix} q_1 & b_1 & 0 & 0 & \cdots \\[1mm] b_1 & q_2 & b_2 & 0 & \cdots \\[1mm] 0 & b_2 & q_3 & b_3 & \\ 0 & 0 & b_3 & q_4 & \ddots\\ \vdots & \vdots & & \ddots & \ddots \end{pmatrix} \end{equation} is the matrix representation of $J$ with respect to the basis $\{e_n\}_{n\in{\mathbb N}}$ (we refer to \cite{MR1255973} for a discussion on matrix representation of unbounded symmetric operators). For $\alpha>1$ fixed, different sequences $\{c_n\}_{n\in{\mathbb N}}$, $\{h_n\}_{n\in{\mathbb N}}$ and $\{q_n\}_{n\in{\mathbb N}}$ define different concrete realizations of $J$. Throughout this paper, we use $J$ to denote anyone of these realizations prescribed by (\ref{eq:weights})--(\ref{eq:matrix-rep}). Since the off-diagonal entries (weights) of (\ref{eq:jm}) grow fast ($\alpha>1$), $J$ is not always self-adjoint. It is known, nevertheless, that certain conditions on the weights ensure self-adjointness when $\{q_n\}_{n\in{\mathbb N}}$ is a bounded sequence. Indeed, the class of operators studied here is a generalization of an example suggested by Kostyuchenko and Mirzoev in \cite{MR1711874} as an illustration of a Jacobi operator whose matrix representation does not satisfy the Carleman criterion \cite{MR0184042} but nevertheless, under certain conditions, can be self-adjoint. Jacobi operators for which $\{c_n\}_{n\in{\mathbb N}}$ is even-periodic and $\{q_n\}_{n\in{\mathbb N}}\equiv\{h_n\}_{n\in{\mathbb N}}\equiv 0$ has been studied in \cite{MR1959871,MR1924991}. For these cases there are formulae in \cite{MR1959871} for the asymptotic behavior of generalized eigenvectors and it is proven there, by means of the Subordinacy Theory \cite{MR915965,MR1179528}, that the spectrum of $J$ is pure point in the self-adjoint case. We carry out the spectral analysis of $J$ on the basis of the asymptotic behavior of its generalized eigenvectors. These asymptotics are obtained by means of a Levinson-type theorem for discrete linear systems. We use transfer matrices to study the generalized eigenequation of $J$ and arrive at a system that turns out to be asymptotically diagonal in the sense that it satisfies the assumptions of the Benzaid-Lutz theorem \cite{MR1002291}. The fact that we can study the generalized eigenequation through a discrete system, for which the Benzaid-Lutz theorem yields the asymptotic behavior of the solutions, is a remarkable property of this class of operators. We impose certain mild conditions over the average behavior of the sequence $\{c_n\}_{n\in{\mathbb N}}$ in order to ensure self-adjointness. These conditions are far less restrictive that those used in previous works and contain them as special cases. Also, these conditions turn out to be sufficient for the absence of continuous spectrum. In passing, we note that in \cite{MR1959871,MR1924991} a much stronger assumption of periodicity of the sequence $\{c_n\}_{n\in{\mathbb N}}$ was required to gain spectral information from the asymptotic behavior of the generalized eigenvectors of operator $J$. As it will become apparent later, the sequences $\{q_n\}_{n\in{\mathbb N}}$ and $\{h_n\}_{n\in{\mathbb N}}$, with the asymptotic behavior specified above, play no role concerning both self-adjointness and spectral properties of $J$. Unlike the methods used in \cite{MR1959871,MR1924991}, our main results do not make use of Subordinacy Theory. Instead we mainly use general facts for Jacobi operators and Spectral Theory. To the knowledge of the authors, the spectral analysis of Jacobi operators corresponding to matrices with rapidly growing weights has been treated in few occasions \cite{MR1959871,MR1977920,MR1924991}. In contrast, literature on Jacobi matrices with slowly growing weights is more abundant (for instance, \cite{MR2078087,MR1707753,MR1764960,MR2077208,MR2091056}). In particular, Jacobi operators with zero main diagonal and $b_n\sim n^\alpha$, with $1/2<\alpha\leq 1$, has been studied in \cite{MR1707753,MR1764960,MR2077208}. In these papers it is proven that, depending on the concrete realization, this kind of Jacobi operators may have either purely absolutely continuous spectrum or absolutely continuous spectrum with a gap where the spectrum is pure point. This paper is organized as follows. In Section~\ref{sec:preliminaries}, we lay down the notation and some preparatory facts. In Section~\ref{sec:asymt-behav-gener}, we derive the asymptotic behavior of solutions of the discrete system associated to the generalized eigenequation for $J$. In Section~\ref{sec:self-adjointness}, we establish sufficient conditions for $J$ to be either self-adjoint (Theorem~\ref{thm:self-adjointness}) or non self-adjoint (Theorem~\ref{thm:nonself-adjointness}), and give examples. Finally, in Section~\ref{sec:absence-cont-spectr}, we characterize the spectrum (Corollary~\ref{pp} and Theorem~\ref{thm:compactness-resolvent}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Preliminaries} \label{sec:preliminaries} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we introduce the notation and main concepts that will be used throughout this paper. The elements of a scalar sequence will be denoted by Latin lowercase letters. For sequences of vectors and matrices we will use Greek lowercase letters and Latin capital letters, respectively. Besides the Hilbert space $l_2(\mathbb{N})=l_2(\mathbb{N}, \mathbb{C})$, we shall consider the space $l_2(\mathbb{N}, \mathbb{C}^d)$ consisting of sequences of $d$-dimensional vectors $\{\xi_n\}_{n\in{\mathbb N}}$ such that $\sum_{n=1}^\infty\norm{\xi_n}_{\mathbb{C}^d}^2$ is convergent, where $\norm{\cdot}_{\mathbb{C}^d}$ denotes some norm in $\mathbb{C}^d$. Should no confusion arise, we will denote $\norm{\xi}_{l_2(\mathbb{C}^d)}$ (for the norm in $l_2(\mathbb{N},\mathbb{C}^d)$) simply as $\norm{\xi}$. Keeping the notation used in the Introduction, the canonical basis in $l_2(\mathbb{N})$ is denoted by $\{e_n\}_{n\in{\mathbb N}}$, while $\{\delta^{(k)}\}_{k=1}^d$ will stand for the canonical basis in $\mathbb{C}^d$. Given a scalar sequence $\{f_n\}_{n\in{\mathbb N}}$, the standard asymptotic notation $O(f_n)$ ($o(f_n)$) as $n\to\infty$, depending on the context, will be used to denote the $n$-th element of a sequence of either scalars, vectors, or matrices, whose norm behaves as $O(f_n)$ ($o(f_n)$) as $n\to\infty$. All products of square matrices will be taken in ``chronological'' order, i.\,e., \begin{equation*} \prod_{s=p}^mA_s=A_mA_{m-1}\dots A_p\,. \end{equation*} Our approach to the spectral analysis of $J$ is based on the study of the asymptotics of solutions of the generalized eigenvalue equation \begin{equation} \label{eq:main-recurrence} b_{n-1}x_{n-1}(\zeta)+ q_nx_n(\zeta)+ b_nx_{n+1}(\zeta)= \zeta x_n(\zeta)\,,\quad n>1\,,\ \zeta\in\mathbb{C}\,. \end{equation} Clearly, if a solution $x(\zeta)=\{x_n(\zeta)\}_{n\in{\mathbb N}}$ satisfies the ``boundary condition'' \begin{equation} \label{eq:boundary} q_1x_1(\zeta)+b_1x_2(\zeta)=\zeta x_1(\zeta) \end{equation} and turns out to be in $\dom (J)$, then $x(\zeta)$ is an eigenvector of $J$ corresponding to the eigenvalue $\zeta$. To simplify somewhat the notation, in what follows we will not indicate the dependency of $x$ on $\zeta$. Equation (\ref{eq:main-recurrence}) can be written as follows \begin{equation} \label{eq:mre} \xi_{n+1}=B_n(\zeta ) \xi_n\,, \qquad n > 1\,,\quad\zeta\in\mathbb{C}\,, \end{equation} where $\xi_n = \biggl( \begin{smallmatrix} x_{n-1} \\[2mm] x_n \end{smallmatrix}\biggr)$ and $B_n(\zeta )= \biggl( \begin{smallmatrix} 0 & 1 \\[2mm] - \frac{b_{n-1}}{b_n} & \frac{\zeta-q_n}{b_n} \end{smallmatrix}\biggr)$ is the so-called transfer matrix. From (\ref{eq:mre}) it follows that \begin{equation} \label{eq:prod} \xi_n= \prod_{k=n_0}^{n-1}B_k(\zeta)\xi_{n_0}\,, \qquad n_0\ge 2\,. \end{equation} Since clearly we have $\|x\|^2<\|\xi\|^2\leq 2\|x\|^2$, the asymptotic behavior (in norm) of solutions of (\ref{eq:mre}) governs the asymptotic behavior of those of (\ref{eq:main-recurrence}) (and vice-versa). There are various methods for obtaining the asymptotic behavior of solutions of discrete linear systems. As already mentioned, the main tools are the so-called discrete Levinson-type theorems. For the purpose of the present work we use the Levinson-type theorem proven by Benzaid and Lutz in \cite{MR1002291}. For the sake of completeness, we state this theorem below. \begin{definition} \label{def:levinson-condition} Fix $k\in\mathbb{N}$ such that $k\leq d$. A sequence $Q=\{Q_n\}_{n= n_0}^\infty$ ($n_0\in\mathbb{N}$) of $d\times d$ diagonal complex matrices, \begin{equation*} Q_n=\diag\{q_n^{(j)}\}_{j=1}^d\,, \end{equation*} is said to satisfy the Levinson condition for $k$, denoted $Q\in\mathcal{L}(k)$, if for every $j=1,\dots,d$ there exists a natural number $N\geq n_0$ such that $q_n^{(j)}\ne 0$, $\forall n \ge N$, and for some constant $M>1$ each $j$ falls into one and only one of two classes $I_1$ or $I_2$, where: \begin{enumerate} \item[(a)] $j\in I_1$ if \begin{equation*} \frac{\abs{\prod_{i=N}^nq_i^{(k)}}} {\abs{\prod_{i=N}^nq_i^{(j)}}} \to \infty\quad\text{ as } n\to\infty\,,\quad\text{ and } \end{equation*} \begin{equation*} \frac{\abs{\prod_{i=n_1}^{n_2}q_i^{(k)}}} {\abs{\prod_{i=n_1}^{n_2}q_i^{(j)}}} >\frac{1}{M}\,, \quad\forall\, n_2\,, n_1\text{ such that } n_2 \ge n_1 \geq N\,. \end{equation*} \item[(b)] $j\in I_2$ if \begin{equation*} \frac{\abs{\prod_{i=n_1}^{n_2}q_i^{(k)}}} {\abs{\prod_{i=n_1}^{n_2}q_i^{(j)}}} < M\,, \quad\forall\, n_2\,, n_1\text{ such that } n_2 \ge n_1 \geq N\,. \end{equation*} \end{enumerate} \end{definition} \begin{theorem}[Benzaid-Lutz] \label{thm:bl} Let $Q=\{Q_n\}_{n= n_0}^\infty$ be a sequence of $d\times d$ diagonal complex matrices, \begin{equation*} Q_n=\diag\{q_n^{(k)}\}_{k=1}^d\,. \end{equation*} Consider also another sequence $R=\{R_n\}_{n= n_0}^\infty$ ($n_0\in\mathbb{N}$) of $d\times d$ complex matrices. Suppose that the following conditions hold: \begin{enumerate}[i.] \item The sequence $Q\in\mathcal{L}(k)$, for all $k=1,\dots,d$,\label{N} \vspace{1mm} \item $\displaystyle{\sum\limits_{n=n_0}^\infty \frac{\norm{R_n}}{|q_n^{(k)}|}<\infty}$, \label{R-condition-2} \vspace{1mm} \item $\det(Q_n+R_n)\ne 0$ for all $n\ge n_0$. \end{enumerate} Then there exists a basis $\varphi^{(k)}$ ($k=1\dots, d$) in the space of solutions of the system \begin{equation*} \varphi_{n+1}=(Q_n+R_n)\varphi_n\,, \qquad n\ge n_0\,, \end{equation*} such that \begin{equation*} \norm{\frac{\varphi_n^{(k)}}{\prod_{i=n_1}^{n-1} q_i^{(k)}} - \delta^{(k)}}\to 0,\quad\text{ as }n\to\infty\,,\quad\text{for }k=1,\dots,d\,. \end{equation*} \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Asymptotic behavior of solutions} \label{sec:asymt-behav-gener} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let us define the matrices \[ A_m(\zeta)= B_{2m+1}(\zeta)B_{2m}(\zeta)\,, \quad \forall m\in\mathbb{N}\,. \] Clearly, system (\ref{eq:mre}) is equivalent to \begin{equation} \label{eq:mreA} \eta_{m+1}=A_m(\zeta)\eta_m\,, \quad \forall m\in\mathbb{N}\,, \end{equation} where $\eta_{m}=\xi_{2m}$. A straightforward computation shows that \[ B_{n+1}(\zeta)B_{n}(\zeta) = - I + \begin{pmatrix} \frac{c_{n}-c_{n-1}+\alpha}{n} & 0\\ 0 & \frac{c_{n+1}-c_{n}+\alpha}{n} \end{pmatrix} + O(n^{-1-\epsilon}) \] as $n\to\infty$, where $I$ denotes the identity matrix and $\epsilon>0$. We notice that the term $O(n^{-1-\epsilon})$ contains all the dependencies on the parameter $\zeta$ and sequences $\{q_n\}_{n\in{\mathbb N}}$ and $\{h_n\}_{n\in{\mathbb N}}$ . Up to the same order in $n$, we may write \[ B_{n+1}(\zeta)B_{n}(\zeta) = - \begin{pmatrix} \exp\frac{c_{n-1}-c_{n}-\alpha}{n} & 0\\ 0 & \exp\frac{c_{n}-c_{n+1}-\alpha}{n} \end{pmatrix}\left[I+O(n^{-1-\epsilon})\right], \] where again the dependency on $\zeta$ is contained exclusively in the term $O(n^{-1-\epsilon})$. Thus, $A_m(\zeta)$ admits the decomposition \begin{equation}\label{a} A_m(\zeta) = G_m + R_m(\zeta)\,, \end{equation} where, for notational convenience, we define \[ G_m:= - \begin{pmatrix} \exp g_m^{(1)} & 0\\ 0 & \exp g_m^{(2)} \end{pmatrix} \] with \[ g_m^{(1)} := \frac{c_{2m-1}-c_{2m}-\alpha}{2m}\quad\mbox{and}\quad g_m^{(2)} := \frac{c_{2m}-c_{2m+1}-\alpha}{2m}\,, \] and the residual term is given by \[ R_m(\zeta):= G_m\,O(m^{-1-\epsilon})\,. \] The r.\ h.\ s.\ of this last identity must be understood as the product of $G_m$ times a matrix that depends on $\zeta$ and whose norm behaves as specified by the last factor as $m\to\infty$. As mentioned earlier, we want to apply Theorem~\ref{thm:bl} to system (\ref{eq:mreA}) in order to obtain the asymptotic behavior of its solutions. In view of (\ref{a}), we must be certain that the sequence $\{G_m\}_{m\in\mathbb{N}}$ satisfies the Levinson condition stated in Definition~\ref{def:levinson-condition}. Since this is not {\em a priori} known, we will always assume it true from now on:\\ \noindent(H.1)\quad $\{G_m\}_{m\in\mathbb{N}}\in\mathcal{L}(k)$ for $k=1,2$.\\ \stepcounter{aux2} Indeed, there are examples of Jacobi matrices of the class discussed in this article whose associated sequences $\{G_m\}_{m\in\mathbb{N}}$ do not satisfy (H.1). However, the following lemma gives criteria for establishing when this hypothesis is fulfilled. \begin{lemma} \label{lem:criteria-for-levinson-cond} Consider an arbitrary monotone increasing sequence of natural numbers $\{n_i\}_{i\in\mathbb{N}}$ such that the sequence $\{\frac{n_{i+1}}{n_i}\}_{i\in\mathbb{N}}$ is bounded. Fix $k\in\{1,2\}$. Then $\{G_m\}_{m\in\mathbb{N}}$ is in $\mathcal{L}(k)$ if one of the following cases is met: \begin{enumerate}[i.] \item There exist constants $C_1$ and $C_2$ such that, for all $i\in\mathbb{N}$, \[ C_1<\sum_{s=1}^{n_i}\frac{c_{2s-1}-2c_{2s}+c_{2s+1}}{2s}n_1$, \begin{equation} \label{eq:estimate-above-through-partition} \sum_{s=1}^{m}\frac{c_{2s-1}-2c_{2s}+c_{2s+1}}{2s} <\sum_{s=1}^{n_i}\frac{c_{2s-1}-2c_{2s}+c_{2s+1}}{2s}+C\,, \end{equation} with $n_i \sum_{s=1}^{n_i}\frac{c_{2s-1}-2c_{2s}+c_{2s+1}}{2s}+\widetilde{C}\,, \end{equation} for any $m>n_1$, where $n_i \left(\sum_{s=1}^{n_{i_2}}\!-\!\sum_{s=1}^{n_{i_1}}\right) \!\frac{c_{2s-1}\!-\!2c_{2s}\!+\!c_{2s+1}}{2s}+\widetilde{C}-C\,. \end{equation*} From (iii) and this inequality it follows that, for any pair of natural numbers $m_1N$ \begin{equation*} \sum_{s=1}^{m}\frac{c_{2s-1}-2c_{2s}+c_{2s+1}}{2s}>K\,. \end{equation*} But this follows from the fact that, due to (iii), for any given $K$ we can choose $i_0$ such that \begin{equation*} \sum_{s=1}^{n_{i_0}}\frac{c_{2s-1}-2c_{2s}+c_{2s+1}}{2s}> K-\widetilde{C}\,. \end{equation*} Then, for any $m>n_{i_0}$, \begin{equation*} \sum_{s=1}^{m}\frac{c_{2s-1}-2c_{2s}+ c_{2s+1}}{2s}>\sum_{s=1}^{n_{i_0}} \frac{c_{2s-1}-2c_{2s}+c_{2s+1}}{2s}+\widetilde{C}>K\,. \end{equation*} This completes the proof. \end{proof} \begin{lemma} \label{lem:asympt-sol-vec-sys} Assume (H.1). Then system (\ref{eq:mreA}) satisfies the assumptions of Theorem~\ref{thm:bl}. That is, there exists a basis $\eta^{(k)}$ ($k=1,2$) in the space of solutions of (\ref{eq:mreA}) such that, for $k=1$ and $k=2$, \begin{equation} \label{eq:basis-asympt-sol} \left\|\eta_m^{(k)}/\prod\limits_{s=n_1}^{m-1}(-1) \exp g_s^{(k)} - \delta^{(k)}\right\|\to 0 \end{equation} as $m\to\infty$. \end{lemma} \begin{proof} We only must verify the fulfillment of conditions (ii) and (iii) of Theorem~\ref{thm:bl}. We have \begin{equation*} \frac{v_1}{2m}\le g_m^{(k)}\le\frac{v_2}{2m} \end{equation*} for $k=1, 2$ and some constants $v_1$ and $v_2$. It follows that \[ \frac{\norm{R_m(\zeta)}}{\exp g_m^{(k)}} \leq r_1 m^{-1-\epsilon}\frac{\norm{G_m}}{\exp g_m^{(k)}} \leq r_2 m^{-1-\epsilon}\exp \frac{v_2-v_1}{2m} \] for some positive constants $r_1$ and $r_2$. This implies that the l.\ h.\ s.\ of the inequality above is summable, hence condition (ii) of Theorem \ref{thm:bl} holds. Finally, since the determinant of each transfer matrix is different from zero, the same is true for the determinant of each $A_m$ (which equals $G_m + R_m$). Thus, condition (iii) of Theorem \ref{thm:bl} also holds. \end{proof} Up to a normalization constant, statement (\ref{eq:basis-asympt-sol}) is equivalent to saying that \begin{equation} \label{asymp} \eta_{m+1}^{(k)} = (-1)^{m} \exp\left(\sum\limits_{s=1}^{m}g_s^{(k)}\right) \left[\delta^{(k)} + \tau_m^{(k)}\right]\,, \end{equation} where $\tau_m^{(k)}$ is $o(1)$ as $m\to\infty$. We will use this expression in the sequel. Solutions of~(\ref{eq:mreA}) obviously depend on $\zeta$ and the specific form of the sequences $\{q_n\}_{n\in{\mathbb N}}$ and $\{h_n\}_{n\in{\mathbb N}}$. However, equation (\ref{eq:basis-asympt-sol}) (or (\ref{asymp})) shows that the main asymptotic term does not. \begin{remark} We have the following consequence of Lemma~\ref{lem:asympt-sol-vec-sys}: Let $\{m_j\}_{j\in{\mathbb N}}$ be a monotone increasing sequence of natural numbers. Recalling that $\eta_m=\xi_{2m}$, from (\ref{asymp}) we have \[ \xi_{2m_j+2}^{(k)}=(-1)^{m_j}\exp\left(\sum_{s=1}^{m_j} g_s^{(k)}\right) \left[\delta^{(k)} + \tau_{m_j}^{(k)}\right]\,. \] Moreover, \[ \xi_{2m_j+2p+2}^{(k)}=(-1)^{m_j+p} \exp\left(\sum_{s=1}^{m_j+p} g_s^{(k)}\right) \left[\delta^{(k)} + \tau_{m_j+p}^{(k)}\right] \] for $p\in{\mathbb N}$. Thus, by splitting the exponential factor and working out residual terms, we obtain \begin{equation} \label{schubert} \xi_{2m_j+2p+2}^{(k)}=(-1)^p\exp\left(\sum_{s=m_j+1}^{m_j+p}g_s^{(k)}\right) \left[\xi_{2m_j+2}^{(k)}+\phi_{j,p}^{(k)}\right]\,, \end{equation} where the norm of $\phi_{j,p}^{(k)}$ equals the product of the norm of $\xi_{2m_j+2}^{(k)}$ times a function $o(1)$ as $j\rightarrow\infty$, uniformly in $p$. \end{remark} We close this section noticing the fact that, in order to obtain the asymptotic behavior of solutions of (\ref{eq:mreA}), it was sufficient to assume the conditions on $\{c_n\}_{n\in{\mathbb N}}$, $\{h_n\}_{n\in{\mathbb N}}$, and $\{q_n\}_{n\in{\mathbb N}}$ given in the Introduction, with the addition of hypothesis (H.1). However, without further assumptions on $\{c_n\}_{n\in{\mathbb N}}$, these asymptotics yields no clue about the decaying behavior of these solutions. This information is necessary to determine whether the operator is self-adjoint or not and, in the latter case, to classify the spectrum (by using, for instance, Subordinacy Theory). Theses issues will be attended, in the following sections, by requiring certain regularity on average of $\{c_n\}_{n\in{\mathbb N}}$ over sequences $\{m_j\}_{j\in{\mathbb N}}$. This will give rise to nice asymptotic behavior for the norm of $\xi_{2m_j+2}^{(k)}$ from which, the asymptotic behavior of the norm of $\xi_{2m+2}^{(k)}$ will be interpolated by resorting to (\ref{schubert}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Self-adjointness} \label{sec:self-adjointness} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% It is well known that, for Jacobi matrices, we have the discrete analogue of Weyl's limit point--limit circle theory for self-adjoint extensions of differential operators. In particular, it is true that if two linearly independent solutions of (\ref{eq:main-recurrence}) are in $l_2(\mathbb{N})$, then $J$ is not self-adjoint (limit circle case). If, however, one solution of (\ref{eq:main-recurrence}) is not in $l_2(\mathbb{N})$ then $J=J^*$ (limit point case) \cite{MR0184042,MR1627806}. Next, we establish sufficient conditions for $J$ to be self-adjoint. \begin{theorem} \label{thm:self-adjointness} Define $w^{(1)}=\min\{0,\inf_{s}\{c_{2s-1}-c_{2s}-\alpha\}\}$ and $w^{(2)}=\min\{0,\inf_{s}\{c_{2s}-c_{2s+1}-\alpha\}\}$, and assume that (H.1) is satisfied. Fix the value of $k$. Consider a monotone increasing sequence $\{m_j\}_{j\in{\mathbb N}}$ of natural numbers that obeys \begin{hypothesis} \sum\limits_{j=1}^\infty\frac{m_{j+1}-m_j}{m_{j}} \left(\frac{m_{j+1}}{m_j}\right)^{w^{(k)}}=\infty\,. \label{cond-2-self-adjoint} \end{hypothesis} Suppose now that, for this sequence, the following holds: \begin{equation} \sum\limits_{s=1}^{m_j}g_s^{(k)} \ge -\frac{1}{2}\log(m_j) + O(1),\quad\mbox{when $j\to\infty$}\,. \label{cond-1-self-adjoint} \end{equation} Then, there exists a solution of (\ref{eq:main-recurrence}) that is not in $l_2(\mathbb{N})$ and, therefore, $J=J^*$. \end{theorem} \begin{proof} We first note that (\ref{cond-2-self-adjoint}) implies that either \[ \sum\limits_{j=1}^\infty\frac{m_{j+1}-m_j-1}{m_{j}} \left(\frac{m_{j+1}-1}{m_j+1}\right)^{w^{(k)}}=\infty\quad\text{or}\quad \sum\limits_{j=1}^\infty\frac{1}{m_{j}}=\infty\,. \] We now prove the assertion under the assumption that $k=1$. The case $k=2$ follows analogously. We note that (\ref{cond-1-self-adjoint}) and (\ref{asymp}) imply \begin{equation} \label{pim} \norm{\xi_{2m_j+2}^{(1)}}^2 \geq r_1 m_{j}^{-1}\,,\quad\forall j\geq j_0, \end{equation} where $r_1$ is some positive constant and $j_0$ is sufficiently large. Moreover, from (\ref{schubert}) we obtain \begin{align} \norm{\xi^{(1)}_{2m_j+2p+2}}^2 &\geq r_2\exp\left(2\sum\limits_{s=m_j+1}^{m_j+p}g_s^{(1)}\right) \norm{\xi^{(1)}_{2m_j+2}}^2\nonumber\\ &\geq r_3\left(\frac{m_j+p}{m_j+1}\right)^{w^{(1)}} \norm{\xi^{(1)}_{2m_j+2}}^2\,,\label{pot} \end{align} for some positive constants $r_2$, $r_3$ and $01$. Now suppose that the inequality \begin{equation} \label{eq:nonself-adj-cond} \sum\limits_{s=1}^{m_j}g_s^{(k)} \leq -\frac{\beta}{2}\log(m_j) + O(1) \quad\text{when $j\to\infty$}, \end{equation} holds for both $k=1$ and $k=2$. Then all solutions of (\ref{eq:main-recurrence}) are in $l_2(\mathbb{N})$ and, therefore, $J\ne J^*$. \end{theorem} \begin{proof} We note first that (\ref{schubert}) and (\ref{eq:nonself-adj-cond}) imply \begin{equation*} \norm{\xi_{2+2m_{j}}^{(k)}}^2 < r_0 m_{j}^{-\beta}\,,\quad\forall j\geq j_1, \end{equation*} where $r_0$ is some positive constant. Without loss we may assume $j_1=1$ to avoid unnecessary complications. Also, (\ref{schubert}) implies \begin{eqnarray*} \norm{\xi^{(k)}_{2m_j+2p+2}}^2 &\le& r_1\exp\left(2\sum\limits_{s=m_j+1}^{m_j+p}g_s^{(k)}\right) \norm{\xi^{(k)}_{2m_j+2}}^2\\ &\leq& r_2\left(\frac{m_j+p}{m_j+1}\right)^{v}\norm{\xi^{(k)}_{2m_j+2}}^2\,, \end{eqnarray*} for $00$ denote the square of such a bound. Now, bearing these facts in mind, we have \begin{align*} \norm{\xi^{(k)}}^2 &\leq r_3 \sum_{m=m_1}^{\infty}\norm{\xi_{2m+2}^{(k)}}^2\\ &= r_3 \sum_{j=1}^{\infty}\sum_{p=0}^{m_{j+1}-m_j-1} \norm{\xi_{2m_j+2p+2}^{(k)}}^2\\ &\leq r_3 \sum_{j=1}^{\infty}\norm{\xi_{2m_j+2}^{(k)}}^2 \left[1+r_2\sum_{j=1}^{m_{j+1}-m_j-1} \left(\frac{m_j+p}{m_j+1}\right)^{v}\right]\\ &\leq r_1r_3\sum_{j=1}^{\infty}\frac{1}{m_j^\beta} + r_1r_2r_3\sum_{j=1}^{\infty}\frac{1}{m_j^\beta} \sum_{p=1}^{m_{j+1}-m_j-1} \left(\frac{m_j+p}{m_j+1}\right)^{v}\,. \end{align*} The first term in the last inequality is clearly convergent, since $m_j\geq j$ and $\beta>1$. The second term is also convergent since (\ref{non-self}) is fulfilled. Thus, we have proved that $\xi^{(1)}$ and $\xi^{(2)}$ belong to $l_2(\mathbb{N},\mathbb{C}^2)$. It follows (by the argumentation given in the proof of Theorem~\ref{thm:self-adjointness}) that $x^{(1)}$ and $x^{(2)}$ belong to $l_2(\mathbb{N})$. Moreover, they are linearly independent as a consequence of the linear independency of $\xi^{(1)}$ and $\xi^{(2)}$. The lack of self-adjointness now follows immediately. \end{proof} \begin{remark} Theorem \ref{thm:nonself-adjointness} does not allow to obtain the converse of Theorem \ref{thm:self-adjointness}. \end{remark} \begin{remark} \label{all-of-them} A fairly large set of sequences $\{m_j\}_{j\in{\mathbb N}}$ satisfies condition (\ref{non-self}), ranging from those of linear growth, i.\ e. $m_j\sim j$, up to those of very fast growth like $m_j\sim j^j$. \end{remark} As we have mentioned previously, Jacobi operators of the form proposed in the present work has been already studied for the case of even-periodic sequences $\{c_n\}_{n\in{\mathbb N}}$. No results are known (to the authors anyway) concerning the odd-periodic case, although it has been suggested that, in this case, these operators would not be self-adjoint \cite{MR1924991}. Theorem~\ref{thm:nonself-adjointness} allows us to prove that assertion easily. \begin{corollary} \label{corollary:non-self} Let $\{c_n\}_{n\in{\mathbb N}}$ be an odd-periodic sequence. Then the operator $J$, given by the matrix representation (\ref{eq:jm}) with entries defined by (\ref{eq:weights}) and (\ref{eq:diagonal}), is not self-adjoint. \end{corollary} \begin{proof} Let $T$ be the period of the sequence. Define $m_j=jT$ for $j\in{\mathbb N}$. Clearly, $\{m_j\}_{j\in{\mathbb N}}$ satisfies (\ref{non-self}) for any $\beta>1$ and $v\geq 0$. Thus, it only remains to verify (H.1) and (\ref{eq:nonself-adj-cond}) for $k=1,2$. We have, \begin{align} \sum_{s=1}^{m_j}g_s^{(1)} &= \frac{1}{2}\sum_{s=1}^{m_j}\frac{c_{2s-1}-c_{2s}-\alpha}{s}\nonumber\\ &= \frac{1}{2}\sum_{n=1}^{j}\sum_{s=m_{n-1}+1}^{m_n}\frac{c_{2s-1}-c_{2s}}{s} - \frac{\alpha}{2}\sum_{s=1}^{m_j}\frac{1}{s}\nonumber\\ &= \frac{1}{2}\sum_{n=1}^{j}\sum_{s=1}^{T}\frac{c_{2s-1}-c_{2s}}{s+(j-1)T} - \frac{\alpha}{2}\log(m_j) + O(1)\,,\label{per} \end{align} where the first term in the last equality follows from the periodicity of $\{c_n\}_{n\in{\mathbb N}}$. Now, for $j\geq 2$, \begin{align*} \sum_{s=1}^{T}\frac{c_{2s-1}-c_{2s}}{s+(j-1)T} &= \sum_{s=1}^{T}\frac{c_{2s-1}-c_{2s}}{(j-1)T}\\ & +\ \sum_{s=1}^{T}(c_{2s-1}-c_{2s}) \left[\frac{1}{s+(j-1)T}-\frac{1}{(j-1)T}\right]. \end{align*} The first term above equals zero because of the odd-periodicity of the sequence, while the second term is $O(j^{-2})$. Therefore, the first term in (\ref{per}) is also $O(1)$ thus yielding the expected inequality (with $\beta=\alpha$) for $\sum_{s=1}^{h_m}g_s^{(1)}$. A similar computation shows that also $\sum_{s=1}^{h_m}g_s^{(2)}$ fulfills the same inequality. Finally, it is clear that case (i) of Lemma~\ref{lem:criteria-for-levinson-cond} holds. \end{proof} The proof of Corollary~\ref{corollary:non-self} tells us that an odd-periodic sequence $\{c_n\}_{n\in{\mathbb N}}$ produces a non-self-adjoint operator because one can find a suitable sequence $\{m_j\}_{j\in{\mathbb N}}$ for which, the contribution of $c_{2s-1}-c_{2s}$ to the l.\ h.\ s.\ of (\ref{eq:nonself-adj-cond}) is nearly canceled (and the same occurs for $c_{2s}-c_{2s+1}$). This idea may be used to construct non-selfadjoint Jacobi operators out of certain non-periodic sequences. As a matter of fact, the following example is defined essentially in the same way as the one discussed previously, having exactly one modification. \begin{example} \label{ex:non-selfadjointness} Consider a sequence $\{c_n\}_{n\in{\mathbb N}}$ defined by two given numbers $c_1$ and $c_2$ arranged as follows: \begin{equation} \label{non-self-adjiont-model} \underbrace{ \underbrace{ c_1\,c_2\,c_1\,c_2\cdots c_1\,c_2\,c_1\,c_2}_{2p_1} c_2\,c_1\,c_2\,c_1\cdots c_2\,c_1\,c_2\,c_1}_{2p_2} c_1\,c_2\,c_1\,c_2\cdots\,, \end{equation} where $\{p_j\}_{j\in{\mathbb Z}^+}$ is a sequence with polynomial growth of the form $p_j=Cj^a$, with $a\in{\mathbb N}\setminus\{1\}$ and $C\in\mathbb{N}$. That is, we use the sequence defined by (\ref{eq:model-dislocations}) with $c_3=c_1$. Alternatively, we may think of this sequence as one obtained from a two-periodic sequence by inserting dislocations at sites $2p_j+1$. We claim that the Jacobi operator defined by (\ref{non-self-adjiont-model}) is not self-adjoint. To prove it, we first define $m_j=p_{2j}$. This sequence satisfies (\ref{non-self}) for any $\beta>1$ as a straightforward computation shows. We can use some of the computations of Example 1. Thus, from (\ref{eq:example-result-1}) and (\ref{eq:example-result-2}) it follows straightforwardly that, for $k=1,2$, \begin{equation*} \sum_{s=m_{n-1}+1}^{m_n}g_s^{(k)}= -\frac{\alpha}{2}\sum_{s=m_{n-1}+1}^{m_n}\frac{1}{s} +O(n^{-2})\,,\quad\text{ as } m_j\to\infty\,. \end{equation*} This yields \begin{equation*} \sum_{s=1}^{m_j}g_s^{(k)}= -\frac{\alpha}{2}\log(m_j) +O(1)\, \end{equation*} for $k=1,2$. From this equation it follows that this example satisfies case (i) of Lemma~\ref{lem:criteria-for-levinson-cond} so (H.1) is satisfied. Also, (\ref{eq:nonself-adj-cond}) holds true. Hence, we have verified that this example fulfills all the hypotheses of Theorem~\ref{thm:nonself-adjointness}. We note that the case $p_m\sim m$ can not be included here since the summability of \begin{equation*} \sum_{j=1}^\infty\sum_{s=m_{j-1}+1}^{m_j}\!\frac{c_{2s-1}-c_{2s}}{s} \end{equation*} is no longer ensured. This is to be expected because, for $p_m\sim m$, the sequence $\{c_n\}_{n\in{\mathbb N}}$ is again even-periodic so the associated Jacobi operator may be self-adjoint (see Remark~\ref{614}). \end{example} Other examples of non-self-adjoint Jacobi operators (of the class discussed in this work) could be devised along similar ideas. We think that the example given above is of some interest because it is derived from an operator defined by an even-periodic sequence and therefore self-adjoint (provided that $|c_1-c_2|\geq \alpha-1$, although this condition plays no role in our example) \cite{MR1959871,MR1924991}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Discreteness of spectrum} \label{sec:absence-cont-spectr} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we show that, if the assumptions of Theorem~\ref{thm:self-adjointness} are fulfilled and $\{m_j\}_{j\in{\mathbb N}}$ obeys certain further conditions, then $J$ has only discrete spectrum. We first prove an easy consequence of inequality (\ref{cond-1-self-adjoint}). \begin{lemma} \label{lemma-bullshit} If (\ref{cond-1-self-adjoint}) is true for $k=1$, then \begin{equation} \label{eq:silly-ineq} \sum_{s=1}^{m_j}g_s^{(2)} < -\frac{\alpha}{2}\log(m_j) + O(1) \quad\text{when }j\to\infty\,. \end{equation} An analogous assertion holds if the roles of $k=1$ and $k=2$ are interchanged. \end{lemma} \begin{proof} We have \begin{equation} \label{bullshit} \sum_{s=1}^{m_j}g_s^{(2)} = -\sum_{s=1}^{m_j}g_s^{(1)} + \sum_{s=1}^{m_j}\frac{c_{2s-1}-c_{2s+1}}{2s} -\alpha\sum_{s=1}^{m_j}\frac{1}{s}\,. \end{equation} Furthermore, \[ \sum_{s=1}^{m_j}\frac{c_{2s-1}-c_{2s+1}}{2s} = \sum_{s=2}^{m_j}\frac{c_{2s-1}}{2}\left(\frac{1}{s}-\frac{1}{s-1}\right) +\frac{c_1}{2}-\frac{c_{2m_j+1}}{2m_j}\,. \] Thus, the second term in (\ref{bullshit}) is $O(1)$ as $j\to\infty$ because the sequence $\{c_n\}_{n\in{\mathbb N}}$ is bounded and $(s^{-1} - (s-1)^{-1})$ is $O(s^{-2})$. Since moreover the last term in (\ref{bullshit}) equals $\alpha\log(m_j)$ up to a term $O(1)$, we obtain \begin{align*} \sum_{s=1}^{m_j}g_s^{(2)} &= -\sum_{s=1}^{m_j}g_s^{(1)} - \alpha\log(m_j) + O(1)\\ &\leq \frac{1}{2}\log(m_j) - \alpha\log(m_j) + O(1)\\ &< - \frac{\alpha}{2}\log(m_j) + O(1)\,, \end{align*} where the last inequality holds since $\alpha>1$. \end{proof} \begin{theorem} \label{arafat} Suppose that the hypotheses of Theorem~\ref{thm:self-adjointness} are satisfied for $k=1$ (then $x^{(1)}$ is not in $l_2(\mathbb{N})$). In addition, assume that $\{m_j\}_{j\in{\mathbb N}}$ satisfies \begin{hypothesis} \label{one-more} \sum_{j=1}^{\infty}\frac{m_{j+1}-m_j}{m_j^\alpha} \left(\frac{m_{j+1}}{m_j}\right)^{\check{v}}<\infty\,, \end{hypothesis} with $\check{v}=\max\{0,\,\sup_s\{c_{2s}-c_{2s+1}-\alpha\}\}$. Then $x^{(2)}$ is in $l_2(\mathbb{N})$. The assertion holds also if we interchange $k=1$ and $k=2$ and take $\check{v}=\max\{0,\,\sup_s\{c_{2s-1}-c_{2s}-\alpha\}\}$. \end{theorem} \begin{proof} By Lemma~\ref{lemma-bullshit} we have (\ref{eq:silly-ineq}) and then, taking into account (\ref{one-more}) and reasoning as in Theorem ~\ref{thm:nonself-adjointness}, we obtain that $\xi^{(2)}$ is in $l_2(\mathbb{N},\mathbb{C}^2)$. \end{proof} \begin{remark} \label{last} In view of Remarks~\ref{divergent} and \ref{all-of-them}, the set of sequences $\{m_j\}_{j\in{\mathbb N}}$ that fulfill both (\ref{cond-2-self-adjoint}) and (\ref{one-more}) is far from being empty. \end{remark} \begin{corollary} \label{pp} $J$ has only pure point spectrum whenever the assumptions of Theorem~\ref{arafat} are met. \end{corollary} \begin{proof} Under the assumptions of the previous theorem we have that $x^{(1)}\not\in l_2(\mathbb{N})$ and $x^{(2)}\in l_2(\mathbb{N})$. This behavior of the basis $\{x^{(1)},\,x^{(2)}\}$ in the space of solutions of (\ref{eq:main-recurrence}) is the same for any $\zeta\in\mathbb{C}$. Thus, we always have a subordinate solution for the generalized eigenequation of the self-adjoint operator $J$. By applying Subordinacy Theory \cite[Theorem 3]{MR1179528}, we conclude that the spectrum is pure point. \end{proof} \begin{remark} Notice that a sequence $\{m_j\}_{j\in{\mathbb N}}$ may satisfy (H.2) but not (H.4). That is, Theorems~\ref{thm:self-adjointness} and \ref{arafat} not necessarily hold true simultaneously. However, in account of what is said in Remark~\ref{last}, that happens for a large set of Jacobi operators $J$. \end{remark} We notice that (\ref{cond-2-self-adjoint}) along with (\ref{eq:silly-ineq}) imply a certain correlation between the slow decay of $x^{(1)}$ and the fast decay of $x^{(2)}$ (or vice-versa). We shall use this to show that the resolvent $(J-\zeta I)^{-1}$ is Hilbert-Schmidt. Unfortunately, (\ref{cond-1-self-adjoint}) is not good enough for that purpose. The next technical result accounts for what is needed. \begin{lemma} \label{life-saver} Suppose that for a certain monotone increasing sequence $\{m_j\}_{j\in{\mathbb{N}}}$ one of the following holds: \begin{enumerate}[i.] \item For any natural $i$ greater than some $i_0\in\mathbb{N}$, \begin{equation*} \sum_{s=m_i+1}^{m_{i+1}}g_s^{(1)} \ge -\frac{1}{2}\log\left(\frac{m_{i+1}}{m_i}\right) + f_i\,, \end{equation*} where $\sum_{i=j}^{l-1}f_i$ is bounded for any $j$ and $l$ ($i_0\le j\abs{x_m}^2$, we have (dependencies on $\zeta$ are dropped from now on) \[ \abs{G_{mn}} < \begin{cases} r_0\norm{\xi_{m}^{(1)}}\norm{\xi_{n}^{(2)}} + r_1\norm{\xi_{m}^{(2)}}\norm{\xi_{n}^{(2)}}&\text{ when } m\le n\,, \\[4mm] r_0\norm{\xi_{m}^{(2)}}\norm{\xi_{n}^{(1)}} + r_1\norm{\xi_{m}^{(2)}}\norm{\xi_{n}^{(2)}}&\text{ otherwise }. \end{cases} \] Since $\xi_m=B_{m-1}\xi_{m-1}$ and $B_n$ is uniformly norm-bounded in $n$, we have $\norm{\xi_{2m+1}}\leq r_2 \norm{\xi_{2m}}$. It thus follows that \begin{eqnarray} \sum_{m=m_0}^{\infty}\sum_{n=n_0}^{\infty}\abs{G_{mn}}^2\!\!\! &<&\!\!\! \sum_{k_0\le k\le j}^\infty\! \left(r_3\norm{\xi_{2k+2}^{(1)}}\norm{\xi_{2j+2}^{(2)}} + r_4\norm{\xi_{2k+2}^{(2)}}\norm{\xi_{2j+2}^{(2)}}\right)^2 \nonumber \\ & &\!\! +\!\sum_{j_0\le j