1$. 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 $0

0$ 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