TITLE: If the degree of $\alpha=5555$, what is the degree of $\alpha^2$?
QUESTION [5 upvotes]: $\alpha$ is an algebraic number with degree $5555$. What is the degree of $\alpha^2$?
Here are my thoughts so far:
I think it is true that $\mathbb{Q} \subseteq \mathbb{Q}[\alpha^2] \subseteq \mathbb{Q}[\alpha]$.
There is a theorem which says that  $|\mathbb{Q}[\alpha]:\mathbb{Q}|=\deg(\alpha)$ and the Tower Rule tells us that $|\mathbb{Q}[\alpha]:\mathbb{Q}|=|\mathbb{Q}[\alpha]:\mathbb{Q}[\alpha^2]||\mathbb{Q}[\alpha^2]:\mathbb{Q}|=\deg(\alpha)=5555$. So we know that $\deg(\alpha^2)$ divides $5555$.
However, I don't know how to narrow down the options. $\deg(\alpha^2)$ could equal any of $1, 5, 11, 55, 101, 505, 1111,5555$ though I'm fairly certain it's not going to be $1$.
Is there a way to deduce $\deg(\alpha^2)$ based on just what we know here or do we need some other knowledge too?
Thanks in advance.

REPLY [7 votes]: The number $5555$ is a red herring. You just need the fact it's odd.
Since $\def\Q{\mathbb{Q}}\Q(\alpha^2)\subseteq\Q(\alpha)$, you have
$$
[\Q(\alpha):\Q]=[\Q(\alpha):\Q(\alpha^2)]\,[\Q(\alpha^2):\Q].
$$
What can the degree of $\alpha$ over $\Q(\alpha^2)$ be? Consider the polynomial $X^2-\alpha^2$ which has coefficients in $\Q(\alpha^2)$, in order to give an answer.

 Since the polynomial has degree $2$, the degree of $\alpha$ over $\Q(\alpha^2)$ is either $1$ (if the polynomial is reducible) or $2$, because certainly $\alpha$ is a root of $X^2-\alpha^2$.

If $[\Q(\alpha):\Q]$ is odd, what more can you say?


TITLE: Show that any root $z$ of $z^4 + z + 3 = 0$ satisfies $|z| > 1$
QUESTION [2 upvotes]: Show that any root $z$ of $z^4 + z + 3 = 0$ satisfies $|z| > 1$ and that any root $z$ of $4z^4 + z + 1 = 0 $ satisfies $ |z| \le 1.$
What I've done so far:
$|z^4+z+3| \le |z|^4+|z|+3$.
If $0<|z|\le1, $ then $3< |z|^4+|z|+3 \le 5$.
How do I then complete the proof?

REPLY [2 votes]: Some comments point to the simplest answer.  If $z^4+z+3=0$ then |z^4+z|=3.  But if $|z|\le 1$ then $|z^4+z|\le |z^4|+|z|=|z|^4+|z|\le 2$, a contradiction.


text:
\section{Extensions}
\label{sec:extensions}

In this section, we present two extensions to the basic framework presented so far. In Section \ref{sec:stochastic rollouts}, we discuss extensions to systems with stochastic dynamics or sensor measurements. In Section \ref{sec:domain shift}, we present an approach that allows us to tackle settings where training and test environments are drawn from different distributions.

\subsection{Stochastic Rollout Functions}
\label{sec:stochastic rollouts}

In our problem formulation in Section \ref{sec:problem formulation}, we assumed that the rollout function $r_w: \mathcal{E} \rightarrow (\mathcal{X} \times \mathcal{U})^H$ is deterministic (i.e., once the environment is fixed, the resulting state-action trajectory obtained by applying a given policy is completely determined). Here we briefly sketch an extension of our framework to settings where the rollout function is stochastic (e.g., due to stochasticity in the dynamics of the system or in sensor measurements). This is made possible by a reinterpretation of the variable $w$. Previously, $w$ corresponded to parameters of the control policy. Suppose now that we think of $w$ as consisting of two components $w \coloneqq [w_\text{int}, w_\text{ext}]$; an ``internal" component $w_\text{int}$ corresponding to parameters of the control policy (just as before), and an additional ``external" component corresponding to uncertain parameters (e.g., external disturbances that the robot might experience). The rollout function now has the following structure: $r_{[w_\text{int}, w_\text{ext}]}: \mathcal{E} \rightarrow (\mathcal{X} \times \mathcal{U})^H$. The stochasticity in $w_\text{int}$ is directly set by us (i.e., by choosing a prior $P_0$ and posterior $P$ as before). However, the stochasticity over $w_\text{ext}$ is beyond our control.

We note that the structure of the resulting problem is identical to the original formulation considered in Section \ref{sec:problem formulation}. The only difference comes from the fact that a portion of the stochasticity in the rollouts is beyond our control. We can thus directly apply Theorem \ref{thm:pac bayes control} in order to obtain an upper bound on the true expected cost. In particular, let $P_0$ and $P$ be the prior and posterior over $w_\text{int}$ (as before) and suppose that the distribution over $w_\text{ext}$ is given by $P_\text{ext}$. Further, assume that $w_\text{int}$ and $w_\text{ext}$ are independent random variables. We can then define $P_0'$ and $P'$ to be the prior and posterior distributions over $w \coloneqq [w_\text{int}, w_\text{ext}]$ and evaluate the ``regularizer" term in the PAC-Bayes bound in Theorem \ref{thm:pac bayes control} by noting that:
\begin{equation}
\KL(P' \| P_0') = \KL(P \| P_0) + \KL(P_\text{ext} \| P_\text{ext}) = \KL(P \| P_0).
\end{equation}
The first equality above follows from the fact that $w_\text{int}$ and $w_\text{ext}$ are independent. In order to evaluate the training cost $C_S(P') = \frac{1}{N} \sum_{E \in S} \underset{w \sim P'}{\EE} [C(r_w; E)]$, we can employ the sampling procedure described in Section \ref{sec:continuous policy space} (i.e., sampling the disturbances $w_\text{ext} \sim P_\text{ext}$ in a manner analogous to how $w$ was sampled in Section \ref{sec:continuous policy space}). Thus, the framework for the deterministic rollout setting can be applied with almost no modifications in order to handle the stochastic rollout case (as long as one can sample disturbances $w_\text{ext}$ and assuming that the disturbances are drawn independently of $w_\text{int}$).

\subsection{Distributionally-Robust Control Policies}
\label{sec:domain shift}

So far, we have assumed that the robot will be tested on environments that are drawn from the same distribution as the training environments. We will now address the setting where this assumption is not valid and learn \emph{distributionally-robust policies} (i.e., policies that are robust to changes in the distribution from which environments are drawn). We will assume that the distribution $\D'$ from which test environments are drawn is bounded in terms of an $f$-\emph{divergence} (see below) from the training distribution $\D$ and formulate a robust version of the PAC-Bayes bound already described.
\begin{definition}[$f$-divergence between $\D'$ and $\D$ \citep{Nguyen10}] For any convex $f(x)$ such that $f(1) = 0$, let
\begin{equation}
D_f(\D'||\D) := \underset{E\sim\D}{\EE}\Bigg{[}f\Bigg{(}\frac{\D'}{\D}\Bigg{)}\Bigg{]}.
\end{equation}
\end{definition}
The $f$-divergences encapsulate a broad class of divergences between distributions and include the KL divergence as a special case (with $f(x) = x\log x$).
We will assume that the test distribution $\D'$ is bounded in terms of an $f$-divergence: $D_f(\D'||\D) \leq \mathcal{B}$ (but no further assumption on $\D'$ will be made). The control policy we learn will have an associated guarantee on \emph{any} test distribution that satisfies this assumption.

\begin{theorem}[$f$-divergence between $\D'$ and $\D$ in terms of $f$ and $f^*$ \citep{Nguyen10}]
	 For a given $f(x)$ and its convex conjugate $f^*(y) \coloneqq \sup_{x \in \mathbb{R}} \big{[}xy - f(x)\big{]}$, we can write the $f$-divergence $D_f(\D'||\D)$ in terms of only $f$ and its conjugate:
\begin{equation}
D_f(\D'||\D) = \sup_{C:\Pi \times \mathcal{E} \rightarrow \mathbb{R}}\bigg{(}\underset{E \sim \D'}{\EE} \ \underset{w \sim P}{\EE} [C(r_w; E)]
 - \underset{E \sim \D}{\EE} \ \underset{w \sim P}{\EE} [f^*(C(r_w; E))]\bigg{)}.
\end{equation}
\end{theorem}

The supremum above is taken over all functions $C$ that result in the expectations in the RHS being finite. Thus, for any particular (cost) function $C$, we obtain a lower bound on the supremum term. This allows us to obtain the following useful corollary.
\begin{corollary} [$f$-Divergence variational inequality] If $D_f(\D'||\D) \leq \mathcal{B}$, then
\label{thm:fdiv var ineq}
\begin{equation}
C_{\D'}(P) := \underset{E \sim \D'}{\EE} \ \underset{w \sim P}{\EE} [C(r_w; E)] \leq \mathcal{B} + \underset{E \sim \D}{\EE} \ \underset{w \sim P}{\EE} [f^*(C(r_w; E))].
\end{equation}
\end{corollary}

Note that this corollary is valid for any $f$-divergence. In particular, it holds when $f(x) = x\log x$, making $f^*(y) = e^{y-1}$. With this choice of $f$, we obtain an upper bound on $C_{\D'}(P)$ in terms of the bound $\mathcal{B}$ on the KL divergence.

\begin{corollary} [KL divergence variational inequality]
\label{cor:kl var ineq}
For $f$-divergence with $f(x) = x\log x$ and $D_f(\D'||\D) \leq \mathcal{B}$, we have $D_f(\D'||\D) = \KL(\D'||\D) \leq \mathcal{B}$ and

\begin{equation}
\label{eq:kl var ineq}
C_{\D'}(P) \leq \mathcal{B} + \underset{E \sim \D}{\EE} \ \underset{w \sim P}{\EE} [e^{C(r_w; E)}] - 1.
\end{equation}
\end{corollary}
While Corollary \ref{cor:kl var ineq} provides a valid inequality in the special case of the KL divergence, a tighter bound can be obtained using the \emph{Donsker-Varadhan (DV)} inequality.

\begin{theorem}[Donsker-Varadhan variational inequality \citep{Donsker75}; Theorem 3.2 in \citep{Gray11}]
\label{thm:dv var ineq}
If $\KL(\D'||\D) \leq \mathcal{B}$, then
\begin{equation}
\label{eq:DK-ineq bound}
C_{\D'}(P) \leq \mathcal{B} + \log \Big{(}\underset{E \sim \D}{\EE} \ \underset{w \sim P}{\EE} [e^{C(r_w; E)}]\Big{)}.
\end{equation}
\end{theorem}

The DV inequality provides a tighter bound than inequality \eqref{eq:kl var ineq} since $x-1 \geq \log(x), \ \forall x > 0$. For the rest of this section, we will specialize our discussion to the KL divergence and use the DV inequality. However, we note that our approach generalizes to any $f$-Divergence by leveraging Corollary \ref{thm:fdiv var ineq}.

As written, inequality \eqref{eq:DK-ineq bound} cannot be used to directly upper bound $C_{\D'}(P)$ since \linebreak $\EE_{E \sim \D} \ \EE_{w \sim P} [e^{C(r_w; E)}]$ is not an observable quantity. However, we can leverage the inequality \eqref{eq:pac inequality control} to obtain an upper bound on $C_{\D'}(P)$ in terms of observable quantities. Since Theorem \ref{thm:pac bayes control} holds for any cost function between $0$ and $1$, we will be able to apply inequality \eqref{eq:pac inequality control} if we replace the cost with an exponentiated one, as long as we rescale to stay between 0 and 1. Thus, if we make the substitution
$$C(r_w; E) \leftarrow \frac{e^{C(r_w; E)}-1}{e-1},$$
we obtain the following bound using inequality \eqref{eq:pac inequality control}:
\begin{equation}
\label{eq:transformed pac-bayes bound}
\underset{E \sim \D}{\EE} \ \underset{w \sim P}{\EE} [e^{C(r_w; E)}] \leq \frac{1}{N} \sum_{E \in S} \underset{w \sim P}{\EE} [e^{C(r_w; E)}] + (e-1)\sqrt{\frac{\KL(P \| P_0) + \log(\frac{2 \sqrt{N}}{\delta})}{2N}}.
\end{equation}
This inequality holds because the transformation keeps the cost between $[0,1]$.
Now, since we assumed that $\KL(\D'||\D) \leq \mathcal{B}$, we can apply Theorem \ref{thm:dv var ineq} to bound $C_{\D'}(P)$.

\begin{corollary}[Distributionally-robust PAC-Bayes bound]
\label{cor:robust PAC-Bayes bound}
For any $\D'$ such that $\KL(\D'||D) \leq \mathcal{B}$ and any $\delta \in (0,1)$, with probability at least $1 - \delta$ over sampled environments $S \sim \D^N$ the following inequality holds:
\begin{equation}
\label{eq:cor robust PAC-Bayes bound}
C_{\D'}(P) \leq \mathcal{B} + \log \Bigg{(} \frac{1}{N} \sum_{E \in S} \underset{w \sim P}{\EE} [e^{C(r_w; E)}] + (e-1)\sqrt{\frac{\KL(P \| P_0) + \log(\frac{2 \sqrt{N}}{\delta})}{2N}} \Bigg{)}.
\end{equation}
\end{corollary}

The RHS of inequality \eqref{eq:cor robust PAC-Bayes bound} gives us an upper bound $C_{\text{PAC}'}$  on $C_{\D'}$. We can thus apply an analogous procedure to Algorithm \ref{a:pac bayes control} to obtain $P^\star_{\text{PAC}'}$ (a distributionally-robust stochastic policy) by minimizing this upper bound and $C^\star_{\text{bound}'}$ (the final distributionally-robust PAC-Bayes bound).

In the finite policy space setting, we can apply a procedure similar to the one employed in Section \ref{sec:finite policy space} to write an REP that minimizes the bound $C_{\text{PAC}'}$. Define:
\begin{equation}
	\hat{C}_e[i,j] \coloneqq e^{C(r_{\pi_j}; E_i)}.
\end{equation}
We then have
\begin{equation}
\frac{1}{N} \sum_{E \in S} \underset{\pi \sim P}{\EE} [e^{C(r_\pi; E)}] = \frac{1}{N} \sum_{i=1}^N \sum_{j=1}^L \hat{C}_e[i,j] p[j] := \bar{C}_ep.
\end{equation}
We can then minimize the bound $C_{\text{PAC}'}$ using an REP analogous to Problem \eqref{opt:pac bayes final}:
\begin{flalign}
		& & \underset{p \in \RR^L, \tau, \lambda}{\textrm{min}} \hspace*{1cm} & \tau &&
		  \\
		& & \text{s.t.} \hspace*{1cm} & \lambda^2 \geq (e-1)^2\frac{\KL(p \| p_0) + \log(\frac{2 \sqrt{N}}{\delta})}{2N} && \nonumber \\
		& & & \lambda = \tau - \bar{C}_ep, \ \lambda \geq 0 && \nonumber \\
		& & &  0 \leq p \leq 1, \ \sum_j p[j] = 1. \nonumber
\end{flalign}
Here $\bar{C}_ep \in [1, e]$, and since $\tau$ upper bounds the true expected cost, we are only interested in values of $\tau \in [1, e]$. Thus an optimal $\lambda$ can be found by searching over $\lambda \in [0, e-1]$, which can then be used to obtain $P_{\text{PAC}'}^\star$ and $C_{\text{PAC}'}$. Additionally, in the continuously-parameterized control policy space case, we can make modifications to Algorithm \ref{a:pac bayes SGD} and equations (\ref{eq:contcase expectation over controllers}$-$\ref{eq:contcase KL inv}) to directly adapt the SGD approach to minimize $C_{\text{PAC}'}$.

Finally, as in Section \ref{sec:computing pac-bayes controllers}, the final distributionally-robust upper bound  $C^\star_{\text{bound}'}$ is not computed as $C_{\text{PAC}'}(P^\star_{\text{PAC}'})$ but with an analogue to the KL inverse in equation \eqref{eq:pac bayes learning kl version}:
\begin{flalign}
		& & \underset{c_{\D'}, c_\D \in [0,1]}{\textrm{max}} \hspace*{1cm} & c_{\D'} && \\
		& & \text{s.t.} \hspace*{1cm} & \KL(C_S(P)||c_\D) \leq \frac{\KL(P \| P_0) + \log(\frac{2 \sqrt{N}}{\delta})}{N} && \nonumber \\
		& & & \KL(c_{\D'}||c_\D) \leq \mathcal{B}.  \nonumber
\end{flalign}
The first constraint is the same as in the non-robust case, and the second constraint accounts for the difference in the training and test distributions. Together these create an REP that can be solved to find $C^\star_{\text{bound}'}$.

config: arxiv


TITLE: If $m$ and $n$ are integers with $\gcd(m,n) = 1$, prove that $\sigma(mn)= \sigma(m)\sigma(n)$.
QUESTION [1 upvotes]: If $m$ and $n$ are integers with $\gcd(m,n) = 1$, prove that $\sigma(mn)= \sigma(m)\sigma(n)$. I am thinking about using the formula for $\sigma(p^k)$ where $p$ is prime.
It follows from the geometric series that
$$\sigma(p^k) = 1 + p + p^2 +\cdots+p^k = \frac{p^{k+1}-1}{p-1}$$
Any Help?

REPLY [0 votes]: Let $m = p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ and $n = q_1^{d_1}q_2^{d_2}...q_i^{d_i}$ for $p, q$ prime and $e, d \in \mathbb{Z}$.
Then we have
$$\sigma(mn) = \sigma(p_1^{e_1}p_2^{e_2}...p_k^{e_k}*q_1^{d_1}q_2^{d_2}...q_i^{d_i})$$
We know the divisors of $p_1^{e_1}$ are $1, p_1, p_1^2, p_1^3,..., p_1^{e_1}.$ This holds for any prime to any power.
Since the gcd$(m,n) = 1$, we also know $m$ and $n$ share no common divisors
So we have $$=[1+p_1+p_1^2+...+p_1^{e_1}][1+p_2+p_2^2+...+p_2^{e_2}]...[1+p_k+p_k^2+...+p_k^{e_k}] * [1+q_1+q_1^2+...+q_1^{d_1}][1+q_2+q_2^2+...+q_2^{d_2}]...[1+q_i+q_i^2+...+q_i^{d_i}]$$
The first half of the statement above is just the divisors of $m$ and the second half is the divisors of $n$
Then
$$= \sigma(m)\sigma(n) $$
The proof is bidirectional


TITLE: How to solve this equation $2.02(T+25)^{5/4}+1.15*10^{-8} (T+273)^4=203.54$
QUESTION [0 upvotes]: How to solve this equation
$2.02(T+25)^{5/4}+1.15*10^{-8} (T+273)^4=203.54$
Here the value of T may lie between -100 to 100 .
The binomial expansion is tedious.

REPLY [0 votes]: If you plot the function, you identify that the solution is close to zero. Then, use Newton's method. Starting from 0, the iterates will successively be 4.06274, 3.98468, 3.98466, 3.98466.


TITLE: Elliptic Curve: point of prime order $n$
QUESTION [1 upvotes]: I am attempting to understand ElGamal Encryption on Elliptic Curves (beginnning of this paper)
I have an elliptic curve $E\left( \mathbb{F}_{11} \right)$ defined by $y^2 = x^3 + 2x + 3$
I need to choose a point $P$ on $E\left(\mathbb{F}_{11}\right)$ of prime order $n$
I haven't been able to find anything to help me understand what is meant by 'prime order $n$'. Can someone point me in the direction of anything useful or explain to me how to find such a point $P$ please?
I am a computer scientist, so assume limited knowledge of any complex maths/theorems - all my knowledge on elliptic curves is self-taught through Googling...

REPLY [0 votes]: I do not have enough reputation to comment, so I answer it here.
The comment says:

Thanks, this makes sense, but if I choose $P=(4,3)$ then I get $\#E=3$ as $3P=\mathcal{O}$, What made you choose P=(2,2)?

Because not all points are generators, however, only the generator point can generate all the points on the curve.
So, $P=(2,2)$ is one generator, but  $P=(4,3)$ is not.
you can read this :       https://www.entrust.com/wp-content/uploads/2014/03/WP_Entrust_Zero-to-ECC_March2014.pdf


text:
\section{Proofs of Main Results}
{The proofs of the main theorems are given next. \ifarxiv Proofs \else Due to space limitations, proofs \fi of auxiliary lemmas are deferred to the \ifarxiv appendix.\else supplementary material~\cite{faradonbeh2018optimality}.\fi}
\subsection{\bf Proof of Theorem \ref{RegretTheorem} and Corollary~\ref{RegretCorollary}}
Given $n \geq 1$, and the linear policy $\policy= \left\{ L_t \right\}_{t=0}^{n-1}$, define the sequence of policies $\policy_0, \cdots, \policy_n$ as follows.
\begin{eqnarray*}
	\policy_0 &=& \left\{ \Lmatrix{\para_0}, \cdots, \Lmatrix{\para_0} \right\}, \\
	\policy_1 &=& \left\{ L_0, \Lmatrix{\para_0}, \cdots, \Lmatrix{\para_0} \right\}, \\
	\vdots \\
	\policy_n &=& \left\{ L_0, L_1, \cdots, L_{n-1} \right\}.
\end{eqnarray*}
Indeed, the policy $\policy_i$ applies the same feedback as $\policy$ at every time $t<i$, and then for $t \geq i$ switches to the optimal policy $\optimalpolicy$. Clearly, $\policy_0=\optimalpolicy$, and $\policy_n=\policy$. Since
\begin{eqnarray} \label{regretdecompos1}
\regret{n}{\policy}= \sum\limits_{k=1}^{n} \sum\limits_{t=0}^{n-1} \left[\instantcost{t}{\policy_k}-\instantcost{t}{\policy_{k-1}}\right],
\end{eqnarray}
it suffices to find $\instantcost{t}{\policy_k}-\instantcost{t}{\policy_{k-1}}$, for $1 \leq k \leq n$, and $0 \leq t \leq n-1$. Fixing $k$, let $\left\{ x(t) \right\}_{t=0}^{n-1}, \left\{ y(t) \right\}_{t=0}^{n-1}$ be the state trajectories under $\policy_k, \policy_{k-1}$, respectively. So, letting $\Dnut=A_0+B_0 \Lmatrix{\para_0}$ and $D_{k-1}=A_0+B_0 L_{k-1}$, we have $x(t) = y(t)$ for $0 \leq t \leq k-1$, as well as $\instantcost{t}{\policy_k} = \instantcost{t}{\policy_{k-1}}$ for $0 \leq t \leq k-2$, and $x(k) = D_{k-1} x(k-1) + w(k)$. Further, if $k \leq t \leq n-1$, then
\begin{eqnarray*}
	y(t) &=& \Dnut^{t-k+1} x(k-1) + \sum\limits_{j=k}^t \Dnut^{t-j} w(j) ,\\
	x(t) &=& \Dnut^{t-k} D_{k-1} x(k-1) + \sum\limits_{j=k}^t \Dnut^{t-j} w(j) .
\end{eqnarray*}
Therefore, we have $x(t)=y(t) + \Dnut^{t-k} \Delta_{k-1} x(k-1)$, for $k \leq t < n$, where
\begin{eqnarray*}
	\Delta_{k-1} = D_{k-1}-\Dnut = B_0 \left(L_{k-1} - \Lmatrix{\para_0}\right).
\end{eqnarray*}
Thus, for we obtain
\begin{eqnarray*}
	&& \instantcost{k-1}{\policy_k}- \instantcost{k-1}{\policy_{k-1}} \\
	&=&x(k-1)' \left( L_{k-1}' R L_{k-1} - \Lmatrix{\para_0}' R \Lmatrix{\para_0} \right) x(k-1).
\end{eqnarray*}
Similarly, denote $P_0=Q+\Lmatrix{\para_0}' R \Lmatrix{\para_0}$, and replace for $x(t)$ to see that if $k \leq t <n$, then
\begin{eqnarray*}
	&&\instantcost{t}{\policy_k}-\instantcost{t}{\policy_{k-1}}\\
	&=&\left( 2 y(t)
	+ \Dnut^{t-k}\Delta_{k-1}x(k-1) \right)' P_0 \Dnut^{t-k} \Delta_{k-1} x(k-1).
\end{eqnarray*}
To proceed, plug-in for $y(t)$ to get \\$\instantcost{t}{\policy_k}-\instantcost{t}{\policy_{k-1}}= x(k-1)' F_{k-1}(t) x(k-1) + \eta_{k-1}(t)$, where $\Delta_{k-1}=D_{k-1}-\Dnut$ leads to
\begin{eqnarray*}
	\eta_{k-1}(t) &=& 2 x(k-1)' \Delta_{k-1}' {\Dnut'}^{t-k}P_0 \sum\limits_{j=k}^t \Dnut^{t-j} w(j), \\
	F_{k-1}(t) &=& D_{k-1}' {\Dnut'}^{t-k}P_0 \Dnut^{t-k} D_{k-1} -  {\Dnut'}^{t-k+1}P_0 \Dnut^{t-k+1}.
\end{eqnarray*}
Next, letting $z_k=\sum\limits_{t=k}^{n-1} \eta_{k-1}(t)$, and \\$G_{k}= L_{k-1}' R L_{k-1} - \Lmatrix{\para_0}' R \Lmatrix{\para_0} + \sum\limits_{t=k}^{n-1} F_{k-1}(t)$,
\\clearly
\begin{eqnarray} \label{regretdecompos1.5}
\sum\limits_{t=0}^{n-1} \left[\instantcost{t}{\policy_k}-\instantcost{t}{\policy_{k-1}}\right]=x(k-1)' G_{k} x(k-1) + z_{k}. \:\:\:\:\:\:
\end{eqnarray}

To proceed, for $0 \leq j \leq n$ let $K_j = \sum\limits_{\ell=n-j}^{\infty}  {\Dnut'}^{\ell} P_0 \Dnut^{\ell}$. So,
\begin{eqnarray*}
	\sum\limits_{t=k}^{n-1} F_{k-1}(t)= D_{k-1}' \left(K_n - K_k \right) D_{k-1} - \Dnut' \left(K_n - K_k \right) \Dnut
\end{eqnarray*}
implies $G_{k} = E_k + H_k$, where
\begin{eqnarray*}
	E_k &=& - D_{k-1}' K_k D_{k-1} + \Dnut' K_k \Dnut,\\
	H_k &=&  L_{k-1}' R L_{k-1} - \Lmatrix{\para_0}' R \Lmatrix{\para_0} \\
	&-& \Dnut' K_n \Dnut+ D_{k-1}' K_n D_{k-1}.
\end{eqnarray*}
The Lyapunov equation (see~\cite{faradonbeh2018stabilization})
\begin{eqnarray} \label{lyapunoveq}
\Kmatrix{\para_0}- \Dnut' \Kmatrix{\para_0} \Dnut = P_0,
\end{eqnarray}
leads to $K_n=\Kmatrix{\para_0}$. Thus, letting $X = L_{k-1} - \Lmatrix{\para_0}$, $ M = B_0'\Kmatrix{\para_0}B_0+R$, since $M\Lmatrix{\para_0}= - B_0' \Kmatrix{\para_0} A_0$, after doing some algebra we get
\begin{eqnarray*}
	H_k &=& \Lmatrix{\para_0}' R X + X' R \Lmatrix{\para_0} + \Dnut' \Kmatrix{\para_0} B_0 X \\
	&+& X' R X + X' B_0' \Kmatrix{\para_0} \Dnut + X' B_0' \Kmatrix{\para_0} B_0 X \\
	&=& X'MX
\end{eqnarray*}
Hence, adding up the terms in~\eqref{regretdecompos1.5}, \eqref{regretdecompos1} implies that
\begin{eqnarray} \label{regretdecompos2}
\regret{n}{\policy} = Z_n+ S_n+ T_n ,
\end{eqnarray}
where $Z_n = \sum\limits_{k=1}^{n}z_k$, $S_n = \sum\limits_{k=0}^{n-1} x(k)'E_{k+1}x(k)$, and $T_n = \sum\limits_{k=0}^{n-1} \norm{M^{1/2}\left(L_{k}-\Lmatrix{\para_0}\right) x(k)}{}^2$. In order to investigate $S_n$, we use the dynamics $x(k)=D_{k-1}x(k-1)+ w(k)$, as well as $\Dnut' K_{k+1} \Dnut = K_{k}$, to get
\begin{eqnarray*}
	x(k)' \Dnut' &K_{k+1}& \Dnut x(k) = x(k-1)' D_{k-1}' K_{k} D_{k-1} x(k-1) \\
	&+& w(k)' K_{k} w(k) + 2 w(k)' K_{k} D_{k-1} x(k-1),
\end{eqnarray*}
for $0 < k < n$. Substituting in the expression for $S_n$, and denoting $w(0)=x(0)$, the telescopic differences vanish:
\begin{eqnarray}
&& S_n + x(n)' \Kmatrix{\para_0} x(n) \notag \\
&=& \sum\limits_{k=0}^{n-1} 2 w(k+1)' K_{k+1} D_{k} x(k)+ \sum\limits_{k=0}^{n} w(k)' K_{k} w(k). \label{Sdecompos}
\end{eqnarray}
Plugging
\begin{eqnarray*}
	D_kx(k)= \sum\limits_{j=0}^k \left( \Dnut^{j+1} w(k-j) + \Dnut^j \Delta_{k-j} x(k-j) \right),
\end{eqnarray*}
as well as $\optstate{n}=\sum\limits_{j=0}^n \Dnut^{n-j}w(j)$, in~\eqref{Sdecompos}, we have $\widetilde{S}_n=S_n + x(n)' K_n x(n) - \optstate{n}' K_n \optstate{n} = \sum\limits_{k=1}^{n} w(k)' K_{k} \xi_{k}$, where
$\xi_{k} = 2 \sum\limits_{\ell=1}^{k} \Dnut^{\ell-1} \Delta_{k-\ell} x(k-\ell)$. Moreover, it is straightforward to show that $Z_n = \sum\limits_{j=1}^{n-1} \zeta_j' w(j)$, where
\\$\zeta_j = 2 \sum\limits_{\ell=1}^{j} \sum\limits_{t=j}^{n-1}  {\Dnut'}^{t-j} P_0 \Dnut^{t-\ell} \Delta_{\ell-1} x(\ell-1)$.
\\Hence, $\zeta_j = \left(K_n - K_j \right) \xi_j$ implies $\widetilde{S}_n+Z_n = \sum\limits_{k=1}^{n} w(k)' K_n \xi_{k}$. Next, we use the following lemma.

\begin{lemm} \cite{lai1982least} \label{Lailemma2}
	Suppose that for all $t \geq 0$, $y(t+1),v(t)$ are $\mathcal{G}_t$ measurable, $\mathcal{G}_t \subseteq \mathcal{G}_{t+1}$, and $\E{v(t+1) | \mathcal{G}_{t}}=0$. Define the martingale $\psi_n = \sum\limits_{t=1}^{n} y(t)'v(t)$, and let $\varphi_n = \sum\limits_{t=1}^{n} \norm{y(t)}{}^2$. If $\sup\limits_{t \geq 0} \E{\norm{v(t+1)}{}^2 \Big| \mathcal{G}_t} < \infty$, then
	\begin{eqnarray*}
		\limsup\limits_{n \to \infty}\left| \psi_n \right| < \infty &\:\:\:\:\text{on}\:\:\:\:& \varphi_\infty < \infty, \\
		\limsup\limits_{n \to \infty} \frac{\psi_n}{\varphi_n^{1/2} \log \varphi_n} = 0 &\:\:\:\:\text{on}\:\:\:\:& \varphi_\infty = \infty.
	\end{eqnarray*}
\end{lemm}
Taking $\mathcal{G}_t = \sigma \left( \left\{ w(i) \right\}_{i=1}^t , \left\{ x(i) \right\}_{i=0}^t \right)$, and $v(t)=w(t)$, $y(t)=\xi_t$, we can use Lemma~\ref{Lailemma2} since Assumption~\ref{momentcondition} holds. So, stability of $\Dnut$ (Proposition~\ref{stabilizable}), and $\eigmin{M}>0$, lead to $\sum\limits_{k=1}^n \norm{\xi_k}{}^2 = \order{T_n}$. Thus, by~\eqref{regretdecompos2}, we get the desired result since $\widetilde{S}_n+Z_n = \order{T_n^{1/2} \log T_n}$.

Next, the first statement in Corollary~\ref{RegretCorollary} follows from Theorem 1 in the work of Lai and Wei~\cite{lai1985asymptotic}. To prove the second result, first observe that $S_\infty=S_n$, $T_\infty=T_n$, and $Z_\infty=Z_n$. Furthermore, note that for $t \geq n$ we have $\instantcost{t}{\policy} = x(t)' P_0 x(t)$, $\instantcost{t}{\optimalpolicy}= \optstate{t}' P_0 \optstate{t}$, as well as
\begin{eqnarray*}
	x(t) &=& \Dnut^{t-n} x(n)+ \sum\limits_{j=n+1}^t \Dnut^{t-j} w(j),\\
	\optstate{t} &=& \Dnut^{t-n} \optstate{n}+ \sum\limits_{j=n+1}^t \Dnut^{t-j} w(j).
\end{eqnarray*}
So, letting
\begin{eqnarray*}
	\delta_n=2\sum\limits_{t=n}^\infty \sum\limits_{j=n+1}^t  \left(x(n)-\optstate{n}\right)' {\Dnut'}^{t-n} P_0 \Dnut^{t-j} w(j),
\end{eqnarray*}
by~\eqref{lyapunoveq} the following holds:
\begin{eqnarray*}
	\sum\limits_{t=n}^{\infty} \left[\instantcost{t}{\policy}-\instantcost{t}{\optimalpolicy}\right] = x(n)'K_n x(n)-\optstate{n}'K_n \optstate{n}+ \delta_n.
\end{eqnarray*}
Finally, \begin{eqnarray} \label{fluctuationmagnitude}
\norm{x(n)-\optstate{n}}{}^2= \norm{\sum\limits_{j=0}^{n-1} \Dnut^{n-1-j} \Delta_{j} x(j)}{}^2 = \order{T_n}, \:\:\:\:\:\:
\end{eqnarray}
together with Lemma~\ref{Lailemma2} imply $\delta_n=\order{T_n^{1/2} \log T_n}$.

\subsection{\bf Proof of Theorem \ref{adaptiveconsistency}}

	First, for an arbitrary $\para \in \mathcal{P}_0$, since $\para \in \nullspace{\para_0}$, we have
	\begin{eqnarray} \label{eqnullspace}
	A+B \Lmatrix{\para_0} = A_0+ B_0 \Lmatrix{\para_0} = D_0.
	\end{eqnarray}
	Next, for an arbitrary fixed unit matrix (in the Frobenius norm) $X \in \R^{r \times p}$, let $L = \Lmatrix{\para_0} + \epsilon X$ be a linear feedback matrix which stabilizes the system of dynamics parameters $\para$. Note that according to Proposition \ref{stabilizable}, $\para \in \levelset{\para_0}$ leads to $\eigmax{\para \extendedLmatrix{\para_0}} <1$. Thus, $\eigmax{A+BL}<1$, as long as $\epsilon$ is sufficiently small.

	Then, applying $L$ to the system $\para$, we get $\avecost{\para}{L} = \tr{ P \left( \epsilon \right) C }$, where $P \left( \epsilon \right)$ is the unique solution of the Lyapunov equation
	\begin{eqnarray} \label{generallyapunoveq}
	P \left( \epsilon \right) - \left( A+B L \right)' P \left( \epsilon \right) \left( A+B L \right) = Q + L'R L.
	\end{eqnarray}
	Note that according to \eqref{lyapunoveq} and \eqref{eqnullspace}, it holds that $P \left( 0 \right) = \Kmatrix{\para_0}$. Letting $\Delta \left( X \right) = \lim\limits_{\epsilon \to 0} {\epsilon}^{-1} \left({ P \left( \epsilon \right) - P \left( 0 \right)}\right)$,
	\eqref{generallyapunoveq} leads to
	\begin{eqnarray} \label{derivativelyapunov}
	\Delta \left( X \right) - D_0' \Delta \left( X \right) D_0= X'N+ N'X,
	\end{eqnarray}
	where $N=R \Lmatrix{\para_0}+ B' \Kmatrix{\para_0} D_0$. Next, $\para \in \levelset{\para_0}$ implies that $\Lmatrix{\para_0}$ is an optimal linear feedback for the system of dynamics parameter $\para$. So, the directional derivative of $\avecost{\para}{L}$ with respect to $L$ is zero in all directions. In the direction of $X$, the derivative is $\tr{ \Delta \left( X \right) C}$. Since all above statements hold regardless of the positive definite matrix $C$, \eqref{derivativelyapunov} and  $\tr{ \Delta \left( X \right) C}=0$ imply $N=0$;
	\begin{eqnarray} \label{conditionconsistency1}
	D_0' \Kmatrix{\para_0} B = - \Lmatrix{\para_0}' R.
	\end{eqnarray}
	Therefore, \eqref{conditionconsistency1} is a necessary condition for $\para \in \mathcal{P}_0$. Note that according to \eqref{lyapunoveq} and \eqref{eqnullspace}, the necessary condition \eqref{conditionconsistency1} implies the necessity of $D_0' \Kmatrix{\para_0} A = \Kmatrix{\para_0} - Q$.

	Further, for every input matrix $B$ which satisfies \eqref{conditionconsistency1}, the transition matrix $A$ will be uniquely determined by \eqref{eqnullspace} as $A=D_0-B\Lmatrix{\para_0}$.

	Conversely, suppose that $B$ is an arbitrary matrix which satisfies \eqref{conditionconsistency1}. Letting $A=D_0-B\Lmatrix{\para_0}$, we show that $\left[A,B\right] = \para \in \mathcal{P}_0$. For this purpose, since the above definition of $A$ automatically leads to $\para \in \nullspace{\para_0}$, it suffices to show $\para \in \levelset{\para_0}$. Writing $Y = B-B_0$, we get $A=A_0-Y \Lmatrix{\para_0}$. Moreover, define $G = A' \Kmatrix{\para_0} A$, $H = B' \Kmatrix{\para_0} A$, $M = B_0' \Kmatrix{\para_0} B_0 +R$, and $S = B_0' \Kmatrix{\para_0}Y + Y' \Kmatrix{\para_0} B_0 + Y' \Kmatrix{\para_0} Y$. Then, we calculate the matrix
	\begin{eqnarray*}
		V &=& Q + G - H' \left( M+S \right)^{-1} H = Q + A'\Kmatrix{\para_0}A \\
		&-& A' \Kmatrix{\para_0}B \left(B'\Kmatrix{\para_0}B+R\right)^{-1} B'\Kmatrix{\para_0}A.
	\end{eqnarray*}
Writing $A,B,G,H$ in terms of $A_0,B_0,M,S,Y$, we have
\begin{eqnarray*}
	V = Q &+& A_0' \Kmatrix{\para_0} A_0 + \Lmatrix{\para_0}' S \Lmatrix{\para_0} - \big[ B_0' \Kmatrix{\para_0} A_0 \\
	&-& S \Lmatrix{\para_0} \big]' \left( M+S \right)^{-1} \big[ B_0' \Kmatrix{\para_0} A_0 - S \Lmatrix{\para_0} \big]
\end{eqnarray*}
Then, using $\left( M+S \right)^{-1} = M^{-1} - \left( M+S \right)^{-1} S M^{-1}$, \eqref{ricatti2}, and $M \Lmatrix{\para_0} = -B_0' \Kmatrix{\para_0} A_0$, $V$ can be written as $V = \Kmatrix{\para_0} + \Lmatrix{\para_0}' S W$, where
\begin{eqnarray*}
W &=& \Lmatrix{\para_0}
- \left( M+S \right)^{-1} \left(S \Lmatrix{\para_0}
+ B_0' \Kmatrix{\para_0} A_0\right) \\
&=& \Lmatrix{\para_0} - \left( M+S \right)^{-1} \left(S +M \right) \Lmatrix{\para_0} = 0 ;
\end{eqnarray*}
i.e., $V= \Kmatrix{\para_0}$ is a solution of the Riccati equation \eqref{ricatti2} for $\para$. According to Proposition \ref{stabilizable}, the solution is unique; which is $\Kmatrix{\para} = \Kmatrix{\para_0}$. Moreover, $\Lmatrix{\para} = - \left( M + S \right)^{-1} H = \Lmatrix{\para_0}$

	shows that $\para \in \levelset{\para_0}$. So far, we have shown that $\para \in \mathcal{P}_0$, if and only if \eqref{eqnullspace} and \eqref{conditionconsistency1} hold. Next, \eqref{conditionconsistency1} is essentially stating that every column of $B-B_0$ (which is a vector in $\R^p$), is orthogonal to the all columns of $\Kmatrix{\para_0} D_0$. This verifies that \eqref{conditionconsistency1} specifies a shifted linear subspace. To find the dimension, since $B$ has $r$ columns, and \eqref{eqnullspace} uniquely determines $A$ in terms of $B$, we get $\dimension{}{\mathcal{P}_0} = \left( p - \rank{\Kmatrix{\para_0} D_0}\right)r$.

	Finally, by positive definiteness of $Q$, \eqref{lyapunoveq} implies $\rank{\Kmatrix{\para_0}}=p$. Further, since $D_0= \left[ I_p - B_0 M^{-1} B_0' \Kmatrix{\para_0} \right] A_0$,

	it suffices to show
	\begin{eqnarray} \label{rankequivalence}
	\rank{I_p - B_0 M^{-1} B_0' \Kmatrix{\para_0}}=p.
	\end{eqnarray}
	If \eqref{rankequivalence} does not hold, there exists $v \in \R^p$ such that $v \neq 0$ and $v = B_0 M^{-1} B_0' \Kmatrix{\para_0} v$. So, $v= B_0 \widetilde{v}$ where $\widetilde{v} = M^{-1} B_0' \Kmatrix{\para_0} v \in \R^r$. Thus,
	\begin{eqnarray*}
	B_0' \Kmatrix{\para_0} B_0 \widetilde{v} = B_0' \Kmatrix{\para_0} v = M \widetilde{v}
	= \left[ B_0' \Kmatrix{\para_0} B_0 + R \right] \widetilde{v},
	\end{eqnarray*}
	or equivalently, $R \widetilde{v}=0$. Positive definiteness of $R$ implies that $\widetilde{v}=0$, which contradicts $B_0 \widetilde{v} \neq 0$. This proves \eqref{rankequivalence}, which completes the proof.

\subsection{\bf Proof of Theorem \ref{RCETheorem}}

	The proof is based on a sequence of intermediate results. First, for $i\geq 1$, let $V_i$ be the (unnormalized) state covariance during the $i$-th episode: $V_i = \sum\limits_{t=\lfloor \rrate^{i-1} \rfloor}^{\lfloor \rrate^{i} \rfloor -1} x(t) x(t)'$.
	\begin{lemm} \label{smallcov}
		For the matrix $V_i$ defined above, the followings hold: $\eigmax{V_m} = \order{\rrate^m}$, $\liminf\limits_{m \to \infty} \rrate^{-m} \eigmin{V_m} \geq \left( \rrate - 1 \right) \eigmin{C}$.
	\end{lemm}
	Then, in order to study the behavior of the least-squares estimate in \eqref{RCEalgo}, define
	\begin{eqnarray*}
		U_i = \sum\limits_{t=0}^{\lfloor \rrate^{i} \rfloor -1} \extendedLmatrix{\estpara{t}} x(t) x(t)' \extendedLmatrix{\estpara{t}}'.
	\end{eqnarray*}
	Note that since the parameter $\estpara{t}$ remains set (not changing) during each episode, $U_i$ can be written in terms of $V_1, \cdots, V_i$ as follows. First, for all $\lfloor \rrate^{i-1} \rfloor \leq t \leq \lfloor \rrate^{i} \rfloor -1$, the parameter estimate $\estpara{t}$ does not change. So, if $t$ belongs to the $i$-th episode, define the linear feedback matrix is $L_i= \Lmatrix{\estpara{t}}$. Letting $\widetilde{L}_{i}=\extendedLmatrix{\estpara{t}}$, we have $U_i = \sum\limits_{j=1}^i \widetilde{L}_{j} V_j \widetilde{L}_{j}'$. Then, the smallest eigenvalue of $U_i$ follows a different lower bound compared to that of $V_i$:
	\begin{lemm} \label{bigcov}
		Define $U_m$ as above. Then, we have $\liminf\limits_{m \to \infty} {\rrate^{-m/2}} \eigmin{U_m} >0 $, and $\eigmax{U_m} = \order{\rrate^{m}}$.
	\end{lemm}
	Next, the following result states that the estimation accuracy is determined by the eigenvalues of $U_i$.
	\begin{lemm} \cite{lai1982least} \label{Lailemma}
		For $n = \lfloor \rrate^m \rfloor$, define $\estpara{n}, \optpara{n}$ according to \eqref{RCEalgo}. Then, we have
		\begin{eqnarray*}
			\Mnorm{\estpara{n} - \optpara{n} - \para_0}{}^2 = \order{ \frac{\log \eigmax{U_m}}{\eigmin{U_m}}}.
		\end{eqnarray*}
	\end{lemm}
	Therefore, Lemma \ref{bigcov} leads to $\Mnorm{\estpara{n}-\optpara{n}-\para_0}{}= \order{n^{-1/4} \log^{1/2}n }$. Using the moment condition in Remark \ref{perturbmomentcondition}, Markov's inequality gives $\PP{\Mnorm{\randommatrix_m}{}> m^{1/4}} = \order{m^{-1-\epsilon/4}}$. Thus, an application of the Borel-Cantelli Lemma leads to $\Mnorm{\randommatrix_m}{}= \order{m^{1/4}}$; i.e., $\Mnorm{\optpara{n}}{} = \order{n^{-1/4} \log^{1/2}n }$. So, we get the desired result about the identification rate: $\Mnorm{\estpara{n}-\para_0}{}= \order{n^{-1/4} \log^{1/2}n }$. To proceed, we present the following auxiliary result which shows that a similar rate holds for the deviations from the optimal linear feedback.
	\begin{lemm} \cite{faradonbeh2017finite} \label{Lipschitzlemma}
		There exist $0<\epsilon_0,\Lipschitz{L}<\infty$, such that for all stabilizable $\para$ satisfying $\Mnorm{\para-\para_0}{}< \epsilon_0$, the following holds: $\Mnorm{\Lmatrix{\para}-\Lmatrix{\para_0}}{} \leq \Lipschitz{L} \Mnorm{\para-\para_0}{}$.
	\end{lemm}
	So, utilizing Lemma~\ref{Lipschitzlemma}, we have
	\begin{eqnarray} \label{randomaccuracy}
	\Mnorm{\Lmatrix{\estpara{n}}-\Lmatrix{\para_0}}{} = \order{ \frac{\log^{1/2}n}{n^{1/4}} + \Mnorm{\optpara{n}}{}}.
	\end{eqnarray}
	On the other hand, since the policy is not being updated during each episode, we can write down the regret in terms of the matrices $V_i$. Henceforth in the proof, suppose that the time $n$ belongs to the $m$-th episode: $\lfloor \rrate^{m-1} \rfloor \leq n < \lfloor \rrate^m \rfloor$. Then, applying Theorem~\ref{RegretTheorem} and Corollary~\ref{RegretCorollary}, we get
	\begin{eqnarray*}
		\regret{n}{\adaptivepolicy} &=& \order{\sum\limits_{i=0}^{m} \left(L_i - \Lmatrix{\para_0} \right) V_i \left(L_i - \Lmatrix{\para_0} \right)' + \rrate^{m/2}}  \\
		&=& \order{\sum\limits_{i=0}^{m} \rrate^i \Mnorm{L_i-\Lmatrix{\para_0}}{}^2 +\rrate^{m/2}},
	\end{eqnarray*}
	where in the last equality above we applied Lemma \ref{smallcov}. Based on the definition of the perturbation $\optpara{n}$ in terms of the random matrix $\randommatrix_m$, define
	\begin{eqnarray*}
		S_m = \sum\limits_{i=0}^{m} i^{1/2}\rrate^{i/2} \Mnorm{\randommatrix_i}{}^2, \:\:\:\:\:\:
		T_m = \sum\limits_{i=0}^{m} i^{3/4}\rrate^{i/2} \Mnorm{\randommatrix_i}{} .
	\end{eqnarray*}
	So, by \eqref{randomaccuracy}, the regret is in magnitude dominated by $S_m,T_m$, and $m \rrate^m/2$: $\regret{n}{\adaptivepolicy} = \order{S_m+T_m+ m \rrate^{m/2}}$. Note that as $m$ and $n$ grow, the magnitudes of $n^{1/2} \log n$ and $m \rrate^{m/2}$ is the same. Finally, the following lemma leads to the desired result:
	\begin{lemm} \label{stochterm}
		For the terms $S_m,T_m$ defined above the followings hold: $S_m = \order{m \rrate^{m/2}}$, $T_m = \order{m \rrate^{m/2}}$.
	\end{lemm}

\subsection{\bf Proof of Theorem \ref{TSTheorem}}

	In this proof, we use the following result.
	\begin{lemm} \label{TScov}
		For the matrix $\Sigma_m$ defined in \eqref{TSalgo2} we have $\liminf\limits_{m \to \infty} \rrate^{-m/2} m^{1/2} \eigmin{\Sigma_m} >0$, $\eigmax{\Sigma_m} = \order{\rrate^{m}}$.
	\end{lemm}
	Hence, since $\mu_m$ is the least-squares estimate, and $\Sigma_m$ is the unnormalized empirical covariance matrix, Lemma~\ref{Lailemma} leads to $\Mnorm{\mu_m-\para_0}{}= \order{\rrate^{-m/4} m^{} }$. Then, because every row of $\estpara{\lfloor \rrate^m \rfloor}-\mu_m$ is a mean zero Gaussian with covariance matrix $\Sigma_m^{-1}$, by Lemma \ref{TScov} we have
	\begin{eqnarray*}
		\sum\limits_{m=0}^\infty \PP{\Mnorm{\estpara{\lfloor \rrate^m \rfloor}-\mu_m}{}> \rrate^{-m/4} m^{}} < \infty.
	\end{eqnarray*}
	Thus, Borel-Cantelli Lemma leads to the desired result about the identification rate: $\Mnorm{\estpara{\lfloor \rrate^m \rfloor}-\para_0}{}= \order{\rrate^{-m/4} m^{} }$. By Lemma \ref{Lipschitzlemma}, a similar rate holds for the linear feedbacks: $\Mnorm{\Lmatrix{\estpara{\lfloor \rrate^m \rfloor}}-\Lmatrix{\para_0}}{}= \order{\rrate^{-m/4} m^{} }$. Finally, plugging in the expression of Theorem~\ref{RegretTheorem}, and utilizing Corollary~\ref{RegretCorollary}, we get the desired result for the regret:
	\begin{eqnarray*}
		\regret{\lfloor \rrate^m \rfloor}{\adaptivepolicy} &=& \order{\sum\limits_{i=0}^{m} \rrate^i \Mnorm{\Lmatrix{\estpara{\lfloor \rrate^m \rfloor }}-\Lmatrix{\para_0}}{}^2 + \rrate^{m/2}} \\
		&=& \order{\sum\limits_{i=0}^{m} \rrate^{i/2} i^2}
		= \order{ \rrate^{m/2} m^2} .
	\end{eqnarray*}

\subsection{\bf Proof of Theorem \ref{GCETheorem}}

	Define $V_i,U_i,L_i,\widetilde{L}_i$ similar to the proof of Theorem \ref{RCETheorem}. Further, for $i\geq 1$, let $n_i= \lfloor \rrate^i \rfloor -1 $ be the end time of episode $i$, and denote
	\\$\loss{i}{}{\para} = \sum\limits_{t=0}^{n_i-1} \norm{x(t+1)- \para \extendedLmatrix{\estpara{t}} x(t)}{}^2$.
	\\Letting $\para_\star= \arg\min\limits_{\para \in \R^{p \times q}} \loss{i}{}{\para}$ for a fixed $i$, it is straightforward to show that
	\begin{eqnarray*}
		\loss{i}{}{\para}= \tr{\left( \para- \para_\star\right) U_i \left( \para- \para_\star\right)' } - \tr{ \para_\star U_i \para_\star'}.
	\end{eqnarray*}
	Therefore, since $\para_0 \in \tempparaspace{0}$, \eqref{GCEalgo} implies that $\loss{i}{}{\estpara{n_i}-\optpara{n_i}} \leq \loss{i}{}{\para_0}$. So, the triangle inequality leads to
	\begin{eqnarray*}
		&& \tr{\left( \estpara{n_i}-\optpara{n_i}-\para_0 \right) U_i \left( \estpara{n_i}-\optpara{n_i}-\para_0 \right)' }\\ &\leq& 4\tr{\left( \para_\star-\para_0 \right) U_i \left( \para_\star-\para_0 \right)' }
	\end{eqnarray*}
	Hence, the normal equation $\left( \para_\star-\para_0 \right) U_i =  \sum\limits_{t=0}^{n_i-1} w(t+1) x(t)'\extendedLmatrix{\estpara{t}}'$, in addition to Lemma~\ref{bigcov} and Lemma~\ref{Lailemma} imply that
	\begin{eqnarray*}
		\tr{\left( \estpara{n_i}-\optpara{n_i}-\para_0 \right) U_i \left( \estpara{n_i}-\optpara{n_i}-\para_0 \right)' } = \order{i}
	\end{eqnarray*}
	Applying Lemma \ref{smallcov}, we obtain
	\begin{eqnarray} \label{GCEproofeq1}
		\sum\limits_{j=0}^i \rrate^{j} \Mnorm{\left( \estpara{n_i}-\optpara{n_i} - \para_0 \right) \widetilde{L}_j}{}^2 = \order{ i } .
	\end{eqnarray}
	Since $\optpara{n_j}= \order{n_j^{-1/2}}$, by Lemma~\ref{Lipschitzlemma} we have $\Mnorm{L_j - \Lmatrix{\estpara{n_j}-\optpara{n_j}}}{} = \order{\rrate^{-j/2}}$. Hence,
	\begin{eqnarray*}
		\sum\limits_{j=0}^i \rrate^{j} \Mnorm{\left( \estpara{n_i} - \optpara{n_i} - \para_0 \right) \extendedLmatrix{\estpara{n_j}-\optpara{n_j}}}{}^2 = \order{ i} .
	\end{eqnarray*}
	Using $\estpara{n_j}-\optpara{n_j} \in \tempparaspace{0}$, \eqref{Sideq} leads to \\$\Mnorm{ \Lmatrix{\estpara{n_i} - \optpara{n_i}} - \Lmatrix{\para_0} }{} = \order{ i^{1/2}\rrate^{-i/2} }$, which by Lemma \ref{Lipschitzlemma} implies that
	\begin{eqnarray} \label{GCEproofeq2}
		\Mnorm{ \Lmatrix{\estpara{n_i}} - \Lmatrix{\para_0} }{} = \order{i^{1/2}\rrate^{-i/2} }.
	\end{eqnarray}
	Thus, we have
	\begin{eqnarray}
		\sum\limits_{t=0}^{n_m-1} \norm{ \left(\Lmatrix{\para_0} - L_t \right) x(t) }{}^2 &=& \order{\sum\limits_{i=0}^{m} \rrate^i \Mnorm{L_i-\Lmatrix{\para_0}}{}^2} \notag \\
		&=& \order{m^2}. \label{GCEdecompos1}
	\end{eqnarray}
	Moreover, putting Assumption~\ref{momentcondition}, Corollary~\ref{RegretCorollary}, \eqref{fluctuationmagnitude}, and \eqref{GCEproofeq2} together, we obtain $\norm{x(n_m)-\optstate{n_m}}{} \norm{\optstate{n_m}}{} = \order{m}$, which in turn leads to
	\begin{eqnarray}\label{GCEdecompos2}
		\optstate{n_m}' \Kmatrix{\para_0} \optstate{n_m} - x(n_m)' \Kmatrix{\para_0} x(n_m) = \order{m}.\:\:\:\:\:\:\:\:\:\:\:\:
	\end{eqnarray}
	Then, \eqref{GCEdecompos1} and \eqref{GCEdecompos2} lead to the desired result for the regret: $\regret{n_m}{\adaptivepolicy} = \order{m^2}$. Further, \eqref{GCEproofeq1} and \eqref{GCEproofeq2} imply that
	\begin{eqnarray*}
	\Mnorm{\left( \estpara{n_m}-\optpara{n_m} - \para_0 \right) \extendedLmatrix{\para_0}}{} = \order{ \rrate^{-m/2} m^{1/2} };
	\end{eqnarray*}
	i.e., $\inf\limits_{\para \in \nullspace{\para_0}} \Mnorm{\estpara{n_m}-\para}{} = \order{\rrate^{-m/2} m^{1/2}}$. Finally, since \eqref{GCEproofeq2} implies a similar result for $\levelset{\para_0}$, the desired result for $\mathcal{P}_0$ holds.


text:
\begin{document}
\title[Generalized Jacobi operators in Krein spaces]
{Generalized Jacobi operators in Krein spaces}
\author{Maxim Derevyagin}

\address{Department of Nonlinear Analysis \\
Institute of Applied Mathematics and Mechanics\\
R.Luxemburg str. 74 \\
83114 Donetsk, Ukraine \\}

\email{derevyagin.m@gmail.com}

\date{\today}

\begin{abstract}
A special class of generalized Jacobi operators which are
self-adjoint in Krein spaces is presented. A description of the
resolvent set of such operators in terms of solutions of the
corresponding recurrence relations is given. In particular,
special attention is paid to the periodic generalized Jacobi
operators. Finally, the spectral properties of generalized Jacobi
operators are applied to prove convergence results for Pad\'e
approximants.
\end{abstract}

\keywords{generalized Jacobi matrix, associated polynomials, Weyl
function, continued fraction, Pad\'e approximant}
\subjclass{Primary 47B36, 47B50; Secondary 30B70, 30E05, 41A21,
47A57}

\maketitle

\section{Introduction}

Let $\mu$ be a positive Borel measure having infinite support
$\supp \mu \subset[a,b]\subset\dR$. To every such a measure $\mu$
there corresponds a linear functional defined on the linear space
$\cP=\spn\{\lambda^k:\,k\in\dZ_+:=\dN\cup\{0\}\}$ by the formula
\[
s_k=\sS(\lambda^k):=\int_{a}^{b}t^kd\mu(t),
\,\,k\in\dZ_+:=\dN\cup\{0\}.
\]
The functional $\sS$ is positive definite on $\cP$, that is,
$\det(s_{i+j})_{i,j=0}^{n}>0$ for all $n\in\dZ_+$ . Besides, the
measure $\mu$ (or, equivalently, the functional $\sS$) generates
the holomorphic function
\begin{equation}\label{mark_asymp}
\widehat{\mu}(\lambda)=\sS_t\left(\frac{1}{t-\lambda}\right)=\int_{a}^{b}\frac{d\mu(t)}{t-\lambda}
=-\frac{s_0}{\lambda}-\frac{s_1}{\lambda^2}-\dots-\frac{s_n}{\lambda^{n+1}}-\dots\quad
(|\lambda|>R),
\end{equation}
where $R$ is large enough. By using the Euclidean algorithm, P.L.
Tchebyshev~\cite{Tcheb} expanded the function $\widehat{\mu}$ into
the following continued fraction
\begin{equation}\label{Jfraction}
\widehat{\mu}(\lambda)\sim
-\frac{1}{\lambda-a_0-\displaystyle{\frac{b_0^2}{\lambda-a_1-\displaystyle{\frac{b_1^2}{\ddots}}}}}=
-\frac{1}{\lambda-a_0}
\begin{array}{l} \\ -\end{array}
\frac{b_0^2}{\lambda-a_1}
\begin{array}{ccc} \\ - \end{array}
\frac{b_{1}^2}{\lambda-a_2}\begin{array}{l} \\
-\cdots \end{array},
\end{equation}
where $a_j$ are real numbers, $b_j$ are positive numbers, and
$\sup\limits_{j\in\dZ_+}\{|a_j|+b_j\}<\infty$. Such continued
fractions are called $J$-fractions~\cite{JTh}. Note that the
coefficients $a_j$ and $b_j$ are uniquely determined by the
coefficients $s_j$ of the Taylor series at infinity
(see~\eqref{mark_asymp}).

It is well-known (see~\cite{Baker},~\cite{JTh},~\cite{NikSor}),
that the  $n$-th convergent $-Q_n(\lambda)/P_n(\lambda)$ of
the continued fraction~\eqref{Jfraction} is characterized by the
following property
\begin{equation}\label{eq:1.12}
-\frac{Q_n(\lambda)}{P_n(\lambda)}=-\frac{s_{0}}{\lambda}-\frac{s_{1}}{\lambda^{2}}-\dots-\frac{s_{2n-1}}{\lambda^{2n}}
+O\left(\frac{1}{\lambda^{2n+1}}\right) \quad(\lambda{\rightarrow
}\infty).
\end{equation}
In other words, the rational function $-Q_n/P_n$ is the $n$-th
diagonal Pad\'e approximant to  $\widehat{\mu}$, that is,
\[
\widehat{\mu}(\lambda)+\frac{Q_n(\lambda)}{P_n(\lambda)}=O\left(\frac{1}{\lambda^{2n+1}}\right)\quad(\lambda{\rightarrow
}\infty).
\]
Further, by using standard argumentation (see~\cite{JTh}), one can
see that the polynomials $P_{n}$ and $Q_{n}$ are solutions of the
three-term recurrence relations
\begin{equation}\label{eq:2.10_d}
b_{j-1}u_{j-1}+a_ju_{j}+b_{j}u_{j+1}=\lambda u_j\quad(j\in\dN),
\end{equation}
with initial conditions
\begin{equation}\label{DiffEq_d}
    P_0(\lambda)=1,\quad P_1(\lambda)=\frac{p_0(\lambda)}{b_0},\quad
     Q_0(\lambda)=0,\quad Q_1(\lambda)=\frac{1}{b_0}.
\end{equation}

On the other hand, to the recurrence relations~\eqref{eq:2.10_d}
(or, equivalently, to the continued fraction~\eqref{Jfraction})
there corresponds a linear bounded self-adjoint operator  in the
space $\ell^2_{[0,\infty)}$. More precisely, that operator is
generated by the following tridiagonal matrix
\[
H=\left(
\begin{array}{cccc}
  a_0 & b_0 &  &  \\
  b_0 & a_1 & b_1 &  \\
      & b_1 & a_2 & \ddots \\
      &     & \ddots & \ddots \\
\end{array}
\right)
\]
and its $m$-function
$m(\lambda)=\left((H-\lambda)^{-1}e,e\right)_{\ell^2_{[0,\infty)}}$,
where $e=(1,0,\dots,0,\dots)^\top\in\ell^2_{[0,\infty)}$,
coincides with $\widehat{\mu}(\lambda)$. So, one has an operator
representation of $\widehat{\mu}$
\[
\widehat{\mu}(\lambda)=\left((H-\lambda)^{-1}e,e\right)_{\ell^2_{[0,\infty)}},\quad
\lambda\in\dC\setminus\dR.
\]

The relations between orthogonal polynomials, Pad\'e approximants,
and Jacobi operators are well-known. These relations allow to use
operator methods to the investigation of orthogonal polynomials
and the Pad\'e approximants.

Note, that the above-mentioned results are also valid if the
underlying functional $\sS$ is not positive and has the property
\begin{equation}\label{reg_fun}
\det(s_{i+j})_{i,j=0}^{n}\ne 0,\quad s_j=\sS(\lambda^j)
\end{equation}
for all $n\in\dZ_+$ (see~\cite{AptKV},~\cite{Becker},~\cite{BK}).
The functional $\sS$ having the property~\eqref{reg_fun} is called
{\it regular}.

In the present paper a similar relations for non-regular
functionals are presented. In fact, the scheme proposed
in~\cite{AptKV},~\cite{BK} for investigation of the convergence of
Pad\'e approximants is generalized here to the non-regular case.
The paper is organized as follows. In Section~2, starting from a
(not necessarily regular) functional on $\cP$, three-term
recurrence relations for associated polynomials are derived. In
Section~3, a special class of generalized Jacobi matrices is
presented. Weyl solutions of the three-term recurrence relations
and Weyl functions are introduced in Section~4. In Section~5,
following the scheme proposed in~\cite{AptKV}, the
characterization of resolvent sets of generalized Jacobi operators
is obtained. Section~6 is concerned with the case of periodic
generalized Jacobi matrices. In Section~7, convergence results for
Pad\'e approximants are proved.

\section{Associated polynomials}

In this section, starting from a (not necessarily regular)
functional on $\cP$, three-term recurrence relations for
associated polynomials are derived.

Let us consider a holomorphic in a neighborhood of infinity
function $\f$ such that
\[
\f^{\sharp}(\lambda):=\overline{\f(\overline{\lambda})}=\f(\lambda).
\]
So, $\f$ has the Taylor expansion at infinity
\[
\displaystyle{\f(\lambda)=-\sum_{j=0}^{\infty}\frac{s_j}{\lambda^{j+1}}},
\]
where $s_j\in\dR$.  To every such a function one can associate a
real linear functional on $\cP$ defined by the formula
\[
\sS(\lambda^k):=\frac{1}{2\pi i}\oint_{|\lambda|=R}
\lambda^k\f(\lambda)d\lambda=s_k\in\dR,
\,\,k\in\dZ_+:=\dN\cup\{0\},
\]
for sufficiently large $R$. Clearly, the functional $\sS$ is not
necessarily regular, that is $\det(s_{i+j})_{i,j=0}^{n}=0$ may
vanish for some $n\in\dZ_+$ (for instance, see~\cite{DD}). In
general case, the functional $\sS$ generates an indefinite inner
product on $\cP$ (see~\cite{vRos},~\cite{AI})
\[
[f,g]_{\sS}:=\sS(f(\lambda)g^{\sharp}({\lambda}))=\frac{1}{2\pi
i}\oint_{|\lambda|=R}f(\lambda)g^{\sharp}({\lambda})\f(\lambda)d\lambda,
\quad f,g\in\cP,
\]
which is degenerate if and only if $\varphi$ is rational. In what
follows we suppose that $\varphi$ is not rational. As in the
regular case, one can associate to $\sS$ the following holomorphic
function
\begin{equation}\label{s_series}
\sS_z\left(\frac{1}{z-\lambda}\right)=\frac{1}{2\pi
i}\oint_{|z|=R} \frac{\f(z)dz}{z-\lambda}
=-\frac{s_0}{\lambda}-\frac{s_1}{\lambda^2}-\dots-\frac{s_n}{\lambda^{n+1}}-\dots\quad(|\lambda|>R).
\end{equation}

 Throughout this paper we suppose that the sequence ${\bf
s}:=\{s_j\}_{j=0}^\infty$ is {\it normalized}, i.e. the first
nonvanishing moment has modulus 1. A number $n_j\in\dN$ is called
{\it a normal index} if $\det (s_{i+k})_{i,k=0}^{n_j-1}\ne 0$.
Since $\varphi$ is not rational, there exists an infinite number
of normal indices (see~\cite[Section~16.10.2]{G}). Let
$n_1<n_2<\dots<n_{j}<\dots$ be a sequence of all normal indices.
By the choice of $n_1$ one has $s_{n_1-1}\ne 0$. Let us set
$\varepsilon_0=s_{n_1-1}$ ($|\varepsilon_0|=1$) and $\f_0:=\f$.
The principal part of the Laurent expansion for ${\displaystyle
-\frac{1}{\f_0}}$ is a polynomial of degree $k_0:=n_1$ with the
leading coefficient $\varepsilon_0$. So, we have
\begin{equation}\label{Q1}
   -\frac{1}{\f_0(\lambda)}=\varepsilon_0p_0(\lambda)+
   b_0^2\f_1(\lambda),
    \quad b_0>0
\end{equation}
where $p_0$ is a monic polynomial of degree $k_0$ and  $\f_1$ is
holomorphic in a neighborhood of infinity. Furthermore, the
function  $\f_1$ satisfies the relation
$\f_1^{\sharp}({\lambda})=\f_1(\lambda)$. Choose $b_0>0$ such that
the sequence ${\bf s}^{(1)}=\{s_j^{(1)}\}_{j=0}^\infty$ defined by
the following expansion
\begin{equation}\label{Q1asympt}
\f_1(\lambda)=
-\frac{s_{0}^{(1)}}{\lambda}-\frac{s_{1}^{(1)}}{\lambda^{2}}-\dots
-\frac{s_{2(n-k_0)}^{(1)}}{\lambda^{2(n-k_0)+1}}-\dots,
\end{equation}
at $\infty$ is normalized. This completes the first step of
expanding the series~\eqref{s_series} into a continued fraction
(see~\cite{Mag1}).

As was shown in~\cite{De}, the set of the normal indices of ${\bf
s}^{(1)}$ coincides with the following sequence
\[
n_2-k_0<\dots<n_{j}-k_0<\dots.
\]
Now one can apply the above reasoning to the function $\f_1$ and
so on. By recursion we obtain the following P-fraction
\begin{equation}\label{ContF2}
-\frac{\varepsilon_0}{p_0(\lambda)}
\begin{array}{l} \\ - \end{array}
\frac{\varepsilon_0\varepsilon_1b_0^2}{p_1(\lambda)}
\begin{array}{ccc} \\ - & \cdots & -\end{array}
\frac{\varepsilon_{j-1}\varepsilon_j b_{j-1}^2}{p_j(\lambda)}
\begin{array}{cc} \\ - & \cdots \end{array},
\end{equation}
where $\e_j=\pm 1$, $b_j>0$ and
$p_{j}(\lambda)={\lambda}^{k_{j}}+p_{k_{j}-1}^{(j)}{\lambda}^{k_{j}-1}+\dots+
p_{1}^{(j)}\lambda+p_{0}^{(j)}$ are real monic polynomials of
degree ${k_j}$ (see also~\cite{De},~\cite{DD1}). Note, that
$n_j=k_0+k_1+\dots+k_{j-1}$.

It also should be mentioned that there exist functions for which
the set $ \{b_j, p_{0}^{(j)}, \dots, p_{k_{j}-1}^{(j)}:
j\in\dZ_+\}$ of coefficients of the P-fraction is not necessarily
bounded. In particular, the Cauchy transform of the signed measure
constructed in~\cite{Stahl} gives such an example.

The continued fraction~\eqref{ContF2} can be considered as a
sequence of the linear-fractional transformations
(see~\cite[Section 5.2]{JTh})
\[
T_j(\omega):=\frac{-\e_j}{p_j(\lambda)+\e_jb_j^2\omega}
\]
having the following matrix representation
\begin{equation}\label{Wj}
\cW_j(\lambda)=\begin{pmatrix}0 & -\frac{\varepsilon_j}{b_j}\\
                            \varepsilon_jb_j &  \frac{p_j(\lambda)}{b_j}
                            \end{pmatrix},\quad j\in\dZ_+.
\end{equation}
The superposition $T_{0}\circ T_{1}\circ\dots \circ T_{j}$ of the
linear-fractional transformations corresponds to the product of
the matrices $\cW_l(\lambda)$
\begin{equation}\label{W}
\cW_{[0,j]}(\lambda)=(w_{ik}^{(j)}(\lambda))_{i,k=1}^2:=\cW_0(\lambda)\cW_1(\lambda)\dots
\cW_j(\lambda).
\end{equation}

To give an explicit formula for $\cW_{[0,j]}$ in terms of  $p_j$,
$b_j$, $\varepsilon_j$, define the polynomials
$P_{j+1}(\lambda)$, $Q_{j+1}(\lambda)$ by the equalities
\begin{equation}\label{PQj}
   \left(
\begin{array}{c}
  -Q_0 \\
  P_0 \\
\end{array}
\right)=\left(
\begin{array}{c}
  0 \\
  1 \\
\end{array}
\right),\quad
\left(
\begin{array}{c}
 -Q_{j+1}(\lambda) \\
 P_{j+1}(\lambda) \\
\end{array}
\right):=\cW_{[0,j]}(\lambda)\left(
\begin{array}{c}
  0 \\
  1 \\
\end{array}
\right), \quad j\in\dZ_+ .
\end{equation}
The relation
$\cW_{[0,j]}(\lambda)=\cW_{[0,j-1]}(\lambda)\cW_{j}(\lambda)$
(see~\eqref{W}) yields
\begin{equation}\label{W0j}
\cW_{[0,j]}(\lambda)\left(
\begin{array}{c}
  1 \\
  0 \\
\end{array}
\right)=
\cW_{[0,j-1]}(\lambda)\left(
\begin{array}{c}
  0 \\
  \varepsilon_jb_j \\
\end{array}
\right)=
\left(
\begin{array}{c}
  -\varepsilon_jb_j Q_j(\lambda) \\
  \varepsilon_jb_j P_j(\lambda) \\
\end{array}
\right), \quad j\in\dN .
\end{equation}
So, the matrix $\cW_{[0,j]}(\lambda)$ has the form
\begin{equation}\label{SolMatr}
\cW_{[0,j]}(\lambda)=\left(
\begin{array}{cc}
  -\varepsilon_jb_j Q_j(\lambda) & -Q_{j+1}(\lambda) \\
  \varepsilon_jb_j P_j(\lambda) & P_{j+1}(\lambda) \\
\end{array}
\right), \quad j\in\dZ_+ .
\end{equation}
Further, the equality
\[
\left(
\begin{array}{c}
 -Q_{j+1}(\lambda) \\
 P_{j+1}(\lambda) \\
\end{array}
\right)=\cW_{[0,j-1]}(\lambda)\cW_{j}(\lambda)\left(
\begin{array}{c}
  0 \\
  1 \\
\end{array}
\right)=
\frac{1}{b_j}\cW_{[0,j-1]}(\lambda)\left(
\begin{array}{c}
  -\varepsilon_j \\
  p_j(\lambda) \\
\end{array}
\right), \quad j\in\dN,
\]
 shows that the polynomials $P_{j}(\lambda)$, $Q_{j}(\lambda)$ are solutions of the difference equation
\begin{equation}\label{eq:2.10}
\varepsilon_{j-1}\varepsilon_j
b_{j-1}u_{j-1}-p_j(\lambda)u_{j}+b_{j}u_{j+1}=0\,\,\,(j\in\dN),
\end{equation}
obeying the initial conditions
\begin{equation}\label{DiffEq}
 \begin{split}
    P_0(\lambda)&=1,\quad P_1(\lambda)=\frac{p_0(\lambda)}{b_0},\\
     Q_0(\lambda)&=0,\quad Q_1(\lambda)=\frac{\varepsilon_0}{b_0}.
\end{split}
\end{equation}
According to~\eqref{SolMatr}, the $(j+1)$-th convergent of the
continued fraction~\eqref{ContF2} is equal to
\[
f_j:=T_{0}\circ T_{1}\circ\dots \circ
T_{j}(0)=-Q_{j+1}(\lambda)/P_{j+1}(\lambda).
\]
The relations~\eqref{SolMatr}, \eqref{W}, and \eqref{Wj} imply the
following statement.
\begin{proposition}[\cite{DD1}]\label{Ostrogr}
The polynomials $P_j$, $Q_j$ satisfy the following generalized
Liouville-Ostrogradsky formula
\begin{equation}\label{Ostrogr2}
\varepsilon_jb_j(Q_{j+1}\left(\lambda)P_{j}(\lambda) -
        Q_{j}(\lambda) P_{j+1}(\lambda) \right)=1\quad(j\in\dZ_+).
\end{equation}
\end{proposition}

\section{Generalized Jacobi matrices}

The main goal of this section is to present a special class of
generalized Jacobi matrices.

 Let $p(\lambda)=p_{n}{\lambda}^{n}+\dots+p_{1}\lambda+p_{0}$
be a monic scalar real polynomial of degree $n$, i.e. $p_{n}=1$.
Let us associate to the polynomial $p$ its symmetrizator  $E_p$
and let the companion matrix $C_p$ be given by
\begin{equation}\label{comp}
E_{p}=\begin{pmatrix}
p_{1}&\dots&p_{n}\\
\vdots&\sddots&\\
p_{n}&&{\bf 0}\\
\end{pmatrix},\quad
C_{p}=\begin{pmatrix}
0&\dots&0&-p_{0}\\
1&&{\bf 0}&-p_{1}\\
&\ddots&&\vdots\\
{\bf 0}&&1&-p_{n-1}\\
\end{pmatrix}.
\end{equation}
As is known, $\det(\lambda-C_p)=p(\lambda)$ and the spectrum
$\sigma(C_p)$ of the companion matrix $C_p$ is simple. The
matrices $E_{p}$ and $C_{p}$ are related by (see~\cite{GLR})
\begin{equation}\label{cb}
C_{p}E_{p}=E_{p}C_{p}^{\top}.
\end{equation}
So, $C_{p}E_{p}$ is a symmetric matrix.

\begin{definition}[\cite{DD}, \cite{KL79}]
Let $p_j$ be real monic polynomials of degree ${k_j}$
\[
p_{j}(\lambda)={\lambda}^{k_{j}}+p_{k_{j}-1}^{(j)}{\lambda}^{k_{j}-1}+\dots+
p_{1}^{(j)}\lambda+p_{0}^{(j)},
\]
and let  $\e_{j}=\pm 1$,  $b_{j}>0$ $(j\in\dN)$. The tridiagonal
block matrix
\begin{equation}\label{mJacobi}
H=\begin{pmatrix}
A_{0}   &\wB_{0}&       &{\bf 0}\\
B_{0}   &A_1    &\wB_{1}&\\
        &B_1    &A_{2} &\ddots\\
{\bf 0} &       &\ddots &\ddots\\
\end{pmatrix}
\end{equation}
where $A_{j}=C_{p_{j}}$ and $k_{j+1}\times k_{j}$ matrices $B_{j}$
and $k_{j}\times k_{j+1}$ matrices $\wB_{j}$ are given by
\begin{equation}\label{bblock}
B_{j}=\begin{pmatrix}
0&\dots&b_{j}\\
\hdotsfor{3}\\
0&\dots&0\\
\end{pmatrix},\,
\wB_{j}= \begin{pmatrix}
0&\dots&\wt b_{j}\\
\hdotsfor{3}\\
0&\dots&0\\
\end{pmatrix} \,
(\wt b_{j}=\e_j\e_{j+1} b_j, \, j=0,\dots, N-1),
\end{equation}
will be called a {\it generalized Jacobi matrix}  associated with
the sequences of polynomials $\{{\e_{j}p_{j}\}}_{j=0}^{\infty}$
and numbers $\{b_j\}_{j=0}^{\infty}$.
\end{definition}
\begin{remark}
The papers~\cite{DD}, \cite{DD1}, and~\cite{KL79} are only
concerned with the case of generalized Jacobi matrices which are
finite rank perturbations of classical Jacobi matrices. In fact,
the generalized Jacobi matrix in question is associated to the
P-fraction~\eqref{ContF2} or, equivalently, to the sequence of
matrices $\cW_j$ having the form~\eqref{Wj}.
\end{remark}

\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}

{\bf From now on, we suppose that}
\begin{equation}\label{assumption1}
\text{there exists }N\in\dN:\quad\deg p_j\le N,\quad j\in\dZ_+,
\end{equation}
\begin{equation}\label{assumption2}
\sup \{b_j, |p_{k}^{(j)}|: j\in\dZ_+, k=0,\dots,k_j-1\}< +\infty.
\end{equation}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{4}

Let $\ell^2_{[0,\infty)}$ denote the Hilbert space of complex
square summable sequences $(w_0,w_1,\dots)$ with the usual inner
product. Setting
\begin{equation}\label{NormInd}
n_{0}=0, \quad n_j=\sum_{i=0}^{j-1}k_{i}\quad(j\in\dN),
\end{equation}
define a standard basis in $\ell^2_{[0, \infty)}$ by the
equalities
\[
e_{j,k}=\{{\delta_{l, n_{j}+k}\}}_{l=0}^{\infty} \quad (j\in\dZ_+;
k=0,\dots,k_{j}-1),\quad e:=e_{0,0}.
\]
 Define the symmetric matrix
$G$ by the equality
\begin{equation}\label{Gram}
G=\mbox{diag}(G_{0},G_1,\dots),\quad
G_{j}=\e_{j}E_{p_{j}}^{-1}\quad (j\in\dZ_+).
\end{equation}
Further, we may identify via usual matrix product the matrix $G$
with an operator on the linear space $\sC_0$ of finite sequences
of $\ell^2_{[0, \infty)}$. Its closure will be also denoted by
$G$. In view of~\eqref{assumption1},~\eqref{assumption2}, the
operator $G$ defined on $\ell^2_{[0,\infty)}$ is bounded and
self-adjoint. Moreover, $G^{-1}$ is a bounded linear operator in
$\ell^2_{[0,\infty)}$.

 Let $H_{[j,l]}$ ($G_{[j,l]}$) be a submatrix of $H$ ($G$), corresponding to the
basis vectors $\{e_{i,k}\}^{k=0,\dots,k_{i}-1}_{i=j,\dots,l}$
$(0\le j\le l < +\infty)$. The matrix $H_{[j,l]}$ will be called a
finite generalized Jacobi matrix.

 Let $\sH_{[0,\infty)}$ be a space of elements of $\ell^2_{[0,\infty)}$
 provided with the following indefinite inner product
\begin{equation}\label{metricInfty}
\left[x,y\right]=(Gx,y)_{\ell^2_{[0,\infty)}}\quad
(x,y\in\ell^2_{[0,\infty)}).
\end{equation}

Let us recall~\cite{AI} that a pair $(\sH, [\cdot,\cdot])$
consisting of a Hilbert space $\sH$ and a sesquilinear form
$[\cdot,\cdot]$ on $\sH\times\sH$ is called {\it a space with
indefinite inner product}. A space with indefinite metric $(\sH,
[\cdot,\cdot])$ is called {\it a Krein space} if the indefinite
scalar product $[\cdot,\cdot]$ can be represented as follows
\[
[x,y]=(Jx,y)_{\sH}\quad x,y\in\sH,
\]
where the linear operator $J$ satisfies the following conditions
\[
J=J^{-1}=J^*.
\]
The operator $J$ is called {\it the fundamental symmetry}. So, one
can see that the space $\sH_{[0,\infty)}$ is the Krein space with
the fundamental symmetry $J=\sg G$ (see~\cite{AI} for details).

\begin{proposition}\label{SymGJM}
Under the assumptions~\eqref{assumption1},~\eqref{assumption2},
the considered generalized Jacobi matrix defines a bounded
self-adjoint operator $H$ (a generalized Jacobi operator) in the
Krein space $\sH_{[0,\infty)}$, that is,
\begin{equation}\label{hermitian}
\left[H x,y\right]=\left[x,H y\right]\quad x,y\in\sH_{[0,\infty)}.
\end{equation}
\end{proposition}

\begin{proof} It is not hard to see that, according to~\eqref{assumption1},~\eqref{assumption2},
the matrix in question generates a bounded operator in
$\sH_{[0,\infty)}$. Relation~\eqref{hermitian} is implied by
~\eqref{cb} (see~\cite{DD} for details).
\end{proof}

  Let us extend the system
$\{P_j(\lambda)\}_{j=0}^{\infty}$,
$\{Q_j(\lambda)\}_{j=0}^{\infty}$ by the equalities
\begin{equation}\label{polynom0}
P_{j,k}(\lambda)={\lambda}^{k}P_{j}(\lambda),\quad
Q_{j,k}(\lambda)={\lambda}^{k}Q_{j}(\lambda)\quad
(j\in\dZ_+;k=0,\dots,k_{j}-1).
\end{equation}
Setting
\[
{\bf
P}_{[l,j]}(\lambda)=(P_{l,0}(\lambda),\dots,P_{l,k_l-1}(\lambda),\dots,
P_{j,0}(\lambda),\dots,P_{j,k_j-1}(\lambda)),
\]
\[
{\bf
Q}_{[l,j]}(\lambda)=(Q_{l,0}(\lambda),\dots,Q_{l,k_l-1}(\lambda),\dots,
Q_{j,0}(\lambda),\dots,Q_{j,k_j-1}(\lambda)).
\]
one can rewrite the system~\eqref{eq:2.10}, \eqref{polynom0} in
the following manner
\begin{equation}\label{Vector10}
{\bf
P}_{[0,j]}(\lambda)(\lambda-H_{[0,j]})=(0,\dots,0,b_jP_{j+1,0}(\lambda))\quad
(j\in\dZ_+),
\end{equation}
\begin{equation}\label{Vector11}
{\bf
Q}_{[0,j]}(\lambda)(\lambda-H_{[0,j]})=(\underbrace{0,\dots,0,-\varepsilon_0}_{k_0},0,\dots,0,b_jQ_{j+1,0}(\lambda))\quad
(j\in\dZ_+).
\end{equation}
Since $Q_{0,0}(\lambda)=\dots=Q_{0,k_0-1}(\lambda)\equiv 0$, the
relation~\eqref{Vector11} reduces to
\begin{equation}\label{Vector12}
{\bf
Q}_{[1,j]}(\lambda)(\lambda-H_{[1,j]})=(0,\dots,0,b_jQ_{j+1,0}(\lambda))\quad
(j\in\dN),
\end{equation}
It follows from~\eqref{Vector10} and \eqref{Vector12} that the
eigenvalues of $H_{[0,j]}$ and $H_{[1,j]}$ coincide with the roots
of $P_{j+1}(\lambda)$ and $Q_{j+1}(\lambda)$, respectively.

\begin{proposition}[\cite{DD}]\label{detf}
The polynomials $P_{j}$ and $Q_j$  $(j\in\dN)$ can be found by the
formulas
\begin{eqnarray}
P_{j}(\lambda)&=&(b_{0}\dots b_{j-1})^{-1}\det(\lambda-H_{[0,j-1]}),\label{polynom1}\\
Q_{j}(\lambda)&=&\varepsilon_0(b_{0}\dots
b_{j-1})^{-1}\det(\lambda-H_{[1,j-1]}).\label{polynom2}
\end{eqnarray}
\end{proposition}
The formulas~\eqref{polynom1} and~\eqref{polynom2} in the
classical case can be found in~\cite[Section 7.1.2]{Be}
and~\cite[Section 6.1]{Atk}. The following statement is an easy
consequence of the recurrence relations~\eqref{eq:2.10}.
\begin{proposition}[\cite{DD},\cite{DD1}]\label{prostota}
Let $j\in\dN$. Then
\begin{enumerate}
\item[i)] The polynomials $P_j$ and $P_{j+1}$  have no common zeros.
\item[ii)] The polynomials $Q_j$ and $Q_{j+1}$ have no common zeros.
\item[iii)] The polynomials $P_j$ and $Q_j$ have no common zeros.
\end{enumerate}
\end{proposition}

Taking into account the equality
$G_{[0,j]}H_{[0,j]}=H_{[0,j]}^\top G_{[0,j]}$ which is implied
by~\eqref{cb} (see~\cite{DD}) and setting
\[
\pi_{[0,j]}(\lambda)=G_{[0,j]}^{-1}{\bf
P}_{[0,j]}(\lambda)^\top,\quad
\xi_{[0,j]}(\lambda)=G_{[0,j]}^{-1}{\bf Q}_{[0,j]}(\lambda)^\top,
\]
one can rewrite~\eqref{Vector10}, \eqref{Vector11} in the form
\begin{equation}\label{Matrix10}
(\lambda-H_{[0,j]})\pi_{[0,j]}(\lambda)=
\varepsilon_jb_jP_{j+1,0}(\lambda)e_{j,0}\quad (j\in\dZ_+),
\end{equation}
\begin{equation}\label{Matrix11}
(\lambda-H_{[0,j]})\xi_{[0,j]}(\lambda)+e_{0,0}=
\varepsilon_jb_jQ_{j+1,0}(\lambda)e_{j,0}\quad (j\in\dZ_+).
\end{equation}
Further, let us set
\begin{equation}\label{defpiksi}
\pi(\lambda)=G^{-1}(P_{0,0}(\lambda),\dots,P_{0,{k_0}-1}(\lambda),\dots)^{\top},\quad
\xi(\lambda)=G^{-1}(Q_{0,0}(\lambda),\dots,Q_{0,{k_0}-1}(\lambda),\dots)^{\top}.
\end{equation}
Now, it follows from~\eqref{Matrix10}-\eqref{defpiksi} that the
following formal equalities hold true
\begin{equation}\label{formal_eq}
(\lambda-H)\pi(\lambda)=0,\quad (\lambda-H)\xi(\lambda)=-e_{0,0}.
\end{equation}

The first equality in~\eqref{formal_eq} allows us to characterize
the point spectrum of  $H$.

\begin{proposition}
$\lambda\in\sigma_p(H)$ if and only if
$\pi(\lambda)\in\ell^2_{[0,\infty)}$.
\end{proposition}
By the definition of $\pi(\lambda)$ and the
assumptions~\eqref{assumption1},~\eqref{assumption2}, we see that
\[
\pi(\lambda)\in\ell^2_{[0,\infty)}\Longleftrightarrow
\sum_{j=0}^{\infty}|P_{j}(\lambda)|^2<+\infty.
\]

\section{Weyl solutions and Weyl functions}

If for some $\lambda\in\dC$ there exists a solution
$\{W_j(\lambda)\}_{j=0}^{\infty}$ of the recurrence
relations~\eqref{eq:2.10} such that
\begin{equation}\label{Weylsolution}
\{W_j(\lambda)\}_{j=0}^{\infty}\in\ell^2_{[0,\infty)}\text{ and
}\{W_j(\lambda)\}_{j=0}^{\infty}\ne\{P_j(\lambda)\}_{j=0}^{\infty}
\end{equation}
then we will say that there exists a Weyl solution
$\{W_j(\lambda)\}_{j=0}^{\infty}$ of the recurrence
relations~\eqref{eq:2.10} at the point $\lambda$. Since
$\{P_j(\lambda)\}_{j=0}^{\infty}$ and
$\{Q_j(\lambda)\}_{j=0}^{\infty}$ are linearly independent
solutions of~\eqref{eq:2.10}, the Weyl solution admits the
following representation
\begin{equation}\label{reprWs}
W_j(\lambda)=Q_{j}(\lambda)+m(\lambda)P_{j}(\lambda),
\end{equation}
where $m(\lambda)$ is a complex number. The following statement
shows the relation between $m(\lambda)$ and the operator  $H$.
\begin{proposition}\label{Wsexplicitform}
Let $\lambda\in\rho(H)$ and let
\begin{equation}\label{Wfexf}
m(\lambda)=[(H-\lambda)^{-1}e,e],\quad e:=e_{0,0}.
\end{equation}
Then the family $\{W_j(\lambda)\}_{j=0}^{\infty}$ given
by~\eqref{reprWs} is the Weyl solution of ~\eqref{eq:2.10} at
$\lambda$. Moreover, $\{W_j(\lambda)\}_{j=0}^{\infty}$ has the
form
\begin{equation}\label{Wsexf}
W_j(\lambda)=[(H-\lambda)^{-1}e,e_{j,0}],\quad j\in\dZ_+.
\end{equation}
\begin{proof}
For $\lambda\in\rho(H)$ the relation~\eqref{formal_eq} implies
that there exists a number $m(\lambda)\in\dC$ such that
\[
\xi(\lambda)+m(\lambda)\pi(\lambda)=(H-\lambda)^{-1}e\in\ell^2_{[0,\infty)}.
\]
Since $G$ is a bounded operator, we have
\begin{equation}\label{l2belong}
G(\xi(\lambda)+m(\lambda)\pi(\lambda))\in\ell^2_{[0,\infty)}.
\end{equation}
Using~\eqref{defpiksi} yields
\[
[(H-\lambda)^{-1}e,e_{j,0}]=(G(\xi(\lambda)+m(\lambda)\pi(\lambda)),e_{j,0})_{\ell^2}=Q_{j}(\lambda)+m(\lambda)P_{j}(\lambda)=:W_j(\lambda)\,(\ne
P_j(\lambda)).
\]
The latter relation means that $\{W_j(\lambda)\}_{j=0}^{\infty}$
is a solution of~\eqref{eq:2.10}. Due to~\eqref{l2belong}, we
obtain that
$\{W_j(\lambda)\}_{j=0}^{\infty}\in\ell^2_{[0,\infty)}$. Now, it
follows from~\eqref{DiffEq} that
\[
W_0(\lambda)=m(\lambda)=[(H-\lambda)^{-1}e,e].
\]
\end{proof}
\end{proposition}

\begin{definition}\label{Weylfunction}
The function $m$ defined by~\eqref{reprWs} is called a Weyl
function of the operator $H$.
\end{definition}
\begin{remark}
It should be mentioned that a general treatment of the Weyl functions
of classical Jacobi matrices in the framework of extension theory of non-densely
defined symmetric operators was proposed in~\cite{MMal92}. A general
treatment of the Weyl functions of symmetric operators in Krein spaces
in the framework of extension theory was presented in~\cite{VDer99}.
The function $m$ defined by~\eqref{Wfexf} on $\rho(H)$ is also
called the $m$-function of $H$ (see~\cite{DD}, \cite{GS}).
\end{remark}

Since $H$ is bounded,  $m$ admits the representation
\begin{equation}\label{mreprinft}
 m(\lambda)= [(H-\lambda)^{-1}e,e]=-\sum_{j=0}^{\infty}\frac{
{s_{j}}}{\lambda^{j+1}},\quad (|\lambda|>\Vert H\Vert)
\end{equation}
where ${s_{i}}=[H^ie,e]$.

Analogously, one can define $m$-functions of shortened generalized
Jacobi matrices.
\begin{definition}\label{trWf}
The function
\begin{equation}\label{SWfun}
    m_{[0,j]}(\lambda)=[(H_{[0,j]}-\lambda)^{-1}e,e]
\end{equation}
is called the $m$-function of $H_{[0,j]}$.
\end{definition}

Making use of the structure of $H_{[0,j]}$, we obtain that
(see~\cite{DD})
\begin{equation}\label{SWfun2}
    m_{[0,j]}(\lambda)=-\e_0\frac{\det(\lambda-H_{[1,j]})}{\det(\lambda-H_{[0,j]})}.
\end{equation}
According to~\eqref{polynom1}, \eqref{polynom2}, the
formula~\eqref{SWfun2} can be rewritten as follows
\begin{equation}~\label{mQP}
m_{[0,j]}(\lambda)=-\frac{Q_{j+1}(\lambda)}{P_{j+1}(\lambda)}.
\end{equation}
It follows from~\eqref{eq:2.10} (see~\cite{DD} for details) that
the $m$-function $m_{[0,j]}(\lambda)$ and the $m$-function
$m_{[1,j]}(\lambda)$ of $H_{[1,j]}$ are related by the equality
\begin{equation}\label{Riccati}
m_{[0,j]}(\lambda)=\frac{-\varepsilon_0}
    {p_0(\lambda)+\varepsilon_0b_0^2m_{[1,j]}(\lambda)},\quad j\in\dN.
\end{equation}

An analogous statement for infinite generalized Jacobi matrices is
an essential ingredient in the proof of the following result.

\begin{theorem}
Under the assumptions~\eqref{assumption1},~\eqref{assumption2},
the generalized Jacobi matrix $H$ is uniquely determined by its
Weyl function $m$.
\end{theorem}
\begin{proof}
By using the Frobenius formula, one can see that the Weyl function
$m$ of $H$ and the Weyl function
\[
m_{[1,\infty)}(\lambda)=[(H_{[1,\infty)}-\lambda)^{-1}e_{1,0},e_{1,0}]
\]
of $H_{[1,\infty)}$ are related by the equality
\begin{equation}\label{Riccatiinfty}
m(\lambda)=\frac{-\varepsilon_0}
    {p_0(\lambda)+\varepsilon_0b_0^2m_{[1,\infty)}(\lambda)},\quad |\lambda|>
\Vert H\Vert\ge\Vert H_{[1,\infty)}\Vert
\end{equation}
(a more detailed reasoning can be found in~\cite{DD}). Further,
consecutive applications of the relation~\eqref{Riccatiinfty}
leads to the $P$-fraction~\eqref{ContF2}. So, one can uniquely
recover the generalized Jacobi matrix $H$.
\end{proof}

\begin{remark}
In the definite case, formulas~\eqref{SWfun2} and~\eqref{mQP} are easy consequences
of the theory developed in~\cite{MMal92}.
In this case, it is well known that the Weyl function determines the classical Jacobi matrix uniquely
(for instance, see~\cite{GS}, \cite{MMal92}).
It is worth to mention that this result as well as Borg
type uniqueness result was recently extended
to the case of normal matrices  (see~\cite{SMal}).
Besides,  a canonical form of a normal matrix
(an analog of Jacobi matrix for selfadjoint matrices)
was also introduced there.
Some inverse problems for finite generalized Jacobi matrices were
considered in~\cite{De06},~\cite{DD}.
\end{remark}

\section{The resolvent set of $H$}

Here, following the scheme proposed in~\cite{AptKV}, the
characterization of resolvent sets $\rho(H)$ of generalized Jacobi
operators is obtained. We begin with an auxiliary lemma which
gives a criterion of the density of $\ran(H-\lambda)$ in
$\ell^2_{[0,\infty)}$.

\begin{lemma}\label{lem_2_1} For $\lambda\in\dC$ the equation
\[
(H-\lambda I)x=e_{j,k}
\]
has a solution $x=x(j,k)\in\ell^2_{[0,\infty)}$ for all
$j,k\in\dZ_+$ iff there exists a Weyl solution at $\lambda$.
\end{lemma}

\begin{proof}
\noindent 1) First, let us consider the case where $j=k=0$. It
follows from~\eqref{formal_eq} that the equation
$(H-\lambda)x=e_{0,0}$ has a solution belonging to
$\ell^2_{[0,\infty)}$ if and only if there exists a number
$m(\lambda)\in\dC$  such that
$\xi(\lambda)+m(\lambda)\pi(\lambda)\in\ell^2_{[0,\infty)}$.
Furthermore, in this case we have
\begin{equation}\label{fres_e}
x=x(0,0)=\xi(\lambda)+m(\lambda)\pi(\lambda).
\end{equation}
\noindent 2) Next, let $k=0$ and $j\in\dN$. According
to~\eqref{Matrix10} and~\eqref{Matrix11}, we see that
\begin{equation}\label{f_2_2}
(H-\lambda)\xi_{[0,j]}(\lambda)=e_{0,0}-\e_jb_jQ_{j+1,0}(\lambda)e_{j,0}+\e_jb_jQ_{j,0}(\lambda)e_{j+1,0},
\end{equation}
\begin{equation}\label{f_2_3}
(H-\lambda)\pi_{[0,j]}(\lambda)=-\e_jb_jP_{j+1,0}(\lambda)e_{j,0}+\e_jb_jP_{j,0}(\lambda)e_{j+1,0},
\end{equation}
Adding~\eqref{f_2_2} multiplied by $-P_j$ and~\eqref{f_2_3}
multiplied by $Q_j$, one obtains
\begin{equation}\label{f_2_4}
(H-\lambda)\left[-P_j(\lambda)\xi_{[0,j]}(\lambda)+Q_j(\lambda)\pi_{[0,j]}(\lambda)\right]=-P_j(\lambda)e_{0,0}-\e_jb_j(P_{j+1}(\lambda)Q_j(\lambda)-Q_{j+1}(\lambda)P_j(\lambda))e_{j,0}.
\end{equation}
Due to~\eqref{Ostrogr2}, \eqref{f_2_4} can be rewritten as follows
\[
(H-\lambda)\left[-P_j(\lambda)\xi_{[0,j]}(\lambda)+Q_j(\lambda)\pi_{[0,j]}(\lambda)\right]+P_j(\lambda)e_{0,0}=e_{j,0}.
\]
The latter relation shows that the equation $(H-\lambda)x=e_{j,0}$
has a solution $x=x(j,0)\in\ell_{[0,\infty)}^{2}$ if and only if
$e_{0,0}\in\ran(H-\lambda)$. Moreover, the solution admits the
representation
\begin{equation}\label{f_2_5}
x=x(j,0)=-P_j(\lambda)\xi_{[0,j]}(\lambda)+Q_j(\lambda)\pi_{[0,j]}(\lambda)+P_j(\lambda)\left[\xi(\lambda)+m(\lambda)\pi(\lambda)\right].
\end{equation}
\noindent 3) Finally, assume that $k\ne 0$. Observe that
\begin{equation}\label{f_2_6}
(H-\lambda)e_{j,0}=e_{j,1}-\lambda e_{j,0},\dots,
(H-\lambda)e_{j,k_j-2}=e_{j,k_j-1}-\lambda e_{j,k_j-2}.
\end{equation}
The chain of equalities~\eqref{f_2_6} imply that the equation
$(H-\lambda)x=e_{j,k}$ has a solution
$x=x(j,k)\in\ell^{2}_{[0,\infty)}$ if and only if
$e_{j,0}\in\ran(H-\lambda)$ (or, equivalently,
$e_{0,0}\in\ran(H-\lambda)$). Besides, the solution $x=x(j,k)$ can
be expressed in the following manner
\begin{equation*}
x=x(j,k)=e_{j,k-1}+\lambda e_{j,k-2}+\dots+
\lambda^k(-P_j(\lambda)\xi_{[0,j]}(\lambda)+Q_j(\lambda)\pi_{[0,j]}(\lambda)+P_j(\lambda)\left[\xi(\lambda)+m(\lambda)\pi(\lambda)\right]).
\end{equation*}
\end{proof}
\begin{remark}
In fact, Lemma~\ref{lem_2_1} gives a way to express the formal
inverse operator $R_{\lambda}$ to $H-\lambda$. Moreover, it is not
so hard to see that the Weyl solution at $\lambda$ exists if and
only if $e\in\ran(H-\lambda)$.
\end{remark}

Now we are ready to prove the main result of the present paper.
\begin{theorem}\label{main_result}
Under the assumptions~\eqref{assumption1},~\eqref{assumption2},
$\lambda\in\rho(H)$ if and only if there exist a Weyl solution
$\{W_j(\lambda)\}_{j=0}^{\infty}$ at $\lambda$ and numbers
$q\in(0,1)$, $C>0$ such that
\begin{equation}\label{estimate1}
|P_i(\lambda)W_j(\lambda)|\le Cq^{n_j-n_i},\quad i\le j.
\end{equation}

\end{theorem}

\begin{proof}
Let us prove the sufficiency. Let
$\cH_k:=\overline{\spn}\{e_{j,k}|j\in\dZ_+\}$ and let
$\cP_{\cH_k}$ be the orthogonal projector onto $\cH_k$ in
$\ell^2_{[0,\infty)}$. We start with proving the boundedness  of
the operator $R_{\lambda}\cP_{\cH_k}$ for any
$k\in\{0,\dots,N-1\}$. First, it is convenient to consider the
operator $GR_{\lambda}\cP_{\cH_0}$. Taking into
account~\eqref{f_2_5}, we obtain
\[
(GR_{\lambda}\cP_{\cH_0}e_{j,0})_{i,k}=(GR_{\lambda}e_{j,0})_{i,k}:=(GR_{\lambda}e_{j,0},e_{i,k})=\begin{cases}
       \lambda^k P_i(\lambda)(Q_j(\lambda)+m(\lambda)P_j(\lambda)),& i\le j;\\
       \lambda^k P_j(\lambda)(Q_i(\lambda)+m(\lambda)P_i(\lambda)),& i>j.\\
 \end{cases}
\]
It is clear that one can represent the operator
$GR_{\lambda}\cP_{\cH_0}$ as the sum of upper and lower triangular
operators:
$GR_{\lambda}\cP_{\cH_0}=R_{\lambda}^{(1)}+R_{\lambda}^{(2)}$. To
be more precise, we choose $R_{\lambda}^{(1)}$ in the following
way
\[
R_{\lambda}^{(1)}e_{j,k}=0\text{ for }k\ne 0,\quad
R_{\lambda}^{(1)}e_{j,0}=y_{j}^{(1)},\text{ where }
\]
\[
(y_{j}^{(1)})_{i,k}=\begin{cases}
       \lambda^k P_i(\lambda)(Q_j(\lambda)+m(\lambda)P_j(\lambda)),& i\le j;\\
       0,& i>j.\\
\end{cases}
\]
Now,  one can prove that $R_{\lambda}^{(1)}$ is bounded. Indeed,
setting
\[
x=\sum\limits_{j=0}^{l}\sum\limits_{k=0}^{k_j-1}x_{j,k}e_{j,k}\text{
and }
y=\sum\limits_{j=0}^{l}\sum\limits_{k=0}^{k_j-1}y_{j,k}e_{j,k}
\]
we get the following relation
\[
\begin{split}
(R_{\lambda}^{(1)}x,y)_{\ell^2}&=\sum_{j=0}^{l}x_{j,0}\left(R_{\lambda}^{(1)}e_{j,0},\sum\limits_{i=0}^{l}\sum_{k=0}^{k_i-1}y_{i,k}e_{i,k}\right)\\&=
\sum_{j=0}^lx_{j,0}\sum_{i=0}^j\sum_{k=0}^{k_i-1}\lambda^k
P_i(\lambda)(Q_j(\lambda)+m(\lambda)P_j(\lambda))\overline{y}_{i,k}.
\end{split}
\]
Thus, \eqref{estimate1} yields
\[
|(R_{\lambda}^{(1)}x,y)_{\ell^2}|\le
C\max\{1,|\lambda|^{N-1}\}\sum_{j=0}^l|x_{j,0}|\sum_{i=0}^j\sum_{k=0}^{k_i-1}q^{n_j-n_i}|{y}_{i,k}|.
\]
Notice that $n_j-n_i=\sum\limits_{l=i+1}^{j}k_l\ge
\sum\limits_{l=i+1}^{j}1=j-i$ and, therefore, we have
\[
\begin{split}
|(R_{\lambda}^{(1)}x,y)_{\ell^2}|&\le
C\max\{1,|\lambda|^{N-1}\}\sum_{j=0}^l\sum_{i=0}^{j}\sum_{k=0}^{k_i-1}q^{{j-i}}|x_{j,0}||{y}_{i,k}|\\&\le
C\max\{1,|\lambda|^{N-1}\}\sum_{s=0}^{l}q^{{s}}\sum_{j=s}^l|x_{j,0}|\sum_{k=0}^{k_{j-s}-1}|{y}_{j-s,k}|\\
&\le C\sqrt{N}\max\{1,|\lambda|^{N-1}\}\sum_{s=0}^{l}q^{{i}}\Vert
x\Vert_{\ell^2}\Vert y\Vert_{\ell^2}
=\widetilde{C}\frac{1-q^{{l+1}}}{1-q}\Vert x\Vert_{\ell^2}\Vert
y\Vert_{\ell^2}.
\end{split}
\]
Hence, $R_{\lambda}^{(1)}$ is a bounded operator. Similarly, one
can prove the boundedness of $R_{\lambda}^{(2)}$. So, we have
proved that $GR_{\lambda}\cP_{\cH_0}$ is bounded. Since $G^{-1}$
is bounded, $R_{\lambda}\cP_{\cH_0}$ is also bounded.

Further, let $k\in\{1,\dots,N-1\}$. From~\eqref{f_2_6} one can
deduce
\[
R_{\lambda}e_{j,k}=e_{j,k-1}+\lambda
R_{\lambda}e_{j,k-1}=V_ke_{j,k}+\lambda R_{\lambda}V_ke_{j,k},
\]
where $V_k: e_{j,k}\mapsto e_{j,k-1}$ is an isometric operator
from $\cH_k$ to $\cH_{k-1}$. If $h\in\cH_k$ then $R_{\lambda}h=V_k
h+\lambda R_{\lambda}V_k h$. So, the boundedness of $V_1$ and
$R_{\lambda}\cP_{\cH_0}$ implies that $R_{\lambda}\cP_{\cH_1}$ is
bounded. Analogously, $R_{\lambda}\cP_{\cH_k}$ is bounded for
$k\in\{2,\dots,N-1\}$. This implies that
$R_{\lambda}=\sum\limits_{i=0}^{N-1}R_{\lambda}\cP_{\cH_i}$ is a
bounded operator. The latter means that the domain of
$R_{\lambda}$ is $\ell^2_{[0,\infty)}$. Since $\lambda$ is not an
eigenvalue, we have $\ker(H-\lambda)=\{0\}$ and
$\ran(H-\lambda)=\ell^2_{[0,\infty)}$. Now, applying the Banach
theorem on inverse operators we obtain that $\lambda\in\rho(H)$.

The necessity of~\eqref{estimate1} follows from~\cite{DMS} and the
relation
\[
(R_{\lambda}e_{j,0})_{i,k_i-1}=\begin{cases}
       P_i(\lambda)(Q_j(\lambda)+m(\lambda)P_j(\lambda)),& i\le j;\\
       P_j(\lambda)(Q_i(\lambda)+m(\lambda)P_i(\lambda)),& i>j,\\
 \end{cases}
\]
which directly follows from~\eqref{f_2_5}.
\end{proof}
\begin{remark}
In the case of nonsymmetric tridagonal operators
Theorem~\ref{main_result} was proved in~\cite{AptKV}.
In fact, we have extended the scheme proposed in~~\cite{AptKV} to
the case of generalized Jacobi matrices.
A similar result for banded matrices with nonvanishing
extreme diagonals in terms of the corresponding vector polynomials
was obtained in~\cite{BO}.
\end{remark}

\section{The Floquet theory}

In the present section, by using Theorem~\ref{main_result}, we
give a description of spectra of periodic generalized Jacobi
operators.
\begin{definition}\label{periodic_gjm}
Let $s\in\dN$. A generalized Jacobi matrix satisfying the
properties
\[
A_{js+k}=A_k,\quad B_{js+k}=B_{k},\quad\e_{js+k}=\e_k,\quad
j\in\dZ_+,\quad k\in\{0,\dots,s-1\}
\]
will be called an $s$-periodic generalized Jacobi matrix. The
corresponding generalized Jacobi operator in $\cH_{[0,\infty)}$
will be also called an $s$-periodic generalized Jacobi operator.
\end{definition}

Evidently, any $s$-periodic generalized Jacobi matrix satisfies
the assumptions~\eqref{assumption1},~\eqref{assumption2} and we
have
\begin{equation}\label{per_mat}
\cW_{js+k}(\lambda)=\cW_k(\lambda),\quad j\in\dZ_+,\quad
k\in\{0,\dots,s-1\}.
\end{equation}
The main tool for analysis of periodic generalized Jacobi
operators is the following matrix
\[
T(\lambda):=\cW_{[0,s-1]}(\lambda)=\begin{pmatrix}
  -\e_{s-1}b_{s-1} Q_{s-1}(\lambda) & -Q_{s}(\lambda) \\
  \e_{s-1}b_{s-1} P_{s-1}(\lambda) & P_{s}(\lambda) \\
\end{pmatrix}.
\]
The matrix $T(\lambda)$ is called {\it the monodromy matrix}.
Using~\eqref{per_mat}, we get the following relation
\begin{equation}\label{f_3_2}
\cW_{[0,js+k-1]}(\lambda)=T^j(\lambda)\cW_{[0,k-1]}(\lambda),
\quad j\in\dZ_+,\quad k\in\{1,\dots,s\}.
\end{equation}
Let $w_1=w_1(\lambda)$ and $w_2=w_2(\lambda)$ be the roots of the
characteristic equation $\det(T(\lambda)-w)=0$. Introduce the
following notations
\[
E:=\{\lambda\in\dC: |w_1(\lambda)|=|w_2(\lambda)|\},\quad
E_p:=\{\lambda\in\dC:
P_{s-1}(\lambda)=0,\,|b_{s-1}Q_{s-1}(\lambda)|>|P_{s}(\lambda)|\}.
\]
Now, we are ready to give a description of spectra of periodic
generalized Jacobi operators.
\begin{theorem}\label{specper}
The spectrum of an $s$-periodic generalized Jacobi operator has
the form
\[
\sigma(H)=E\cup E_p,\quad \sigma_p(H)=E_p.
\]
\end{theorem}

\begin{proof}
Since $\det T(\lambda)\equiv 1$, we have that
\[
w_1(\lambda)w_2(\lambda)=1.
\]

\noindent{\it Step 1.} First, let us prove that
\[
\{\lambda\in\dC: P_{s-1}(\lambda)\ne
0,\,|w_1(\lambda)|\ne|w_2(\lambda)|\}\subset\rho(H).
\]
To be definite, assume that $|w_1(\lambda)|>|w_2(\lambda)|$. In
this case, we see that
\begin{equation}\label{f_3_2s}
T(\lambda)=
\begin{pmatrix}
x_1&x_2\\
x_3&x_4\\
\end{pmatrix}
\begin{pmatrix}
w_1(\lambda)&0\\
0&w_2(\lambda)\\
\end{pmatrix}
\begin{pmatrix}
x_4&-x_2\\
-x_3&x_1\\
\end{pmatrix},\quad
\det\begin{pmatrix}
x_1&x_2\\
x_3&x_4\\
\end{pmatrix}=1.
\end{equation}
Now, \eqref{f_3_2} can be rewritten in the form
\begin{equation}\label{f_3_3}
\cW_{[0,js+k-1]}(\lambda)=
\begin{pmatrix}
x_1&x_2\\
x_3&x_4\\
\end{pmatrix}
\begin{pmatrix}
w_1^j&0\\
0&w_2^j\\
\end{pmatrix}
\begin{pmatrix}
x_4&-x_2\\
-x_3&x_1\\
\end{pmatrix}
\cW_{[0,k-1]}(\lambda).
\end{equation}
Further, \eqref{f_3_3} is reduced to the following relation
\begin{equation}\label{f_3_4}
\cW_{[0,js+k-1]}(\lambda)=
\begin{pmatrix}
x_1x_4w_1^j-x_2x_3w_2^j & -x_1x_2w_1^j+x_1x_2w_2^j\\
x_3x_4w_1^j-x_4x_3w_2^j & -x_3x_2w_1^j+x_1x_4w_2^j\\
\end{pmatrix}
\cW_{[0,k-1]}(\lambda).
\end{equation}
Multiplying~\eqref{f_3_4} by the vector $(1\,0)^{\top}$ we obtain
\[
\left(
\begin{array}{c}
  -\e_{k-1}b_{k-1} Q_{js+k-1}(\lambda) \\
  \e_{k-1}b_{k-1} P_{js+k-1}(\lambda) \\
\end{array}
\right)=
\begin{pmatrix}
x_1x_4w_1^j-x_2x_3w_2^j & -x_1x_2w_1^j+x_1x_2w_2^j\\
x_3x_4w_1^j-x_4x_3w_2^j & -x_3x_2w_1^j+x_1x_4w_2^j\\
\end{pmatrix}
\left(
\begin{array}{c}
  -\e_{k-1}b_{k-1} Q_{k-1}(\lambda) \\
  \e_{k-1}b_{k-1} P_{k-1}(\lambda) \\
\end{array}
\right).
\]
Thus, for the polynomials $Q_{\cdot}$ one has
\begin{equation}\label{f_3_5}
Q_{js+k-1}(\lambda)=-w_1^j(x_1x_4Q_{k-1}(\lambda)+x_1x_2P_{k-1}(\lambda))+w_2^j(x_2x_3Q_{k-1}(\lambda)+x_1x_2P_{k-1}(\lambda)).
\end{equation}
Similarly, for the polynomials $P_{\cdot}$ we have
\begin{equation}\label{f_3_6}
P_{js+k-1}(\lambda)=-w_1^j(x_3x_4Q_{k-1}(\lambda)+x_3x_2P_{k-1}(\lambda))+w_2^j(x_4x_3Q_{k-1}(\lambda)+x_1x_4P_{k-1}(\lambda)).
\end{equation}
Note that $x_3\ne 0$. Indeed,  if $x_3=0$ then according
to~\eqref{f_3_2s} we would have that $P_{s-1}(\lambda)=0$. Now,
formulas~\eqref{f_3_5} and~\eqref{f_3_6} yield
\begin{equation}\label{f_3_7}
Q_{js+k-1}(\lambda)-\frac{x_1}{x_3}P_{js+k-1}=-w_2^j\left(\frac{x_1}{x_3}P_{k-1}(\lambda)+Q_{k-1}(\lambda)\right).
\end{equation}
For brevity, define $C_k(\lambda):=-\left(\displaystyle{\frac{x_1}{x_3}}P_{k-1}(\lambda)+Q_{k-1}(\lambda)\right)$.
Then the equality~\eqref{f_3_7} can be rewritten in the following way
\begin{equation}\label{f_3_8s}
Q_{js+k-1}(\lambda)-\frac{x_1}{x_3}P_{js+k-1}=(\widetilde{w}_2)^{js+k-1}C_k(\lambda),\quad
\widetilde{w}_2=w_2^{1/s},\quad k\in\{1,\dots,s\}.
\end{equation}
Since $|w_2|<1$, it follows from~\eqref{f_3_8s} that
$\displaystyle{\xi(\lambda)-\frac{x_1}{x_3}}\pi(\lambda)\in\ell^2_{[0,\infty)}$.
A linear independence of $\pi(\lambda)$ and $\xi(\lambda)$,
and~\eqref{f_3_6},~\eqref{f_3_5}  imply
$\pi(\lambda)\notin\ell^2_{[0,\infty)}$. Now we are ready to
verify the condition~\eqref{estimate1} of
Theorem~\ref{main_result}. Let us assume that $i<j$. Then
\[
\left|P_i(\lambda)\left[Q_j(\lambda)-\frac{x_1}{x_3}P_j(\lambda)\right]\right|=|(-\widetilde{w}_1^if_1^{(k_1)}(\lambda)+\widetilde{w}_2^if_2^{(k_1)}(\lambda))
\widetilde{w}_2^jC_{k_2}(\lambda)|,
\,\widetilde{w}_1=\widetilde{w}_2^{-1},\, k_1,k_2\in\{1,\dots,s\},
\]
where $k_1\equiv i(\text{mod} s)$ and $k_2\equiv j(\text{mod} s)$.
Since
$(n_j-n_i)/N=\left(\sum\limits_{l=i+1}^{j}k_l\right)/N\le\left(\sum\limits_{l=i+1}^{j}N\right)/N=j-i$
we have that $(n_j-n_i)/N\le j-i$. Thus, one obtains
\[
\left|P_i(\lambda)\left[Q_j(\lambda)-\frac{x_1}{x_3}P_j(\lambda)\right]\right|\le|\widetilde{w}_2|^{j-i}C(\lambda)\le
q^{n_j-n_i}C(\lambda),
\]
where
$C(\lambda)=\sup\limits_{k_,k_2\in\{1,\dots,s\}}\{(|f_1^{(k_1)}(\lambda)|+|f_2^{(k_2)}(\lambda)|)|C_{k_2}(\lambda)|\}$
and $q=\widetilde{w}_2<1$.

\noindent {\it Step 2.} Let us show that
\[
\{\lambda\in\dC:
P_{s-1}(\lambda)=0,\,|b_{s-1}Q_{s-1}(\lambda)|>|P_{s}(\lambda)|\}\subset\sigma_p(H).
\]
In this case the monodromy matrix can be represented as follows
\[
T(\lambda)=\begin{pmatrix}
  -\e_{s-1}b_{s-1} Q_{s-1}(\lambda) & -Q_{s}(\lambda) \\
    0                          & P_{s}(\lambda) \\
\end{pmatrix}=
\begin{pmatrix}
1&x_1\\
0&1\\
\end{pmatrix}
\begin{pmatrix}
w_1(\lambda)&0\\
0&w_2(\lambda)\\
\end{pmatrix}
\begin{pmatrix}
1&-x_1\\
0&1\\
\end{pmatrix},
\]
where $w_1(\lambda)=-\e_{s-1}b_{s-1} Q_{s-1}(\lambda)$ and
$w_2(\lambda)=P_{s}(\lambda)$. Further, we have that
\[
\left(
\begin{array}{c}
  -\e_{k-1}b_{k-1} Q_{js+k-1}(\lambda) \\
  \e_{k-1}b_{k-1} P_{js+k-1}(\lambda) \\
\end{array}
\right)=
\begin{pmatrix}
w_1^j & w_1^jx_1-w_2^jx_1\\
0     & w_2^j\\
\end{pmatrix}
\left(
\begin{array}{c}
  -\e_{k-1}b_{k-1} Q_{k-1}(\lambda) \\
  \e_{k-1}b_{k-1} P_{k-1}(\lambda) \\
\end{array}
\right),
\]
that is, the following formulas hold true
\[
P_{js+k-1}(\lambda)=w_2^jP_{k-1}(\lambda),
\]
\[
Q_{js+k-1}(\lambda)=-w_1^jQ_{k-1}(\lambda)+(w_1^j-w_2^j)x_1P_{k-1}(\lambda).
\]
Since $|w_2(\lambda)|<|w_1(\lambda)|$,
$G\pi(\lambda)\in\ell^2_{[0,\infty)}$ and, therefore,
$\pi(\lambda)\in\ell^2_{[0,\infty)}$. So, we have proved that
$E_p\subset\sigma(H)$.

\noindent{\it Step 3.} By the same reasoning as in the Step 1 it
can be shown that
\[
\{\lambda\in\dC:
P_{s-1}(\lambda)=0,\,|b_{s-1}Q_{s-1}(\lambda)|>|P_{s}(\lambda)|\}
\subset\rho(H).
\]

\noindent{\it Step 4.} We complete this proof by proving the
following inclusion
\[
\{\lambda\in\dC: |w_1(\lambda)|=|w_2(\lambda)|\}\subset\sigma(H).
\]
Since $w_1w_2=1$ we have that $|w_1(\lambda)|=|w_2(\lambda)|=1$.
If $T$ is diagonalizable  then, due to~\eqref{f_3_5},
\eqref{f_3_6}, there exist numbers $\alpha_k^{(1)}(\lambda),
\alpha_k^{(2)}(\lambda),\beta_k^{(1)}(\lambda),\beta_k^{(2)}(\lambda)\in\dC$
such that
\begin{equation}\label{f_3_8}
Q_{js+k-1}(\lambda)=w_2^{-j}\alpha_k^{(1)}+w_2^j\alpha_k^{(2)},\quad
j\in\dZ_+,\quad k\in\{1,\dots,s\}.
\end{equation}
\begin{equation}\label{f_3_9}
P_{js+k-1}(\lambda)=w_2^{-j}\beta_k^{(1)}+w_2^j\beta_k^{(2)},\quad
j\in\dZ_+,\quad k\in\{1,\dots,s\}.
\end{equation}
Since the sequences $Q_{\cdot}$ and $P_{\cdot}$ are linearly
independent, the sequence $\{Q_j+mP_j\}_{j=0}^{\infty}$ is nonzero
for any $m\in\dC$. Also, observe that the sequence
$\{|w^j\alpha+\beta|\}_{j=1}^{\infty}$ ($\alpha\ne 0$, $|w|=1$)
converges only for $w=1$. Taking into account these observations
one can conclude that the series
\[
\sum_{j=0}^{\infty}|Q_j(\lambda)+mP_j(\lambda)|^2=
\sum_{i=0}^{\infty}\sum_{k=0}^{s-1}|\alpha_k^{(1)}+m\beta_k^{(1)}+w_2^{2i}(\alpha_k^{(2)}+m\beta_k^{(2)})|^2
\]
does not converge for any number $m\in\dC$. So, we have that
$\xi(\lambda)+m\pi(\lambda)\notin\ell^2_{[0,\infty)}$ for any
$m\in\dC$. Similarly, one can conclude that
$\pi(\lambda)\notin\ell^2_{[0,\infty)}$.

If $T$ is similar to a Jordan block then $w_1=w_2$ and $w_1w_2=1$.
So, $w_1=\pm 1$. First, let us consider the case $w_1=1$. The
monodromy matrix takes the form
\[
T(\lambda)=
\begin{pmatrix}
x_1&x_2\\
x_3&x_4\\
\end{pmatrix}
\begin{pmatrix}
1&1\\
0&1\\
\end{pmatrix}
\begin{pmatrix}
x_4&-x_2\\
-x_3&x_1\\
\end{pmatrix}.
\]
Further,~\eqref{f_3_2} yields
\[
\cW_{[0,js+k-1]}(\lambda)=
\begin{pmatrix}
x_2x_4-x_1x_3j-x_2x_3 & -x_2^2+x_1^2j-x_1x_2\\
-x_3^2j               & -x_3x_2+x_3x_1j+x_4x_1\\
\end{pmatrix}
\cW_{[0,k-1]}(\lambda).
\]
Thus for any $m\in\dC$ the vectors $\xi(\lambda)+m\pi(\lambda)$
and $\pi(\lambda)$ do not belong to $\ell^2_{[0,\infty)}$.
Analogously, one can consider the case $w_1=-1$.
\end{proof}
\begin{remark}
A description of spectra of classical Jacobi operators was
obtained in~\cite{Ger}.
\end{remark}
Let us remind that $w_1$ and $w_2$ are the roots of the equation
\begin{equation}\label{f_3_10}
w^2-(P_s(\lambda)-\e_{s-1}b_sQ_{s-1}(\lambda))w+1=0.
\end{equation}
Due to~\eqref{f_3_10}, one has
\begin{equation}\label{f_3_11}
w_1+w_2=P_s(\lambda)-\e_{s-1}b_sQ_{s-1}(\lambda),
\end{equation}
\begin{equation}\label{f_3_12}
w_1w_2=1.
\end{equation}
\begin{remark}[\cite{BK}]\label{rem_3_3}
Formulas~\eqref{f_3_11},~\eqref{f_3_12} allow us to give another
description of $E$
\begin{equation}\label{h_ex_1}
E=\{\lambda\in\dC:
(P_s(\lambda)-\e_{s-1}b_sQ_{s-1}(\lambda))\in[-2,2]\}.
\end{equation}
\end{remark}

\begin{example} Let us consider the following difference equation
\begin{equation}\label{ex_1}
\frac{1}{2}u_{j+1}-\lambda^2u_j-\frac{1}{2}u_{j-1}=0\quad
(j\in\dN).
\end{equation}
Clearly, the three-term recurrence relations~\eqref{ex_1} generate
the following $1$-periodic generalized Jacobi matrix
\[
H=\begin{pmatrix}
A   & B & {\bf 0}\\
B   & A &\ddots    \\
{\bf 0} &  \ddots&\ddots\\
\end{pmatrix},\quad
A=\begin{pmatrix}
0&0\\
1&0\\
\end{pmatrix},\quad
B=\begin{pmatrix}
0&\frac{1}{2}\\
0&0\\
\end{pmatrix}.
\]
In this case, by easy calculations, we have
\[
P_0(\lambda)=1,\quad P_1(\lambda)=2\lambda^2,\quad
Q_0(\lambda)=0,\quad Q_1(\lambda)=2.
\]
Since $P_0(\lambda)=1\ne 0$, it follows from Theorem~\ref{specper}
that $\sigma_p(H)=\emptyset$. Next, according to
Theorem~\ref{specper} and~\eqref{h_ex_1}, we have
\[
\sigma(H)=\{\lambda\in\dC: 2\lambda^2\in[-2,2]\}=[-1,1]\cup[-i,i].
\]
\end{example}

 By the same reasoning as in~\cite{BK}, one can prove the
following statement on the structure of the set $E$.
\begin{proposition}[\cite{BK}]\label{structure}
The compact set $E$ has no interior points. The open set
$D:=\dC\setminus E$ is connected. The functions $w_1$ and $w_2$
are single-valued in  $D$.
\end{proposition}

\section{Pad\'e approximants}

Our goal in this section is to prove convergence results for
Pad\'e approximants.

\begin{definition}[\label{Pade}{\rm\cite{NikSor}}] The $[L/M]$ Pad\'e approximant
to the function
$\f(\lambda)=\displaystyle{-\sum\limits_{j=0}^{+\infty}\frac{s_j}{{\lambda}^{j+1}}}$
is defined as a ratio
\[
f^{[L/M]}(\lambda)=\frac{A^{[L/M]}\left(\frac{1}{\lambda}\right)}
{B^{[L/M]}\left(\frac{1}{\lambda}\right)}
\]
of two polynomials $A^{[L/M]}$, $B^{[L/M]}$ of formal degree $L$
and $M$, respectively, such that $B^{[L/M]}(0)\ne 0$ and
\[
\sum_{j=0}^{+\infty}\frac{s_j}{{\lambda}^{j+1}}+f^{[L/M]}(\lambda)=
O({\lambda}^{-(L+M+1)})\quad(\lambda\to\infty).
\]
\end{definition}

In the case $L=M=n$, the $[n/n]$ Pad\'e approximant is also called
{\it the $n$-th diagonal Pad\'e approximant}.

Let us consider the Weyl function
$m(\lambda)=[(H-\lambda)^{-1}e,e]$, where the corresponding matrix
$H$ satisfies the assumptions~\eqref{assumption1},
\eqref{assumption2}. The representation~\eqref{SWfun} of
$m_{[0,j-1]}$ yields
\begin{equation}\label{Qasympt-1}
m_{[0,j-1]}(\lambda)=- \frac{Q_j(\lambda)}{P_j(\lambda)}=
-\frac{s_{0}}{\lambda}-\frac{s_{1}}{\lambda^{2}}-\dots-\frac{s_{2n_j-2}}{\lambda^{2n_j-1}}
+O\left(\frac{1}{\lambda^{2n_j}}\right) \quad(\lambda\to\infty),
\end{equation}
where ${s_{i}}=[H^ie,e]$ ($i=0,\dots,2n_j-2$). Moreover, it was
shown in~\cite{DD} that  $m_{[0,j-1]}(\lambda)$ has the following
asymptotic expansion
\begin{equation}\label{Qasympt+1}
m_{[0,j-1]}(\lambda)=
-\sum_{i=0}^{2n_j-2+k_j}\frac{s_{i}}{\lambda^{i+1}}
+O\left(\frac{1}{\lambda^{2n_j+k_j}}\right)
\quad(\lambda\to\infty),
\end{equation}
where ${s_{i}}=[H^ie,e]$ ($i=0,\dots,2n_j-2+k_j$). The latter
means that the rational function
\begin{equation}\label{PadeAp}
f^{[n_j/n_j]}(\lambda)=m_{[0,j-1]}(\lambda)=\frac{\displaystyle
A^{[n_j/n_j]}\left({1}/{\lambda}\right)} {\displaystyle
B^{[n_j/n_j]}\left({1}/{\lambda}\right)}=
\frac{\displaystyle-\frac{1}{\lambda^{n_j}}Q_j(\lambda)}{\displaystyle\frac{1}{\lambda^{n_j}}P_j(\lambda)}\quad
(j=1,2,\dots)
\end{equation}
is the $[n_j/n_j]$ Pad\'e approximant to  $m$. Besides, it follows
from the Pad\'e theorem (see~\cite[Theorem~1.4.3]{Baker}), that
for $L$ and $M$ satisfying
\[
L\ge n_j,\,\,\,M\ge n_j,\,\,\, L+M\le 2n_j+{k_j}-1
\]
the $[L/M]$ Pad\'e approximants coincide with $f^{[n_j/n_j]}$, and
for $L$, $M$ satisfying
\[
L\le n_j+{k_j}-1,\,\,\,M\le n_j+{k_j}-1,\,\,\, L+M\ge 2n_j+{k_j}
\]
the $[L/M]$ Pad\'e approximants do not exist  (for details
see~\cite{DD}).

In what follows we need the following definition.

\begin{definition}\label{num_ran}
The set $\Theta(H):=\{(Hy,y)_{\ell^2}: \Vert y\Vert=1\}\subset\dC$
is called a numerical range of the operator $H$.
\end{definition}
Clearly, the numerical range of a bounded operator is a bounded
set. By the Hausdorff theorem we have that
$\sigma(H)\subset\overline{\Theta(H)}$ (see~\cite{Kat}).
\begin{theorem}\label{un_loc_con}
Let $m(\lambda)=[(H-\lambda)^{-1}e,e]$ be the Weyl function of $H$
satisfying~\eqref{assumption1}, \eqref{assumption2}. Then there
exists a subsequence of diagonal Pad\'e approximants
$f^{[n_j/n_j]}$ to $m$, which converges to $m$ locally uniformly
in $\dC\setminus\overline{\Theta(H)}$.
\end{theorem}
\begin{proof}
First, note that $\Theta(H_{[0,n]})\subset\Theta(H)$, and,
therefore,
$\overline{\Theta(H_{[0,n]})}\subset\overline{\Theta(H)}$. As a
consequence, we have that if
$\lambda\in\dC\setminus\overline{\Theta(H)}$ then
$\lambda\in\rho(H_{[0,n]})$ for all $n\in\dZ_+$. Let
$\lambda\in\dC\setminus\overline{\Theta(H)}$. Then
$\ran(H-\lambda)=\ell^2_{[0,\infty)}$ and for any finite vector
$\phi$ we have
\[
(H_{[0,j]}-\lambda)^{-1}\phi\to(H-\lambda)^{-1}\phi,\quad
j\to+\infty.
\]
Thus, one can obtain
\begin{equation}\label{conv_pa}\begin{split}
f^{[n_j/n_j]}(\lambda)=m_{[0,j-1]}(\lambda)=[(H_{[0,j-1]}-\lambda)^{-1}e,e]\\=((H_{[0,j-1]}-\lambda)^{-1}e,Ge)_{\ell^2}\to((H-\lambda)^{-1}e,Ge)_{\ell^2}=m(\lambda).\end{split}
\end{equation}
According to~\cite[p.336 Theorem 3.2]{Kat} one has
\[
\Vert(H_{[0,j]}-\lambda)^{-1}\Vert\le
\frac{1}{\dist(\lambda,\Theta(H_{[0,n]}))}
\le\frac{1}{\dist(\lambda,\Theta(H))}.
\]
Hence, the family  $f^{[n_j/n_j]}=m_{[0,j]}$ is uniformly bounded
on compact sets in $\dC\setminus\overline{\Theta(H)}$. Actually,
the following estimates hold true
\[
|m_{[0,j]}(\lambda)|=|((H_{[0,j]}-\lambda)^{-1}e,Ge)_{\ell^2}|\le\Vert(H_{[0,j]}-\lambda)^{-1}e\Vert_{\ell^2}\Vert
Ge\Vert_{\ell^2}\le \frac{\Vert
Ge\Vert_{\ell^2}}{\dist(\lambda,\Theta(H))}.
\]
So, the family  $f^{[n_j/n_j]}$ is uniformly bounded and,
therefore, $f^{[n_j/n_j]}$ is precompact. Now, to complete the
proof it is sufficient to apply~\eqref{conv_pa} and the Vitali
theorem.
\end{proof}
\begin{remark}
In the proof of Theorem~\ref{un_loc_con} we used the method
proposed in~\cite{BK} for complex Jacobi matrices. Note that
Theorem~\ref{un_loc_con} is a generalization of~\cite[Theorem
1]{Gonc82}. More precisely, we do not suppose existence of all
diagonal Pad\'e approximants and all poles of the existed diagonal
Pad\'e approximants belong to the convex set
$\overline{\Theta(H)}$.
\end{remark}

 Further, following the scheme proposed in~\cite{AptKV}, we find
out a behavior of  the associated polynomials at the points of the
resolvent set.

\begin{proposition}\label{pol_as}
For any $\lambda\in\rho(H)$ the following inequality holds true
\begin{equation}\label{n_pol_as}
\limsup_{j\to+\infty}|P_j(\lambda)|^{1/j}>1.
\end{equation}
\end{proposition}
\begin{proof} In fact, the proof is in line with that in~\cite{AptKV}.
However, we give the proof here for the convenience of the
readers. Using~\eqref{Ostrogr2} and~\eqref{reprWs}, one can see
\begin{equation}\label{PA_h_1}
P_j(\lambda)(\e_j b_j W_{j+1}(\lambda))-P_{j+1}(\lambda)(\e_j b_j
W_j(\lambda))=1.
\end{equation}
Due to~\eqref{assumption1}, \eqref{assumption2},
and~\eqref{estimate1}, one has
\begin{equation}\label{PA_h_2}
|\e_j b_j W_{j}(\lambda)|\le C_1{q_1}^{j},\quad 0<q_1<1.
\end{equation}
It follows from~\eqref{PA_h_1} and~\eqref{PA_h_2} that the
sequence $P_j(\lambda)$ can not be majorized by a geometric
sequence $p^j$ with $p<1/q_1$, that is, for any $p<1/q_1$ and any
positive constant $C_2$ the inequality
\[
|P_j(\lambda)|\le C_2p^j
\]
is not satisfied for an infinite number of indices $j$. We thus
have
\[
|P_{j_k}(\lambda)|\ge C_2p^{j_k}.
\]
If we choose $p$ such that $1/q_1>p>1$ we obtain the required
result.
\end{proof}

From these statements one can deduce the following result on
convergence of diagonal Pad\'e approximants.

\begin{theorem}\label{pointwise}
Under the assumptions~\eqref{assumption1}, \eqref{assumption2},
for any $\lambda\in\rho(H)$ there exists a subsequence of diagonal
Pad\'e approximants to  $m(\lambda)=[(H-\lambda)^{-1}e,e]$, which
converges to $m(\lambda)$ at $\lambda$.
\end{theorem}
\begin{proof}
From~\eqref{n_pol_as} we get that there exists a subsequence $j_k$
such that
\begin{equation}\label{f_subs}
|P_{j_k}(\lambda)|\ge C_2p^{j_k},\quad p>1.
\end{equation}
Further, Theorem~\ref{main_result} and~\eqref{f_subs} imply the
relation
\[
\left|m(\lambda)+\frac{Q_{j_k}(\lambda)}{P_{j_k}(\lambda)}\right|=
\left|\frac{W_{j_k}(\lambda)}{P_{j_k}(\lambda)}\right|\le
C_3\left(\frac{q}{p}\right)^{j_k},
\]
which completes the proof.
\end{proof}

\noindent{\bf Acknowledgments}. I express my gratitude to
Professor V.A.~Derkach for reading the manuscript and giving
helpful comments. I also thank the anonymous referee for helpful remarks.