\section{\textbf{Introduction}} 
The spectral analysis of operators defined over sequence spaces has been treated by many
researchers worldwide. The literature contains the spectrum and fine spectrum of several classes of Toeplitz operators \cite{t3,bir,duren}, Ces{\`a}ro operators \cite{cesaro3,cesaro2,cesaro1}, Rhaly operators \cite{rhaly2,rhaly1}, operators generated by various difference equations \cite{diff1,diff2,diff3,patra2019some,patra2020spectrum}, etc. For a detailed review, one may refer to the survey articles \cite{baliarsingh2021survey,yecsilkayagil2019survey} and the references therein. In particular, numerous mathematicians have focused their research on an important class of tridiagonal matrix, known as Jacobi matrix  \cite{bairamov2001non,janas2003similarity,janas1998point,}. The Jacobi matrix $J$ is generated by the difference equation 
\[(Jy)_n=a_{n-1}y_{n-1}+b_ny_n+a_ny_{n+1},~~ y_n\in \mathbb{C^N}\]
  with certain initial conditions where $\{a_n\}$ and $\{b_n\}$ are complex sequences. In this paper, we focus on the spectral properties of a class of Jacobi-like penta-diagonal band matrices defined over the sequence space $\ell_p (1 < p < \infty)$ where the entries in the non-zero bands form sequences with two limit points.



Let $\ell_{p}$ represents the Banach space of  $p$-absolutely summable sequences of real or complex numbers with the norm 
\[ \norm{x}_{p}=\left( \sum_{n=1}^{\infty}|x_n|^p\right)^{\frac{1}{p}}.\] 
Also let $\mathcal{D}_p$ denotes the set of all diagonal operators on $\ell_p.$ For any operator $T \in \mathcal{D}_p,$ $\mbox{diag}(T)$ represents the sequence in the diagonal of $T.$ In this work we investigate the spectral properties of a class of operators $T$ defined over $\ell_p$ represented by the following form:
\[T = S_r^2D_1 + D_2S_{\ell}^2 + D_3\]
where $S_r, S_{\ell} : \ell_p \to \ell_p$, denotes the right shift operator, left shift operator respectively and $D_1, D_2, D_3 \in \mathcal{D}_p$ with $\mbox{diag}(D_1) = \{c_n\},$ $\mbox{diag}(D_2) = \{b_n\}$ and $\mbox{diag}(D_3)=\{a_n\}$. We further assume that the subsequences $\{a_{2n-1}\},$ $\{b_{2n-1}\}$, $\{c_{2n-1}\}$ converges to the real numbers $r_1,$ $s_1,$ $s_1$ respectively and $\{a_{2n}\},$ $\{b_{2n}\}$, $\{c_{2n}\}$ converges to the real numbers $r_2,$ $s_2,$ $s_2$ respectively where $s_1 \neq 0$ and $s_2 \neq 0$.  

Our focus is to investigate the spectral properties of the operator $T$ using compact perturbation technique. Let us consider another operator $T_0$ over $\ell_p$ defined by
\[T_0 = S_r^2D_1^\prime + D_2^\prime S_{\ell}^2 + D_3^\prime\] 
where $D_1^\prime, D_2^\prime, D_3^\prime \in \mathcal{D}_p$ with 
\[\mbox{diag}(D_1^\prime) =\mbox{diag}(D_2^\prime) =  \{s_1, s_2, s_1, s_2, \cdots\}, \ \mbox{diag}(D_3^\prime) = \{r_1, r_2, r_1, r_2, \cdots\}.\]
Using the properties of compact operators, it can be proved that $T - T_0$ is a compact operator over $\ell_{p}$. Both the operators $T$ and $T_0$ can be represented by the following penta-diagonal matrices
\[T=\begin{pmatrix}
	a_1 &0& b_1 &0&0&\cdots &  \\
	0 & a_2&0 & b_2& 0&\cdots&\\
	c_1&0&a_3&0&b_3&\cdots&\\
	0&c_2&0&a_4&0&\cdots\\
	0&0&c_3&0&a_5&\cdots\\
	\vdots  &\vdots & \vdots  & \vdots &\vdots&\ddots
	
	\end{pmatrix},
	T_0=
		\begin{pmatrix}
			r_1 & 0 &s_1&0&0&\cdots &  \\
			0 & r_2 & 0& s_2&0& \cdots&\\
			s_1&0&r_1&0&s_1&\cdots&\\
			0&s_2&0&r_2&0&\cdots&\\
			0&0&s_1&0&r_1&\cdots\\
			\vdots  & \vdots  & \vdots &\vdots&\vdots& \ddots
		\end{pmatrix}.
	\]
 
  We obtain the spectrum, fine spectrum and the sets of various spectral subdivisions of the operator $T_0.$ It is interesting to note that the spectrum of $T_0$ is given by \[[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2],\] which is also its essential spectrum with no eigenvalues. Later we investigate how the spectrum of $T$ and $T_0$ are related. Few results on the essential spectrum of $T$ (which is identical to the essential spectrum of $T_0$) being devoid of its eigenvalues are also derived. This helps us to characterize the point spectrum of $T.$ Theory of difference equations plays an important role in our study. We use various results on the asymptotic behaviour of solutions of difference equations to demonstrate the findings of our paper. For more details on difference equations one can refer \cite{difference}. 

 
 
 
The remainder of paper is organized as follows: section 2 is devoted to introduce some terminologies and results which are relevant to our work. Section 3 contains the results on the spectrum and fine spectrum of $T_0$ over $\ell_p.$ The spectral properties of $T$ are discussed in section 4.
 

 \section{\bf Preliminaries}
  Let $X$ and $Y$ are Banach spaces and for any operator $T:X \to Y$, $N(T)$ and $R(T)$ denote the null space and range space of $T$ respectively. The operator $T^*:Y^* \to X^*$ is called the adjoint operator and defined by
 \begin{equation*}
 	(T^{*}f)(x)=f(Tx) \quad \text{for all}~ f \in Y^{*} \text{and}~ x\in X
 \end{equation*}
 where $X^*$, $Y^*$ are the dual spaces of $X$ and $Y$ respectively. $B(X)$ denotes the set of all bounded linear operators from $X$ to itself. For any $T\in B(X)$, the resolvent set $ \rho(T, X)$ of $T$ is the set of all $\lambda $ in the complex plane such that $(T-\lambda I)$ has a bounded inverse in $X$ where $I$ is the identity operator defined over $X$. The complement of resolvent set in the complex plane $\mathbb{C} $ is called the spectrum of $T$ and it is denoted by $ \sigma(T, X)$. The spectrum $\sigma(T,X)$ can be partitioned into three disjoint sets which are 
 \begin{enumerate}
 	\item[(i)] the point spectrum, denoted by $\sigma_p(T, X)$, is the set of all such $\lambda \in \mathbb{C}$ for which $(T-\lambda I)^{-1}$ does not exist. An element $\lambda\in\sigma_p(T , X)$ is called an eigenvalue of $T$, 
 	\item[(ii)] the continuous spectrum, denoted by $\sigma_c(T , X)$, is the set of all such $\lambda \in\mathbb{C}$ for which $(T-\lambda I)^{-1}$ is exists, unbounded and the domain of $(T-\lambda I)^{-1}$ is dense in $X$ but $R(T-\lambda I)\neq X$,
 	\item[(iii)] the residual spectrum, denoted by $\sigma_r(T, X)$, is the set of all such $\lambda \in\mathbb{C}$ for which $(T-\lambda I)^{-1}$ exists (and may be bounded or not) but the domain of $(T-\lambda I)^{-1}$ is not dense in $X$.
 \end{enumerate} 
These three disjoint sets are together known as fine spectrum and their union becomes the whole spectrum. There are some other important subdivisions of the spectrum such as approximate point spectrum $\sigma_{app}(T, X)$, defect spectrum $\sigma_{\delta}(T, X)$ and compression spectrum $\sigma_{co}(T, X)$, defined by

\begin{eqnarray*}
	\sigma_{app}(T, X)&=&\{\lambda \in \mathbb{C}:~~ (T - \lambda I)~~ \text{is not bounded below}\},\\
	\sigma_{\delta}(T,X)&=&\left\lbrace \lambda \in \mathbb{C}: (T-\lambda I ) \text{ is not surjective} \right\rbrace,\\
	\sigma_{co}(T,X)&=&\left\lbrace  \lambda \in \mathbb{C}:\overline{R(T-\lambda I)} \neq X\right\rbrace.
\end{eqnarray*}
 The sets which are defined above also forms subdivisions of spectrum of $T$  which are not necessarily disjoint as follows
 \begin{eqnarray*}
 	\sigma(T,X)&=&\sigma_{app}(T,X) \cup \sigma_{co}(T,X),\\
 	\sigma(T,X)&=&\sigma_{app}(T,X) \cup \sigma_{\delta}(T,X).
 \end{eqnarray*}
 
   An operator $T \in B(X)$ is said to be Fredholm operator if $R(T)$ is closed and  $\dim(N(T))$, $\dim(X/R(T))$ are finite. In this case the number
   \begin{equation*}
   	 \dim(N(T))- \dim(X/R(T))
   \end{equation*} is called the index of the Fredholm operator $T$. The essential spectrum of $T$ is defined by the set \[ \sigma_{ess}(T,\ell_{p})=\left\lbrace \lambda \in \mathbb{C}:(T-\lambda I)~ \text{is not a Fredholm operator}\right\rbrace.\]
   If $T$ is a Fredholm operator and $K \in B(X)$ is a compact operator then $T+K$ is also a Fredholm operator with same indices. Since compact perturbation does not effect the Fredholmness and index of a Fredholm operator, we have \[\sigma_{ess}(T, X)=\sigma_{ess}(T+K, X).\]
   
   
   
   
For any isolated eigenvalue $\lambda$ of $T,$ the operator $P_T$ which is defined by
\[P_T(\lambda) = \frac{1}{2 \pi i} \int_\gamma  (\mu I - T)^{-1} d \mu,\]   
is called the Riesz projection of $T$ with respect to $\lambda$ where $\gamma$ is positively orientated circle centred at $\lambda$ with sufficiently small radius such that it excludes other spectral values of $T.$ An eigenvalue $\lambda$ of $T$ is said to be a discrete eigenvalue if it is isolated and the rank of the associated Riesz projection is finite. The rank of the Riesz projection is called the algebraic multiplicity of $\lambda$. The set of all such eigenvalues with finite multiplicities is called the discrete spectrum of $T$ and it is denoted by $\sigma_d(T,X).$ This type of eigenvalues sometimes referred as eigenvalues with finite type.


   
In the following proposition, we mention some inclusion relation of spectrum of a bounded linear operator and its adjoint operator.
\begin{proposition} \label{p3.1} \cite[p.195]{basar2012summability}
	If $X$ is a Banach space and $T\in B(X)$, $T^*\in B(X^*)$ then the spectrum and subspectrum of $T$ and $T^*$ are related by the following relations: 
	\begin{itemize}
	\item[(a)] $\sigma(T^*,X^*)=\sigma(T,X).$
	\item[(b)] $\sigma_c(T^*,X^*)\subseteq\sigma_{app}(T,X).$
	\item[(c)] $\sigma_{app}(T^*,X^*)=\sigma_{\delta}(T,X).$
	\item[(d)] $\sigma_{\delta}(T^*,X^*)=\sigma_{app}(T,X).$
	\item[(e)] $\sigma_p(T^*,X^*)=\sigma_{co}(T,X).$
	\item[(f)] $\sigma_{co}(T^*,X^*)\supseteq\sigma_{p}(T,X).$
	\item[(g)] $ \sigma(T,X)=\sigma_{app}(T,X)\cup\sigma_{p}(T^*,X^*)$=$\sigma_p(T,X)\cup\sigma_{app}(T^*,X^*).$
	\end{itemize}
\end{proposition} 
Here we record few lemmas related to the boundness of an infinite matrix defined over sequence spaces, which are useful to our research.
\begin{lemma}\cite[p. 253]{choudhary1989functional} \label{l1}
The matrix $A=(a_{nk})$ gives rise to a bounded linear operator $T \in B(\ell_{1})$ from $\ell_1$ to itself if and only if the supremum of $\ell_1$ norms of  the columns of $A$ is bounded.
\end{lemma}
\begin{lemma}\cite[p. 245]{choudhary1989functional} \label{l2}
The matrix $A=(a_{nk})$ gives rise to a bounded linear operator $T \in B(\ell_\infty)$  from $\ell_\infty$ to itself if and only if the supremum of $\ell_1$  norms of the rows of $A$ is bounded.
\end{lemma}
\begin{lemma}\cite[p. 254]{choudhary1989functional}\label{L3.5}
The matrix $A=(a_{nk})$ gives rise to a bounded linear operator $T\in B(\ell_p)(1<p<\infty)$ if $T \in B(\ell_1)\cap B(\ell_{\infty}).$
\end{lemma}

\section{\bf Spectra of $T_0$}

  It is already mentioned that we study the spectral properties of $T$ by using the spectral properties of $T_0$ and compact perturbation technique. In this section we derive the spectrum and fine spectrum of $T_0$. The notation $\|T\|_p$ denotes the operator norm of an operator $T \in B(\ell_{p})$ where $1 \leq p \leq \infty$.  
\begin{theorem} The operator
	$T_0:\ell_p \rightarrow \ell_p$ is a bounded linear operator which satisfies the following inequality
  $$	\left(\frac{|r_1|^p+|r_2|^p+|s_1|^p+|s_2|^p}{2}\right)^{\frac{1}{p}}\leq \norm{T_0}_{p}\leq \left( 3^{p-1}\left(|r_1|^p+2|s_1|^p+|r_2|^p+2|s_2|^p\right)\right)^{\frac{1}{p}}.$$
\end{theorem}
\begin{proof}
As linearity of $T_0$ is trivial, we omit it. Let $e=(1,1,0,0,...) \in\ell_{p}$. Then $T_{0}(e)=(r_1,r_2,s_1,s_2,0,...)$ and one can observe that
\begin{equation*}
\frac{\norm{T_0(e)}_{p}}{\norm{e}_{p}}=\left( \frac{|r_1|^p+|r_2|^p+|s_1|^p+|s_2|^p}{2}\right)^{\frac{1}{p}}.
\end{equation*}
This proves \[ \left( \frac{|r_1|^p+|r_2|^p+|s_1|^p+|s_2|^p}{2}\right)^{\frac{1}{p}}\leq \norm{T_0}_{p}.\]



\noindent Also, let $x=\{x_n\}\in \ell_p$ and $x_n=0$ if $n \leq0$.
Then,
\begin{align*} 
\norm{T_0(x)}_{p}^p=&\sum_{n=1}^{\infty}|s_{1}x_{2n-3}+r_1x_{2n-1}+s_1x_{2n+1}|^p+	\sum_{n=1}^{\infty}|s_2x_{2n-2}+r_2x_{2n}+s_2x_{2n+2}|^p\\
\leq& \sum_{n=1}^{\infty}(|s_{1}x_{2n-3}|+|r_1 x_{2n-1}|+|s_1 x_{2n+1}|)^p+	\sum_{n=1}^{\infty}(|s_2x_{2n-2}|+|r_2x_{2n}|+|s_2x_{2n+2}|)^p\\
\end{align*}
 By Jensen’s inequality we get,
\begin{align*}
\norm{T_0(x)}_{p}^p \leq & 3^{p-1}\sum_{n=1}^{\infty}\left( |s_1x_{2n-3}|^p+|r_1x_{2n-1}|^p+|s_1x_{2n+1}|^p\right)\\ +& 3^{p-1}\sum_{n=1}^{\infty}\left( |s_2x_{2n-2}|^p+|r_2x_{2n}|^p+|s_2x_{2n+2}|^p\right)\\
\leq& 3^{p-1}\left(|r_1|^p+2|s_1|^p+|r_2|^p+2|s_2|^p\right){\norm{x}_p^p}.  
\end{align*}
This implies,
$$\norm{T_0} \leq \left( 3^{p-1}\left(|r_1|^p+2|s_1|^p+|r_2|^p+2|s_2|^p\right)\right)^{\frac{1}{p}}.$$ This completes the proof. 
\end{proof}
The following theorem proves the non-existence of eigenvalues of the operator $T_0$ in $\ell_p$.
\begin{theorem}\label{t1.2}
The point spectrum of $T_0$ is given by $\sigma_p(T_0,\ell_p)=\emptyset.$
\end{theorem}
\begin{proof}
Consider $(T_0 - \lambda I)x=0$ for $\lambda\in\mathbb{C}$ and $x=\{x_n\}\in \mathbb{C^N}$. This gives the following system of equations

	\begin{eqnarray*}
		\begin{rcases*}
		(r_{1} - \lambda) x_{1}+s_{1} x_{3}&=0\\
		(r_{2} - \lambda) x_{2}+s_{2} x_{4}&=0 \\
		s_1 x_1 + (r_1 - \lambda) x_3 + s_1 x_5 &= 0\\
		s_2x_2+(r_2 - \lambda) x_4+s_2x_6&=0 \\
		                       &\vdots \\
    	s_1 x_{2n-1}+(r_1 - \lambda) x_{2n+1}+s_1 x_{2n+3}&=0\\
		s_2 x_{2n}+(r_2-\lambda)x_{2n+2}+s_2 x_{2n+4}&=0\\
		                             &\vdots
	\end{rcases*}
\end{eqnarray*}
We assume that $x_1 \neq 0$ and $x_2 \neq 0$ otherwise $x_n = 0$ for all $n \in \mathbb{N}.$ Let us consider two sequences $\{y_n\}$ and $\{z_n\}$ where  $y_n=x_{2n-1}$ and $z_n=x_{2n}$, $n \in \mathbb{N}$ respectively. Then the system of equations of $(T_0-\lambda I)x=0$ reduces to
 \begin{equation}\label{3.1}
  y_{n}+p_1y_{n+1}+y_{n+2}=0,

 \end{equation}
 \begin{equation} \label{3.2}
 z_{n}+p_2z_{n+1}+z_{n+2}=0,\\

 \end{equation}
where $p_1=\frac{r_1-\lambda}{s_1}$, $p_2=\frac{r_2-\lambda}{s_2}$, $n \in \mathbb{N}\cup \{0\}$ and $y_0=z_0=0$. The general solution of the difference equation (\ref{3.1}) is given by 
\begin{equation} \label{eqq1}
y_n = \begin{cases}
& (c_1 + nc_2) (-1)^n, \ \ \mbox{if} \ p_1 = 2\\
& c_1+ n c_2, \ \ \mbox{if} \ p_1 = -2\\
& c_1 \alpha_1^n + c_2 \alpha_2^n, \ \ \mbox{if} \ p_1 \notin \{-2,2\}
\end{cases}
\end{equation}
where $c_1,$ $c_2$ are arbitrary constants and $\alpha_1,$ $\alpha_2$ are the roots of the polynomial 
\begin{equation} \label{ch1}
y^2+p_1y+1=0
\end{equation}
which is called the characteristic polynomial of (\ref{3.1}). The following two equalities 
\begin{equation*}
\alpha_1\alpha_2=1  \text{ and }  \alpha_1+\alpha_2= -p_1
\end{equation*}
are useful.
There are three cases to be considered.\\

\noindent Case 1: If $p_1=2$ (i.e., $\lambda =r_1-2s_1 $). In this case the general solution of (\ref{3.1}) is
 \begin{equation*}
 	y_{n}=(c_1+c_2n)(-1)^{n}, \ n \in \mathbb{N}\cup\{0\}
 \end{equation*}
with the initial condition $y_{0}=0$ which gives $c_1 = 0.$ This reduces the solution as $y_n = nc_2 (-1)^n.$ This also implies $c_2 = -y_1$ and the solution in this case is 
\[y_n = ny_1 (-1)^{n+1}, ~~ n \in \mathbb{N}.\]


\noindent Case 2: If $p_1=-2$ (i.e., $\lambda =r_1+2s_1 $). Similar as Case 1, the solution reduces to
\[y_n = ny_1,~~ n \in \mathbb{N}.\]


\noindent Case 3: If $p_1 \notin \{-2, 2\}.$ The general solution of (\ref{3.1}) is given by
 \begin{equation*}
 	y_{n}=c_1 \alpha_1^n+c_2 \alpha_2^n.
 \end{equation*}
With the help of initial condition $y_0=0$ and by using the equalities $\alpha_1\alpha_2=1$, $\alpha_1+\alpha_2= -p_1$, one can obtain that $c_2 = -c_1$ and the solution reduces to 
\[y_n = \frac{\alpha_1^n - \alpha_2^n}{\alpha_1 - \alpha_2} y_1,~~ n \in \mathbb{N}.\]
If $y_1 \neq 0$ then  $\{y_n\} \notin \ell_p$ in Case 1 and Case 2. In Case 3 $\{y_n\} \in \ell_p$ if and only if $|\alpha_1| <1$ and $|\alpha_2|<1$ which can not be the case since $\alpha_1 \alpha_2  = 1.$ Hence in all the three cases $\{y_n\} \in \ell_p$ if and only if $y_1 = 0$ and this leads to the trivial solution of the difference equation (\ref{3.1}). Hence there is no non-trivial solution of ({\ref{3.1}}).

Similarly for the difference equation (\ref{3.2}), the general solution $\{z_n\}$ is of the form
\begin{equation} \label{eqq2}
z_n = \begin{cases}
& (d_1 + nd_2) (-1)^n, \ \ \mbox{if} \ p_2 = 2\\
& d_1+ n d_2, \ \ \mbox{if} \ p_2 = -2\\
& d_1 \beta_1^n + d_2 \beta_2^n, \ \ \mbox{if} \ p_2 \notin \{-2,2\}
\end{cases}
\end{equation}
where $d_1,$ $d_2$ are arbitrary constants and $\beta_1,$ $\beta_2$ are the roots of the polynomial 
\begin{equation} \label{ch2}
z^2+p_2z+1=0
\end{equation}
which is called the characteristic polynomial of difference equation (\ref{3.2}). In a similar way, it can be proved that $\{z_n\} \in \ell_p$ if and only if $z_1 = 0$ and this leads to the trivial solution of (\ref{3.2}). Hence, there does not exist any non-trivial solution of the system $(T_0 - \lambda I)x = 0$ such that $x \in \ell_p.$ This proves the required result.
\end{proof}

\begin{remark}
The solution $x = \{x_n\}$ of the system $Tx = \lambda x,$ which are obtained in terms of the sequences $\{y_n\}$ and $\{z_n\}$ in the equations (\ref{eqq1}) and (\ref{eqq2}) respectively, actually depends on the unknown $\lambda.$ Therefore, instead of writing $x_n(\lambda),$ we write $x_n$ for the sake of brevity throughout this paper except in Theorem \ref{t4.12} where the dependency of the solutions on $\lambda$ is vital. 
\end{remark}


It is well known that the adjoint operator of $T_0$ is $T_0^*$ which is defined over sequence space $\ell_p^*$ where $\ell_p^*$ denotes the dual space of $\ell_p$ which is isomorphic to $\ell_q$ where $\frac{1}{p}+ \frac{1}{q}=1$.
\begin{corollary}\label{c1.3}
	The point spectrum of adjoint operator $T_0^*$ over the sequence space $\ell_p^*$ is given by
	 $\sigma_p(T_0^*,\ell_p^*)=\emptyset.$
\end{corollary}
\begin{proof}
It is well known that the adjoint operator $ {T_0}^*:\ell_p^* \to \ell_p^* $, is represented by transpose of the matrix $T_0$. Since $T_0$ is represented by a symmetric matrix, using the same argument as Theorem \ref{t1.2}, it is easy to prove that $\sigma_p(T_0^*,\ell_p^*)=\emptyset.$
\end{proof}
\begin{corollary}
	The residual spectrum of $T_0$ over the sequence space $\ell_{p}$ is given by
$\sigma_r(T_0,\ell_p)=\emptyset$.
\end{corollary}
\begin{proof}
	We know that the operator $T$ has a dense range if and only if $T^*$ is one to one \cite[p.197]{basar2012summability}. Using this we have the following relation
\[\sigma_r(T_0,\ell_p)=\sigma_p(T_0^*,\ell_p^*)\setminus\sigma_p(T_0,\ell_p).\]
Hence, $\sigma_r(T_0,\ell_p)=\emptyset.$
\end{proof}

Following that, we obtain the spectrum of $T_0$.
\begin{theorem}
	The spectrum of $T_0$ is given by
	\begin{equation*}
	\sigma(T_0,\ell_p) =\left[r_{1}-2s_1, r_1+2s_1\right] \cup [r_2-2s_2, r_2+2s_2].
	\end{equation*}
\end{theorem}



\begin{corollary} 
The continuous spectrum of $T_0$ is given by
	$$\sigma_c(T_0,\ell_p)=\left[r_{1}-2s_1, r_1+2s_1\right] \cup [r_2-2s_2, r_2+2s_2].$$
	\end{corollary}
	\begin{proof}
		It is evident that $\sigma(T_0,\ell_{p})$ is the disjoint union of $\sigma_p(T_0,\ell_{p})$,  $\sigma_r(T_0,\ell_{p})$ and  $\sigma_c(T_0,\ell_{p})$, we have 
		\begin{eqnarray*} 
			\sigma(T_0,\ell_p)=\sigma_c(T_0,\ell_p).
		\end{eqnarray*}
		Hence, $ \sigma_c(T_0,\ell_p)=\left[r_{1}-2s_1, r_1+2s_1\right] \cup [r_2-2s_2, r_2+2s_2]$.
	\end{proof}
	
\begin{corollary}\label{ess}
Essential spectrum of $T_0$ is given by $$\sigma_{ess}(T_0,\ell_{p})=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2].$$
\end{corollary}	
\begin{proof}
It is well-known that $\sigma_c(T_0, \ell_p) \subseteq \sigma_{ess}(T_0,\ell_p)$ and we have $$\sigma_{ess}(T_0,\ell_p)\subseteq \sigma(T_0,\ell_{p}) = \sigma_c(T_0,\ell_{p})\subseteq \sigma_{ess}(T_0, \ell_{p}).$$ The desired result is obvious.
\end{proof}	
Using the relations which are mentioned in Proposition \ref{p3.1} we can easily obtain the following results.
\begin{corollary}
The compression spectrum, approximate point spectrum and defect spectrum of $T_0$ are as follows
\item[(i)] $\sigma_{co}(T_0,\ell_p)=\emptyset$.
\item[(ii)]$\sigma_{app}(T_0,\ell_p)=\left[r_{1}-2s_1, r_1+2s_1\right] \cup [r_2-2s_2, r_2+2s_2]$.
\item[(iii)] $\sigma_{\delta}(T_0,\ell_p)=\left[r_{1}-2s_1, r_1+2s_1\right]\cup[r_2-2s_2, r_2+2s_2]$. 
\end{corollary}



\section{\bf Spectra of $T = T_0 + K$}


This section contains the spectral properties of $T$ which can be expressed as $T=T_0+K$  where $K=T-T_0$ is represented by the following matrix
\begin{equation*}
	K=
	\begin{pmatrix}
		a_1-r_1 & 0 &b_1-s_1&0&0&\cdots &  \\
		0 & a_2-r_2 & 0& b_2-s_2&0& \cdots&\\
		c_1-s_1&0&a_3-r_1&0&b_3-s_1&\cdots&\\
		0&c_2-s_2&0&a_4-r_2&0&\cdots&\\
		0&0&c_3-s_1&0&a_5-r_1&\cdots\\
		\vdots  & \vdots  & \vdots &\vdots&\vdots& \ddots
	\end{pmatrix}.
\end{equation*}
The following result proves the compactness of $K$ on $\ell_{p}$.

\begin{theorem}\label{theorem1.6}
  The operator $K$ is a compact operator on $\ell_{p}$.
 
\end{theorem}
\begin{proof}
	The operator $K$ on $\ell_{p}$ can be represented by the following infinite matrix
	 \begin{equation*}
		K=
		\begin{pmatrix}
			u_1 & 0 &v_1&0&0&\cdots &  \\
			0 & u_2 & 0& v_2&0& \cdots&\\
			w_1&0&u_3&0&v_3&\cdots&\\
			0&w_2&0&u_4&0&\cdots&\\
			\vdots  & \vdots  & \vdots &\vdots&\vdots& \ddots
		\end{pmatrix},
	\end{equation*}
where, $\{u_n\}$, $\{v_n\}$ and $\{w_n\}$ are null sequences, which are defined as follows,
 
\begin{equation*}
	u_n=\begin{cases*}
		a_n-r_1,~~ \mbox{$n$ is odd}\\
		a_n-r_2, ~~ \mbox{$n$ is even},
	\end{cases*}~~~~\qquad
	v_n=\begin{cases*}
		b_n-s_1,~~ \mbox{ $n$ is odd}\\
		b_n-s_2,~~ \mbox{$n$ is even}
	\end{cases*}
\end{equation*}
and
\begin{equation*}
w_n=\begin{cases*}
	c_n-s_1,~~ \mbox{ $n$ is odd}\\
	c_n-s_2,~~ \mbox{$n$ is even}.
\end{cases*}
\end{equation*}
Let $x=\{x_1,x_2,x_3...\}\in \ell_p$. 
 We construct a sequence of compact operators $\left\lbrace K_n\right\rbrace $ such that for $i \in \mathbb{N}$, \\
\begin{equation*}
(K_n(x))_i=\begin{cases}
	(Kx)_i,~~ i=1,2,...n\\
	0,~~ \text{otherwise.}
\end{cases}	
\end{equation*}



For $n \geq2$,
\begin{align*}
\norm{(K-K_n)x}_{p}
= & \left( \sum_{k=n-1}^{\infty}|w_kx_k+u_{k+2}x_{k+2}+v_{k+2}x_{k+4}|^p\right) ^{\frac{1}{p}}\\
\leq & \left( \sup_{k\geq n-1}|w_k|\right)\norm{x}_p + \left( \sup_{k \geq n-1}|u_k|\right)\norm{x}_p+ \left( \sup_{k\geq n-1}|v_k|\right) \norm{x}_p
\end{align*} 
This implies,
$$ \|{K-K_n}\|_p ~\leq  \sup_{k \geq n-1}|w_k|+\sup_{k \geq n-1 }|u_k|+\sup_{k \geq n-1}|v_k|.$$
Thus, $\{K_n\}$ converges to $K$ as $n\to\infty$ in operator norm and hence $K$ is a compact operator over $\ell_p$.
\end{proof}







Next we derive an inclusion relation between $\sigma(T_0, \ell_{p})$ and $\sigma(T,\ell_{p})$.
\begin{theorem}\label{t4.11}
The spectrum of $T$ satisfies the following inclusion relation $$\sigma(T_0,\ell_p) \subseteq \sigma(T,\ell_p)$$ and $\sigma(T,\ell_p)\setminus \sigma(T_0,\ell_p)$ contains finite or countable number of eigenvalues of $T$ of finite type with no accumulation point in $\sigma(T,\ell_p)\setminus \sigma(T_0,\ell_p)$.  
\end{theorem}


\begin{corollary}\label{ess1}
	$\sigma_{ess}(T, \ell_{p})=\sigma(T_0, \ell_{p})=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]$.
\end{corollary}
\begin{proof}
	As we are aware of that compact perturbation does not effect the Fredholmness and index of a Fredholm operator. Therefore, $\sigma_{ess}(T_0,\ell_{p})=\sigma_{ess}(T,\ell_{p})$. Hence, by using Corollary \ref{ess}, we have 
	\begin{equation*}
		\sigma_{ess}(T, \ell_{p})=\sigma(T_0, \ell_{p})=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2].
	\end{equation*}
\end{proof}
We now focus on the point spectrum of $T$. First we analyze the eigenvalues of $T$ lying in $\sigma_{ess}(T, \ell_{p})=\sigma(T_0, \ell_p)$, in particular we derive sufficient conditions for the absence of point spectrum on $\sigma_{ess}(T, \ell_p)$. In Theorem \ref{poT}, sufficient conditions are provided in terms of the rate of convergence of the sequences $\{a_{2n-1}\}, \ \{a_{2n}\},  \ \{b_{2n-1}\}, \ \{b_{2n}\}, \ \{c_{2n-1}\}$ and $\{c_{2n}\}$. Sufficient conditions of absence of point spectrum on $\sigma_{ess}(T, \ell_{p})$ are also provided in Theorem \ref{sc} in terms of the entries of the matrix $T$.

\begin{theorem}\label{poT}
If the convergence of the sequences $\{a_{2n-1}\}, \ \{a_{2n}\}, \ \{b_{2n-1}\}, \ \{b_{2n}\}, \\ \{c_{2n-1}\}$ and $\{c_{2n}\}$ are exponentially fast then $$\sigma_{ess}(T, \ell_{p}) \cap \sigma_{p}(T,\ell_{p})= \emptyset.$$
\end{theorem}


In the next theorem, we apply transfer matrix approach as discussed in \cite{janas2003similarity, janas1998point} . This enables us to examine the sufficient condition for the absence of point spectrum in essential spectrum of $T$ in terms of the entries of matrix $T$.
\begin{theorem}\label{sc}
	If \(\lambda \in \sigma_{ess}(T,\ell_{p})\) satisfies either of the following conditions\\
	(i) $ \sum_{n=1}^{\infty} \prod_{j=1}^n \left[\frac{1}{2} \left(P_j(\lambda)-\sqrt{P_j(\lambda)^2-\left| \frac{2c_{2j+1}}{b_{2j+1}}\right|^2}\right) \right]^{\frac{1}{2}}=+\infty$ \\
	\\ or\\
	(ii) $ \sum_{n=1}^{\infty} \prod_{j=1}^n \left[\frac{1}{2} \left( Q_j(\lambda)-\sqrt{Q_j(\lambda)^2-\left| \frac{2c_{2j+2}}{b_{2j+2}}\right|^2}\right) \right]^{\frac{1}{2}}=+\infty$\\
	where,
	$$  P_j(\lambda)= \left| \frac{c_{2j-1}}{b_{2j+1}}\right| ^2 +\left| \frac{a_{2j+1-\lambda}}{b_{2j+1}}\right|^2+1,~~Q_j(\lambda)= \left| \frac{c_{2j}}{b_{2j+2}}\right| ^2 +\left| \frac{a_{2j+2-\lambda}}{b_{2j+2}}\right|^2+1, $$
	then \(\lambda \notin \sigma_{p}(T,\ell_{p})\).
\end{theorem}
Now we focus our study on the point spectrum of $T$.
Under the sufficient conditions as mentioned in previous two results, we have
$\sigma_p(T, \ell_{p})\cap \sigma(T_0, \ell_{p})=\emptyset$. In this case, all the eigenvalues of $T$ are lying outside the set $\sigma(T_0, \ell_p)$.
To characterize the eigenvalues, let $Tx=\lambda x$, $x \in \mathbb{C^N}$ and $\lambda \in \sigma(T_0,\ell_{p})^c$ where $\sigma(T_0,\ell_{p})^c$ denotes the complement of $\sigma(T_0,\ell_{p})$. From equations (\ref{point_diff}) and (\ref{point_diff1}) in Theorem \ref{poT}, we have the following system 
 \begin{equation} \label{point_diff4}
 	c_{2n-1}y_n+(a_{2n+1}-\lambda) y_{n+1}+b_{2n+1}y_{n+2}=0,
 \end{equation}
 \begin{equation}\label{point_diff5}
 	c_{2n}z_n+(a_{2n+2}- \lambda)z_{n+1}+b_{2n+2}z_{n+2}=0,
 \end{equation}
where $n \in \mathbb{N}\cup\{0\}$ with $y_0=z_0=0$ and $y_n=x_{2n-1}$, $z_n=x_{2n}$.
Clearly each of the difference equations (\ref{point_diff4}) and (\ref{point_diff5}) have two fundamental solutions. Let $\{y_n^{(1)}(\lambda),y_n^{(2)}(\lambda)\}$ and $\{z_n^{(1)}(\lambda),z_n^{(2)}(\lambda)\}$ are the sets of fundamental solutions of the equations  (\ref{point_diff4}) and (\ref{point_diff5}) respectively. Under this setting we have the following result:
\begin{theorem}\label{t4.12}
If either of the sufficient conditions mentioned in Theorem \ref{poT} and Theorem \ref{sc} hold true then the point spectrum of $T$ is given by
\[\sigma_p(T,\ell_p)=\left\lbrace \lambda \in \mathbb{C}: y_{0}^{(1)}(\lambda) =0\right\rbrace \cup \left\lbrace \lambda \in \mathbb{C}:{z_0}^{(1)}(\lambda)=0\right\rbrace .\]
\end{theorem}



\begin{remark}\label{remark1}
	The adjoint operator $ {T}^*:\ell_p^*\to \ell_p^* $, is represented by transpose of the matrix $T$ and dual of $\ell_p$ is isomorphic to $\ell_q$ where $\frac{1}{p}+ \frac{1}{q}=1$ and $1<q<\infty$. Similar as $T$, the operator $T^*$ can also be written as 
	\begin{equation*}
		T^*=T_0+K^t,
	\end{equation*}
where $K^t$ denotes the transpose of $K$ and $K^t$ is also a compact operator. Since $\sigma(T, \ell_{p})=\sigma(T^*, \ell_{p}^*)$, Theorem \ref{t4.11} implies $$\sigma(T_0, \ell_{p})\subseteq \sigma(T^*, \ell_{p}^*),$$ and using similar argument of the proof of Theorem \ref{t4.11} it can be obtain that $\sigma(T^*,\ell_{p}^*)\setminus \sigma(T_0,\ell_p)$ contains finite or countable number of eigenvalues of $T^*$ of finite type with no accumulation point in $\sigma(T^*,\ell_{p}^*)\setminus \sigma(T_0,\ell_p)$. Assuming similar hypothesis on the rate of convergence of sequences in Theorem \ref{poT},
 we can prove that $$\sigma_{ess}(T^*, \ell_{p}^*)\cap \sigma_p(T^*,\ell_{p}^*)=\emptyset$$
 and this implies, the point spectrum of $T^*$ is lying outside of the region \[[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2].\]
Now similar as Theorem \ref{t4.12},
let $\{g_n^{(1)}(\lambda), g_n^{(2)}(\lambda)\}$ and $\{h_n^{(1)}(\lambda), h_n^{(2)}(\lambda)\}$ are the sets of fundamental solutions of the following difference equations respectively
\begin{eqnarray*}
	b_{2n-1}g_n+(a_{2n+1}-\lambda)g_{n+1}+c_{2n+1}g_{n+2}=0,\\
	b_{2n}h_n+(a_{2n+2}-\lambda)h_{n+1}+c_{2n+2}h_{n+2}=0,
\end{eqnarray*}
 which are obtained from $T^*f= \lambda f$,
$f \in\ell_{p}^*$ and $g_n(\lambda)=f_{2n-1}(\lambda)$, $h_n(\lambda)=f_{2n}(\lambda)$. Also, $g_0(\lambda)=h_0(\lambda)=0$.
This leads us to the following result 
\begin{equation*}
\sigma_p(T^*, \ell_{p}^*)=\left\lbrace \lambda \in \mathbb{C}:g_0^{(1)}(\lambda)=0\right\rbrace \cup \left\lbrace \lambda \in \mathbb{C}:h_0^{(1)}(\lambda)=0\right\rbrace.
\end{equation*}
Eventually, we obtain that
$$\sigma(T^*,{\ell_p^*})=[r_1-2s_1, r_1+2s_1]\cup [r_2-2s_2, r_2+2s_2] \cup S_2. $$
where,
\begin{equation*}
	S_2=\left\lbrace \lambda \in \mathbb{C}:g_0^{(1)}(\lambda)=0\right\rbrace \cup \left\lbrace \lambda \in \mathbb{C}:h_0^{(1)}(\lambda)=0\right\rbrace. 
\end{equation*}
Since, $\sigma(T,\ell_{p})=\sigma(T^*,\ell_{p}^*)$ and $S_1,S_2$ both sets are disjoint from $[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]$ we have, $S_1=S_2$.
  
\end{remark}
 Using the observations in Remark \ref{remark1} and Proposition \ref{p3.1}, we can summarize all the results of spectrum and various spectral subdivisions of the operator $T$ in the following theorem,
	\begin{theorem}
		If the convergence of the sequences $\{a_{2n-1}\}, \ \{a_{2n}\}, \ \{b_{2n-1}\}, \ \{b_{2n}\},\\ \ \{c_{2n-1}\}$ and $\{c_{2n}\}$ are exponentially fast then the following hold,
	\begin{itemize}
		\item[(i)] The spectrum of $T$ on $\ell_p$ is
		\[\sigma(T,\ell_p)=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.\]
		\item[(ii)] The point spectrum of $T$ on $\ell_{p}$ is 
		 \[\sigma_p(T,\ell_p)=\left\lbrace \lambda \in \mathbb{C}: y_{0}^{(1)}(\lambda) =0\right\rbrace \cup \left\lbrace \lambda \in \mathbb{C}:{z_0}^{(1)}(\lambda)=0\right\rbrace .\]
		 \item[(iii)] The residual spectrum of $T$ on $\ell_{p}$ is 
		 	\[\sigma_r(T, \ell_p)=\emptyset.\]
		 \item[(iv)] The continuous spectrum of $T$ on $\ell_{p}$ is 
		 \[\sigma_c(T,\ell_{p})=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2].\]
		 \item[(v)] The essential spectrum of $T$ on $\ell_{p}$ is
		 \[\sigma_{ess}(T,\ell_{p})=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2].\]
		 \item[(vi)] The discrete spectrum of $T$ on $\ell_{p}$ is
		 \[\sigma_d(T, \ell_{p})=\left\lbrace \lambda \in \mathbb{C}: y_{0}^{(1)}(\lambda) =0\right\rbrace \cup \left\lbrace \lambda \in \mathbb{C}:{z_0}^{(1)}(\lambda)=0\right\rbrace.\]
		\item [(vii)] The compression spectrum of $T$ on $\ell_{p}$ is \[\sigma_{co}(T,\ell_p)=\left\lbrace \lambda \in \mathbb{C}: y_{0}^{(1)}(\lambda) =0\right\rbrace \cup \left\lbrace \lambda \in \mathbb{C}:{z_0}^{(1)}(\lambda)=0\right\rbrace.\]
		\item[(viii)] The approximate spectrum of $T$ on $\ell_{p}$ is \[\sigma_{app}(T,\ell_p)=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.\]
		\item[(ix)] The defect spectrum of $T$ on $\ell_{p}$ is \[\sigma_{\delta}(T,\ell_p)=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.\]
	\end{itemize}	
\end{theorem} 
\begin{proof} The proofs of the above statements are given below. 
	\begin{enumerate}
			\item[(i)] It is well known that $\sigma_p(T,\ell_p)\subseteq \sigma(T,\ell_p)$ and $\sigma(T_0,\ell_p)\subseteq \sigma(T,\ell_p)$. This implies, $$[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1\subseteq \sigma(T,\ell_p).$$ Also by using Theorem (\ref{t4.11}) we get, $$\sigma(T,\ell_p)\subseteq[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.$$ Hence,
		$$\sigma(T,\ell_p)=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.$$
		\item[(ii)]  Result has been proved in Theorem \ref{t4.12}.
		\item [(iii)] We already aware of that $\sigma_r(T,\ell_p)=\sigma_p(T^*,\ell_p^*)\setminus\sigma_p(T,\ell_p)$.
		Hence, $$\sigma_r(T, \ell_p)=\emptyset.$$
		\item [(iv)] Spectrum of an operator is the disjoint union of point spectrum, residual spectrum and continuous spectrum. By using this result we can obtain the desired result.
		\item [(v)]
		The required result has been proved in Corollary \ref{ess1}.
	
	
		\item [(vi)] We already proved that the point spectrum of $T$ is disjoint from $[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]$, and by Theorem \ref{t4.11} we have, every element of $\sigma_{p}(T, \ell_{p})$ is of finite type. Hence,
		\[\sigma_d(T, \ell_{p})=\left\lbrace \lambda \in \mathbb{C}: y_{0}^{(1)}(\lambda) =0\right\rbrace \cup \left\lbrace \lambda \in \mathbb{C}:{z_0}^{(1)}(\lambda)=0\right\rbrace.\]
		\item [(vii)] By part (e) of Proposition \ref{p3.1}, the desired result is obvious.
		\item [(viii)] Clearly, $$\sigma_{app}(T, \ell_{p})\subseteq \sigma(T,\ell_p)=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.$$ Also, we know that point spectrum is always a subset of approximate point spectrum.  By using this fact and with the help of part (g) of Proposition \ref{p3.1}, we have 
		\[[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1 \subseteq \sigma_{app}(T, \ell_{p}).\]
		Hence,\[\sigma_{app}(T, \ell_{p})=\sigma(T, \ell_{p})=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.\]
		\item [(ix)] By part (c) of Proposition \ref{p3.1}, we have 
		\[\sigma_{app}(T^*, \ell_{p}^*)=\sigma_{\delta}(T,\ell_{p}).\]
		Clearly, $$\sigma_{app}(T^*, \ell_{p}^*)\subseteq \sigma(T^*,\ell_{p}^*)=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.$$
		As $\sigma_{p}(T^*, \ell_{p}^*)\subseteq\sigma_{app}(T^*, \ell_{p}^*)$ thus 
		$S_1\subseteq\sigma_{app}(T^*, \ell_{p}^*)$.  By using this fact and with the help of part (g) of Proposition \ref{p3.1}, we have 
		\[[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1 \subseteq \sigma_{app}(T^*, \ell_{p}^*).\] Therefore, $\sigma_{app}(T^*, \ell_{p}^*)=\sigma(T,\ell_{p})$. Hence,
		\[\sigma_{\delta}(T,\ell_p)=[r_1-2s_1,r_1+2s_1]\cup[r_2-2s_2,r_2+2s_2]\cup S_1.\]
	\end{enumerate}
\end{proof}

\section*{\bf Disclosure statement}
No potential conflict of interest was reported by the authors.

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