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Introduction
The lower triangular matrix A t = (c nk ) defined by c nk = t n-k /(n + ),  < t ≤  is called a discrete generalized Cesàro operator. The matrix reduces to the Cesàro matrix by setting t = . In , Rhaly [] showed that the discrete generalized Cesàro operator A t on the  Hilbert space was a bounded compact linear operator and computed its spectrum. Also in [], lower bounds for these classes were obtained under certain restrictions on p ( < p < ∞) by Rhoades. In this article, we show that this operator is a compact linear operator, calculate its spectrum and get two subdivisions of this spectrum on the p ( < p < ∞) sequence space.

Boundedness of discrete generalized Cesàro operator
In , Rhaly [] showed that the discrete generalized Cesàro operator A t on the Hilbert space  is a bounded linear operator. We will show that A t is a bounded linear operator on p ( < p < ∞).

Theorem  ([] (Hardy inequalities))
If p > , a n ≥ , and A n = a  + a  + · · · + a n , then unless all a n 's are , Theorem  A t ∈ B( p ) and A t B( p ) ≤ p p- for  < t < , where  < p < ∞.
Proof Using Theorem , since  < t < , we have Hence we get A t ∈ B( p ) and A t ≤ p p - .

Compactness of discrete generalized Cesàro operator
Compact linear operators have a great deal of application in practice. For instance, they play a central role in the theory of integral equations and in various problems of mathematical physics.
Disentangling the historical development of the spectral theory of compact linear operators is particularly hard because many of the results were originally proved early in the twentieth century for integral equations acting on particular Banach spaces of functions. These operators behave very much like familiar finite dimensional matrices without necessarily having finite rank. For a compact linear operator, spectral theory can be treated fairly completely in the sense that Fredholm's famous theory of linear integral equations may be extended to linear functional equations Txλx = y with a complex parameter λ. This generalized theory is called the Riesz-Schauder theory.

Definition  ([]
) Let X and Y be normed spaces. An operator T : X → Y is called a compact linear operator (or completely continuous linear operator) if T is linear and if, for every bounded subset M of X, the image T(M) is relatively compact, that is, the closure T(M) is compact.
From the definition of compactness of a set, we readily obtain a useful criterion for the operator.

Theorem  ([]
) Let X and Y be normed spaces and T : X → Y be a linear operator. Then T is compact if and only if it maps every bounded sequence (x n ) in X onto a sequence (Tx n ) in Y which has a convergent subsequence.
The following theorem makes it easy to show the compactness of a linear operator over a normed space.

Theorem  ([]) Let X and Y be normed spaces and T : X → Y be a linear operator. Then:
(a) If T is bounded and dim T(X) < ∞, the operator T is compact.
The following is important as a tool for proving compactness of a given operator as the uniform operator limit of a sequence of compact linear operators.

Theorem  ([])
Let (T n ) be a sequence of compact linear operators from a normed space X into a Banach space Y . If (T n ) is uniformly operator convergent, say, if T n -T → , then the limit operator T is compact.
In , Rhaly [] showed that the discrete generalized Cesàro operator A t on the Hilbert space  is a compact linear operator. We show that A t is a compact linear operator on p ( < p < ∞).

Theorem  A t is a compact linear operator over
For ∀r ∈ N, we obtain that dim(A r ) = r +  < ∞. Hence, from Theorem , for all r ∈ N, the operator A r is compact on p . With triangular inequality and Hölder's inequality, for all x ∈ p , we have Then we get Hence, we obtain After that, we get Thus, from the Raabe test, ∞ n= c n converges, and therefore ∞ k=n c k →  (for n → ∞).
Thus, A t is the compact linear operator over p ( < p < ∞) for  < t <  from Theorem .

Spectrum of discrete generalized Cesàro operator
Definition  Let X = {} be a complex normed space and T : D(T) → X be a linear operator with domain D(T) ⊂ X. A number λ ∈ C that provides the following conditions is called the regular value of T, and the set of all regular values of T will be denoted by ρ(T) and it is called the resolvent set of T: Furthermore, the spectrum σ (T) naturally splits into three disjoint sets, some of which may be empty. The discrete splitting of the spectrum can be defined as the point spectrum, the continuous spectrum and the residual spectrum as follows.

Definition  ([])
(a) The point spectrum or discrete spectrum σ p (T) is the set such that R λ (T) does not exist. A λ ∈ σ p (T) is called an eigenvalue of T.
(b) The continuous spectrum σ c (T) is the set such that R λ (T) exists and satisfies (R) but not (R), that is, R λ (T) is unbounded.
(c) The residual spectrum σ r (T) is the set such that R λ (T) exists (and may be bounded Spectral theory is an important part of functional analysis. It plays a crucial role in many branches of mathematics such as function theory, complex analysis, differential and integral equations, control theory and also in numerous applications as they are intimately related to the stability of the underlying physical systems. For more information on spec- The following theorem tells us that the point spectrum of a compact linear operator is not complicated. In fact, we also know that each spectral value λ =  of a compact linear operator is an eigenvalue from the next theorem. The spectrum of a compact linear operator largely resembles the spectrum of an operator on a finite dimensional space.

Theorem  ([])
A compact linear operator T : X → X on a normed space X has the following properties: (a) The set of the eigenvalues of T is countable (perhaps finite or even empty).
(b) λ =  is the only possible point of accumulation of that set.
(c) Every spectral value λ =  is an eigenvalue. We determine the spectrum of A t on p ( < p < ∞). Let S := {  n : n = , , . . .}. In this section, we will compute the spectrum of the generalized discrete generalized Cesàro matrix, the compact linear operator A t , where  < t < .

Proof Let
where x = θ . In this case, equations Hence, we have (x n ) = (t n x  ) ∈ p . Therefore, the eigenvector corresponding to λ = , from the second equation in (.). If x  = , then λ =   . Hence, we obtain from the other equations in (.). Then, since x n+ x n p = n +  n p t p → t p < , we have n |x n | p < ∞, that is, x = (x n ) ∈ p . Thus, the eigenvector corresponding to λ =   is x = (, x  , tx  , t  x  , . . .) ∈ p , i.e., λ = / ∈ σ p (A t , p ).
(iii) If x m is the first nonzero component of the sequence x = (x n ), then from mth equation in (.), i.e., In this case, we have x m+n = (m + )(m + ) · · · (m + n) n! t n x m for all n ≥  from other equations in (.). Since t ∈ (, ), We will use the following lemma to find the adjoint on the p ( < p < ∞) sequence space of a linear transform.

Lemma  ([], p. ) If A ∈ B( p ) ( < p < ∞), then A can be represented by an infinite matrix and A * , which is an element of B( q )
, where  p +  q = , can be represented by the transpose of A matrix.
The adjoint matrix of A t on p ( < p < ∞) is as follows: Lemma  The adjoint operator over p (p > ) of the matrix A t can be given as its transposition. That is, the matrix (A t ) * = (a * nk ) is given by Proof Let x = θ and A * t x = λx. Then, for all n ≥ , the equations are realized from Lemma . Therefore  / ∈ σ p (A * t , q ) because if λ =  then x n =  for all n = , , , . . . . Hence, we get where x  = . If λ =  m for an integer m, then we have n |x n | q < ∞ because x n =  for every n ≥ m, so that, x = (x n ) ∈ q is obtained. Hence, we get λ =  m ∈ σ p (A * t , * p ∼ = q ) for all integers m. Let λ =  m for all integers m. Since n |x n | q series is divergent. So, there is no other eigenvalue, i.e., we have Proof Since dim p = ∞, we have  ∈ σ (A t , p ) from Theorem . Also, since A t is a compact linear operator by Theorem , each nonzero spectral value of A t is an eigenvalue from Theorem . Therefore, σ (A t , p ) = S ∪ {} is obtained from Theorem .

The fine spectrum of discrete generalized Cesàro operator on p (1 < p < ∞)
Let X be a Banach space, B(X) denotes the collection of all bounded linear operators on X and T ∈ B(X). Then there are three possibilities for R(T), the range of T: (I) R(T) = X, (II) R(T) = X, but R(T) = X, Table 1 Goldberg's decomposition of the spectrum (

1) (2) (3) R(λ; T) exists and is bounded R(λ; T) exists and is unbounded R(λ; T) does not exists
(III) R(T) = X, and three possibilities for T - : () T - exists and is continuous, If these possibilities are combined in all possible ways, nine different states are created. These are labeled by I  , I  , I  , II  , II  , II  , III  , III  , III  . For example, let an operator be in state III  . Then R(T) = X and T - exist and T - is unbounded. From the closed graph theorem, I  is empty (see []).
Applying the Goldberg classification to the operator T λ := λI -T, we have If λ is a complex number such that T λ = λI -T ∈ I  or T λ = λI -T ∈ II  , then λ ∈ ρ(T, X).
All scalar values of λ not in ρ(T, X) comprise the spectrum of T. The further classification of σ (T, X) gives rise to the fine spectrum of T. That is, σ (T, X) can be divided into the subsets I  σ (T, X), I  σ (T, X), II  σ (T, X), II  σ (T, X), III  σ (T, X), III  σ (T, X), III  σ (T, X). For example, if T λ = λI -T is in a given state, III  (say), then we write λ ∈ III  σ (T, X).
We can summarize the above in Table . This classification of the spectrum is called the Goldberg classification. Let us give the theorems that will help the Goldberg classification. The fine spectra of bounded linear operators defined by some particular limitation matrices over some sequence spaces were first discussed in [, -] and [].
Then the spectra and fine spectra of some operators have been studied by various authors [-] and are still being studied.
We will examine the fine spectrum of a discrete generalized Cesàro operator on p ( < p < ∞), which is compact in this section.
The operator A * t is - because  / ∈ σ p (A * t , q ). Hence, we have R(A t ) = p from Theorem . If A t x = y, we obtain y n =  n +  n k= t n-k x k . Therefore, we get x  = y  and x n = (n + )y ntny n- from (n + )y n = t n x  + t n- x  + · · · + tx n- + x n , tny n- = t t n- x  + t n- x  + · · · + x n- .
Hence A t is not onto, that is, R(A t ) = p . Therefore, A t ∈ II. As a consequence, A t ∈ II  or A t ∈ II  . We have A t / ∈ II  because  ∈ σ (A t , p ). Then we have A t ∈ II  , i.e.,  ∈ II  σ (A t , p ).

Subdivision of the spectrum of discrete generalized Cesàro operator on p (1 < p < ∞)
Given a bounded linear operator T in a Banach space X, we call a sequence (x k ) in X a Weyl sequence for T if x k =  and Tx k →  as k → ∞.
In what follows, we call the set By the definitions given above, we can write Table . This separation of the spectrum of some operator has been studied by various authors in [, -, ] and is still being studied.
Theorem  For  < t <  and  < p < ∞, we have theorem was given in []. The spectra and spectral separation of this operator over the other sequence spaces are left as clear problems.