# On the Fine Spectrum of the Operator Defined by the Lambda Matrix over the Spaces of Null and Convergent Sequences

Authors: Medine Yeşilkayagil; Feyzi Başar
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's
classification of the operator defined by the lambda matrix over the sequence spacesc0 and c. As a new
development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix
operator Λ on the sequence spaces c0 and c. Finally, we present a Mercerian theorem. Since the matrix Λ is
reduced to a regular matrix depending on the choice of the sequence (λk) having certain properties and its
spectrum is firstly investigated, our work is new and the results are comprehensive.


---

## Body

## 1. Introduction

LetX and Y be Banach spaces, and let T:X→Y also be a bounded linear operator. By R(T), we denote the range of T; that is,
(1)R(T)={y∈Y:y=Tx,x∈X}.
By B(X), we also denote the set of all bounded linear operators on X into itself. If X is any Banach space and T∈B(X) then the adjointT* of T is a bounded linear operator on the dual X* of X defined by (T*f)(x)=f(Tx) for all f∈X* and x∈X.LetX≠{θ} be a nontrivial complex normed space and T:D(T)→X a linear operator defined on a subspace D(T)⊆X. We do not assume that D(T) is dense in X or that T has closed graph {(x,Tx):x∈D(T)}⊆X×X. By the statement “Tis invertible,” it is meant that there exists a bounded linear operator S:R(T)→X for which ST=I on D(T) and R(T)¯=X, such that S=T-1 is necessarily uniquely determined and linear; the boundedness of S means that T must be bounded below, in the sense that there is M>0 for which ∥Tx∥≥M∥x∥ for all x∈D(T). Associated with each complex number, α is the perturbed operator
(2)Tα=αI-T
defined on the same domain D(T) as T. The spectrumσ(T,X) consists of those α∈ℂ, the complex field, for which Tα is not invertible, and the resolvent is the mapping from the complement σ(T,X) of the spectrum into the algebra of bounded linear operators on X defined by α↦Tα-1.


## 2. The Subdivisions of Spectrum

In this section, we define the parts of spectrum called point spectrum, continuous spectrum, residual spectrum, approximate point spectrum, defect spectrum, and compression spectrum. There are many different ways to subdivide the spectrum of a bounded linear operator. Some of them are motivated by applications to physics, in particular, quantum mechanics.


### 2.1. The Point Spectrum, Continuous Spectrum, and Residual Spectrum

The name resolvent is appropriate sinceTα-1 helps to solve the equation Tαx=y. Thus, x=Tα-1y provided that Tα-1 exists. More importantly, the investigation of properties of Tα-1 will be basic for an understanding of the operator T itself. Naturally, many properties of Tα and Tα-1 depend on α, and the spectral theory is concerned with those properties. For instance, we are interested in the set of all α's in the complex plane such that Tα-1 exists. Boundedness of Tα-1 is another property that will be essential. We will also ask for what α's the domain of Tα-1 is dense in X, to name just a few aspects. A regular valueα of T is a complex number such that Tα-1 exists and is bounded and whose domain is dense in X. For our investigation of T, Tα, and Tα-1, we need some basic concepts in the spectral theory which are given, as follows (see [1, pages 370-371]).Theresolvent setρ(T,X) of T is the set of all regular values α of T. Furthermore, the spectrum σ(T,X) is partitioned into the following three disjoint sets.Thepoint (discrete) spectrumσp(T,X) is the set such that Tα-1 does not exist. An α∈σp(T,X) is called an eigenvalue of T.Thecontinuous spectrumσc(T,X) is the set such that Tα-1 exists and is unbounded, and the domain of Tα-1 is dense in X.Theresidual spectrumσr(T,X) is the set such that Tα-1 exists (and may be bounded or not) but the domain of Tα-1 is not dense in X.Therefore, these three subspectra form a disjoint subdivision(3)σ(T,X)=σp(T,X)∪σc(T,X)∪σr(T,X).
To avoid trivial misunderstandings, let us say that some of the sets defined above may be empty. This is an existence problem which we will have to discuss. Indeed, it is well known that σc(T,X)=σr(T,X)=∅ and the spectrum σ(T,X) consists of only the set σp(T,X) in the finite-dimensional case.


### 2.2. The Approximate Point Spectrum, Defect Spectrum, and Compression Spectrum

In this subsection, following Appell et al. [2], we define three more subdivisions of the spectrum called approximate point spectrum, defect spectrum, and compression spectrum.Given a bounded linear operatorT in a Banach space X, we call a sequence (xk) in X a Weyl sequence for T if ∥xk∥=1 and ∥Txk∥→0, as k→∞. Then, the approximate point spectrumσap(T,X)ofT is defined by
(4)σap(T,X):={α∈ℂ:thereexistsaWeylsequenceforαI-T}.
Moreover, the subspectrum
(5)σδ(T,X):={α∈ℂ:αI-Tisnotsurjective}
is called the defect spectrum of T.The two subspectra given by (4) and (5) form a (not necessarily disjoint) subdivision
(6)σ(T,X)=σap(T,X)∪σδ(T,X)
of the spectrum. There is another subspectrum,
(7)σco(T,X)={α∈ℂ:R(αI-T)¯≠X}
which is often called compression spectrum in the literature. The compression spectrum gives rise to another (not necessarily disjoint) decomposition
(8)σ(T,X)=σap(T,X)∪σco(T,X)
of the spectrum. Clearly, σp(T,X)⊆σap(T,X) and σco(T,X)⊆σδ(T,X). Moreover, comparing these subspectra with those in (3), we note that
(9)σr(T,X)=σco(T,X)∖σp(T,X),σc(T,X)=σ(T,X)∖[σp(T,X)∪σco(T,X)].Sometimes it is useful to relate the spectrum of a bounded linear operator to that of its adjoint. Building on classical existence and uniqueness, results for linear operator equations in Banach spaces and their adjoints are also useful.Proposition 1 (see [2, Proposition 1.3, page 28]).
Spectrum and subspectrum of an operatorT∈B(X) and its adjoint T*∈B(X*) are related by the following relations: (a)
σ(T*,X*)=σ(T,X),(b)
σc(T*,X*)⊆σap(T,X),(c)
σap(T*,X*)=σδ(T,X),(d)
σδ(T*,X*)=σap(T,X),(e)
σp(T*,X*)=σco(T,X),(f)
σco(T*,X*)⊇σp(T,X),(g)
σ(T,X) = σap(T,X) ∪σp(T*,X*) = σp(T,X)∪σap(T*,X*).The relations (c)–(f) show that the approximate point spectrum is in a certain sense dual to the defect spectrum, and the point spectrum is dual to the compression spectrum. The equality (g) implies, in particular, thatσ(T,X)=σap(T,X) if X is a Hilbert space and T is normal. Roughly speaking, this shows that normal (in particular, self-adjoint) operators on the Hilbert spaces are most similar to matrices in finite dimensional spaces (see [2]).


### 2.3. Goldberg's Classification of Spectrum

IfX is a Banach space and T∈B(X), then there are three possibilities for R(T):  (A)
R(T)=X,(B)
R(T)≠R(T)¯=X,(C)
R(T)¯≠X, and (1)
T-1 exists and is continuous,(2)
T-1 exists but is discontinuous,(3)
T-1 does not exist.If these possibilities are combined in all possible ways, nine different states are created. These are labelled by:A1, A2, A3, B1, B2, B3, C1, C2, and C3. If an operator is in state C2, for example, then R(T)¯≠X and T-1 exists but is discontinuous (see [3]). Figure 1 due to Wenger [4] may be useful for the readers.Figure 1
State diagram forB(X) and B(X*) for a non-reflective Banach space X.Ifα is a complex number such that Tα∈A1 or Tα∈B1, then α∈ρ(T,X). All scalar values of α 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 A2σ(T,X)=∅, A3σ(T,X), B2σ(T,X), B3σ(T,X), C1σ(T,X), C2σ(T,X), and C3σ(T,X). For example, if Tα is in a given state, C2 (say), then we write α∈C2σ(T,X).By the definitions given above, we can illustrate subdivision (3) in Table 1.Table 1
Subdivision of spectrum of a linear operator.
1
2
3
T
α
-
1 exists
T
α
-
1 exists
T
α
-
1
and is bounded
and is unbounded
does not exist
A
R
(
α
I
-
T
)
=
X
α
∈
ρ
(
T
,
X
)
—
α
∈
σ
p
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
c
(
T
,
X
)
α
∈
σ
p
(
T
,
X
)
B
R
(
α
I
-
T
)
¯
=
X
α
∈
ρ
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
r
(
T
,
X
)
α
∈
σ
r
(
T
,
X
)
α
∈
σ
p
(
T
,
X
)
C
R
(
α
I
-
T
)
¯
≠
X
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
co
(
T
,
X
)
α
∈
σ
co
(
T
,
X
)
α
∈
σ
co
(
T
,
X
)Observe that the case in the first row and the second column cannot occur in a Banach spaceX, by the closed graph theorem. If we are not in the third column, that is, if α is not an eigenvalue of T, we may always consider the resolvent operator Tα-1 (on a possibly “thin” domain of definition) as “algebraic inverse” of αI-T.By asequence space, we understand a linear subspace of the space ω=ℂℕ of all complex sequences which contain ϕ, the set of all finitely nonzero sequences, where ℕ={0,1,2,…}. We write ℓ∞, c, c0, and bv for the spaces of all bounded, convergent, null, and bounded variation sequences which are the Banach spaces with the sup-norm ∥x∥∞=supk∈ℕ|xk| and ∥x∥bv=∑k=0∞|xk-xk+1|, respectively, while ϕ is not a Banach space with respect to any norm. Also by ℓp, we denote the space of all p-absolutely summable sequences which is a Banach space with the norm ∥x∥p=(∑k=0∞|xk|p)1/p, where 1≤p<∞.Letμ and ν be two sequence spaces, and let A=(ank) be an infinite matrix of complex numbers ank, where k,n∈ℕ. Then, we say that A defines a matrix transformation from μ into ν, and we denote it by writing A:μ→ν if for every sequence x=(xk)∈μ, the sequence Ax={(Ax)n}, the A-transform of x, is in ν, where
(10)(Ax)n=∑k=0∞‍ankxkforeachn∈ℕ.
By (μ:ν), we denote the class of all matrices A such that A:μ→ν. Thus, A∈(μ:ν) if and only if the series on the right side of (10) converges for each n∈ℕ and each x∈μ, and we have Ax={(Ax)n}n∈ℕ∈ν for all x∈μ.Throughout this paper, letλ=(λk) be a strictly increasing sequence of positive reals tending to infinity; that is,
(11)0<λ0<λ1<λ2<⋯,limk→∞λk=∞.
Following Mursaleen and Noman [20], we define the matrix Λ=(λnk) of weighted mean relative to the sequence λ by
(12)λnk={λk-λk-1λn,0≤k≤n0,k>n
for all k,n∈ℕ. It is easy to show that the matrix Λ is regular and is reduced, in the special case λk=k+1 for all k∈ℕ, to the matrix C1 of Cesàro mean of order one. Introducing the concept of Λ-strong convergence, several results on Λ-strong convergence of numerical sequences and Fourier series were given by Móricz [21]. Since we have
(13)Qn=∑‍k=0nqk=λn,rnk=qkQn=λk-λk-1λn=λnk
in the special case qk=λk-λk-1 for all k∈ℕ, the matrix Λ is also reduced to the Riesz means Rq=(rnk) with respect to the sequence q=(qk). We say that a sequence x=(xk)∈ω is λ-convergent if Λx∈c. In particular, we say that x is a λ-null sequence if Λx∈c0 and we say that x is λ-bounded if Λx∈ℓ∞.Lemma 2 (see [22, Theorem 1.3.6, page 6]).
The matrixA = (ank) gives rise to a bounded linear operator T∈B(c) from c to itself if and only if (1)
the rows ofA are in ℓ1 and their ℓ1 norms are bounded;(2)
the columns ofA are in c;(3)
the sequence of row sums ofA is in c.The operator norm ofT is the supremum of the ℓ1 norms of the rows.Corollary 3.
Λ
:
c
→
c is a bounded linear operator with the norm ∥Λ∥(c:c)=1.Lemma 4 (see [22, Example 8.4.5.A, page 129]).
The matrixA=(ank) gives rise to a bounded linear operator T∈B(c0) from c0 to itself if and only if  (1)
the rows ofA are in ℓ1 and their ℓ1 norms are bounded,(2)
the columns ofA are in c0.The operator norm ofT is the supremum of the ℓ1 norms of the rows.Corollary 5.
Λ
:
c
0
→
c
0 is a bounded linear operator with the norm ∥Λ∥(c0:c0)=1.We give a short survey concerned with the spectrum of the linear operators defined by some triangle matrices over certain sequence spaces. Wenger [4] examined the fine spectrum of the integer power of the Cesàro operator in c and Rhoades [5] generalized this result to the weighted mean methods. The fine spectrum of the operator on the sequence space ℓp was studied by González [23], where 1<p<∞. The  spectrum of the Cesàro operator on the sequence spaces c0 and bv were also investigated by Reade [6], Akhmedov and Başar [7], and Okutoyi [8], respectively. The fine  spectrum of the Rhaly operators on the sequence spaces c0 and c were examined by Yıldırım [9]. Furthermore, Coşkun [10] has studied the spectrum and fine spectrum for p-Cesàro operator acting on the space c0. Besides, de Malafosse [11] and Altay and Başar [12], respectively, studied the spectrum and the fine spectrum of the difference operator on the sequence spaces sr and c0, c, where sr denotes the Banach space of all sequences x=(xk) normed by ∥x∥sr=supk∈ℕ|xk|/rk, (r>0). Altay and Karakuş [24] determined the fine spectrum of the Zweier matrix which is a band matrix as an operator over the sequence spaces ℓ1 and bv. In 2010, Srivastava and Kumar [16] determined the spectra and the fine spectra of the double sequential band matrix Δν on ℓ1, where Δν is defined by (Δν)nn=νn and (Δν)n+1,n=-νn for all n∈ℕ, under certain conditions on the sequence ν=(νk) and they have just generalized these results by the double sequential band matrix Δuv defined by Δuvx=(unxn+vn-1xn-1)n∈ℕ for all n∈ℕ (see [18]). Altun [25] studied the fine spectra of the Toeplitz operators, which are represented by upper and lower triangular n-band infinite matrices, over the sequence spaces c0 and c. Later, Karakaya and Altun determined the fine spectra of upper triangular double-band matrices over the sequence spaces c0 and c, in [26]. Quite recently, Akhmedov and El-Shabrawy [15] obtained the fine spectrum of the double sequential band matrix Δa,b, defined as a double-band matrix with the convergent sequences a~=(ak) and b~=(bk) having certain properties, over the sequence space c. The fine spectrum with respect to Goldberg’s classification of the operator B(r,s,t) defined by a triple band matrix over the sequence spaces ℓp and bvp with 1<p<∞ has recently been studied by Furkan et al. [14]. Quite recently, Karaisa and Başar [19] have determined the fine spectrum of the upper triangular triple band matrix B′(r,s,t) over the sequence space ℓp, where 0<p<∞. At this stage, Table 2 may be useful.Table 2
Spectrum and fine spectrum of some triangle matrices in certain sequence spaces.
σ
(
A
,
λ
)
σ
p
(
A
,
λ
)
σ
c
(
A
,
λ
)
σ
r
(
A
,
λ
)
Refer to
σ
(
C
1
p
,
c
)
—
—
—
[4]
σ
(
W
,
c
)
—
—
—
[5]
σ
(
C
1
,
c
0
)
—
—
—
[6]
σ
(
C
1
,
c
0
)
σ
p
(
C
1
,
c
0
)
σ
c
(
C
1
,
c
0
)
σ
r
(
C
1
,
c
0
)
[7]
σ
(
C
1
,
b
v
)
—
—
—
[8]
σ
(
R
,
c
0
)
σ
p
(
R
,
c
0
)
σ
c
(
R
,
c
0
)
σ
r
(
R
,
c
0
)
[9]
σ
(
R
,
c
)
σ
p
(
R
,
c
)
σ
c
(
R
,
c
)
σ
r
(
R
,
c
)
[9]
σ
(
C
1
p
,
c
0
)
—
—
—
[10]
σ
(
Δ
,
s
r
)
—
—
—
[11]
σ
(
Δ
,
c
0
)
—
—
—
[11]
σ
(
Δ
,
c
)
—
—
—
[11]
σ
(
Δ
(
1
)
,
c
)
σ
p
(
Δ
(
1
)
,
c
)
σ
c
(
Δ
(
1
)
,
c
)
σ
r
(
Δ
(
1
)
,
c
)
[12]
σ
(
Δ
(
1
)
,
c
0
)
σ
p
(
Δ
(
1
)
,
c
0
)
σ
c
(
Δ
(
1
)
,
c
0
)
σ
r
(
Δ
(
1
)
,
c
0
)
[12]
σ
(
B
(
r
,
s
)
,
ℓ
p
)
σ
p
(
B
(
r
,
s
)
,
ℓ
p
)
σ
c
(
B
(
r
,
s
)
,
ℓ
p
)
σ
r
(
B
(
r
,
s
)
,
ℓ
p
)
[13]
σ
(
B
(
r
,
s
)
,
b
v
p
)
σ
p
(
B
(
r
,
s
)
,
b
v
p
)
σ
c
(
B
(
r
,
s
)
,
b
v
p
)
σ
r
(
B
(
r
,
s
)
,
b
v
p
)
[13]
σ
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
σ
p
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
σ
c
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
σ
r
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
[14]
σ
(
B
(
r
,
s
,
t
)
,
b
v
p
)
σ
p
(
B
(
r
,
s
,
t
)
,
b
v
p
)
σ
c
(
B
(
r
,
s
,
t
)
,
b
v
p
)
σ
r
(
B
(
r
,
s
,
t
)
,
b
v
p
)
[14]
σ
(
Δ
a
,
b
,
c
)
σ
p
(
Δ
a
,
b
,
c
)
σ
c
(
Δ
a
,
b
,
c
)
σ
r
(
Δ
a
,
b
,
c
)
[15]
σ
(
Δ
ν
,
ℓ
1
)
σ
p
(
Δ
ν
,
ℓ
1
)
σ
c
(
Δ
ν
,
ℓ
1
)
σ
r
(
Δ
ν
,
ℓ
1
)
[16]
σ
(
Δ
u
v
2
,
c
0
)
σ
p
(
Δ
u
v
2
,
c
0
)
σ
c
(
Δ
u
v
2
,
c
0
)
σ
r
(
Δ
u
v
2
,
c
0
)
[17]
σ
(
Δ
u
v
,
ℓ
1
)
σ
p
(
Δ
u
v
,
ℓ
1
)
σ
c
(
Δ
u
v
,
ℓ
1
)
σ
r
(
Δ
u
v
,
ℓ
1
)
[18]
σ
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
σ
p
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
σ
c
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
σ
r
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
[19]In this work, our purpose is to determine the fine spectrum of the operatorΛ over the sequence spaces c0 and c with respect to Goldberg’s classification. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator Λ over the spaces c0 and c. Finally, we state and prove a Mercerian theorem.


## 2.1. The Point Spectrum, Continuous Spectrum, and Residual Spectrum

The name resolvent is appropriate sinceTα-1 helps to solve the equation Tαx=y. Thus, x=Tα-1y provided that Tα-1 exists. More importantly, the investigation of properties of Tα-1 will be basic for an understanding of the operator T itself. Naturally, many properties of Tα and Tα-1 depend on α, and the spectral theory is concerned with those properties. For instance, we are interested in the set of all α's in the complex plane such that Tα-1 exists. Boundedness of Tα-1 is another property that will be essential. We will also ask for what α's the domain of Tα-1 is dense in X, to name just a few aspects. A regular valueα of T is a complex number such that Tα-1 exists and is bounded and whose domain is dense in X. For our investigation of T, Tα, and Tα-1, we need some basic concepts in the spectral theory which are given, as follows (see [1, pages 370-371]).Theresolvent setρ(T,X) of T is the set of all regular values α of T. Furthermore, the spectrum σ(T,X) is partitioned into the following three disjoint sets.Thepoint (discrete) spectrumσp(T,X) is the set such that Tα-1 does not exist. An α∈σp(T,X) is called an eigenvalue of T.Thecontinuous spectrumσc(T,X) is the set such that Tα-1 exists and is unbounded, and the domain of Tα-1 is dense in X.Theresidual spectrumσr(T,X) is the set such that Tα-1 exists (and may be bounded or not) but the domain of Tα-1 is not dense in X.Therefore, these three subspectra form a disjoint subdivision(3)σ(T,X)=σp(T,X)∪σc(T,X)∪σr(T,X).
To avoid trivial misunderstandings, let us say that some of the sets defined above may be empty. This is an existence problem which we will have to discuss. Indeed, it is well known that σc(T,X)=σr(T,X)=∅ and the spectrum σ(T,X) consists of only the set σp(T,X) in the finite-dimensional case.


## 2.2. The Approximate Point Spectrum, Defect Spectrum, and Compression Spectrum

In this subsection, following Appell et al. [2], we define three more subdivisions of the spectrum called approximate point spectrum, defect spectrum, and compression spectrum.Given a bounded linear operatorT in a Banach space X, we call a sequence (xk) in X a Weyl sequence for T if ∥xk∥=1 and ∥Txk∥→0, as k→∞. Then, the approximate point spectrumσap(T,X)ofT is defined by
(4)σap(T,X):={α∈ℂ:thereexistsaWeylsequenceforαI-T}.
Moreover, the subspectrum
(5)σδ(T,X):={α∈ℂ:αI-Tisnotsurjective}
is called the defect spectrum of T.The two subspectra given by (4) and (5) form a (not necessarily disjoint) subdivision
(6)σ(T,X)=σap(T,X)∪σδ(T,X)
of the spectrum. There is another subspectrum,
(7)σco(T,X)={α∈ℂ:R(αI-T)¯≠X}
which is often called compression spectrum in the literature. The compression spectrum gives rise to another (not necessarily disjoint) decomposition
(8)σ(T,X)=σap(T,X)∪σco(T,X)
of the spectrum. Clearly, σp(T,X)⊆σap(T,X) and σco(T,X)⊆σδ(T,X). Moreover, comparing these subspectra with those in (3), we note that
(9)σr(T,X)=σco(T,X)∖σp(T,X),σc(T,X)=σ(T,X)∖[σp(T,X)∪σco(T,X)].Sometimes it is useful to relate the spectrum of a bounded linear operator to that of its adjoint. Building on classical existence and uniqueness, results for linear operator equations in Banach spaces and their adjoints are also useful.Proposition 1 (see [2, Proposition 1.3, page 28]).
Spectrum and subspectrum of an operatorT∈B(X) and its adjoint T*∈B(X*) are related by the following relations: (a)
σ(T*,X*)=σ(T,X),(b)
σc(T*,X*)⊆σap(T,X),(c)
σap(T*,X*)=σδ(T,X),(d)
σδ(T*,X*)=σap(T,X),(e)
σp(T*,X*)=σco(T,X),(f)
σco(T*,X*)⊇σp(T,X),(g)
σ(T,X) = σap(T,X) ∪σp(T*,X*) = σp(T,X)∪σap(T*,X*).The relations (c)–(f) show that the approximate point spectrum is in a certain sense dual to the defect spectrum, and the point spectrum is dual to the compression spectrum. The equality (g) implies, in particular, thatσ(T,X)=σap(T,X) if X is a Hilbert space and T is normal. Roughly speaking, this shows that normal (in particular, self-adjoint) operators on the Hilbert spaces are most similar to matrices in finite dimensional spaces (see [2]).


## 2.3. Goldberg's Classification of Spectrum

IfX is a Banach space and T∈B(X), then there are three possibilities for R(T):  (A)
R(T)=X,(B)
R(T)≠R(T)¯=X,(C)
R(T)¯≠X, and (1)
T-1 exists and is continuous,(2)
T-1 exists but is discontinuous,(3)
T-1 does not exist.If these possibilities are combined in all possible ways, nine different states are created. These are labelled by:A1, A2, A3, B1, B2, B3, C1, C2, and C3. If an operator is in state C2, for example, then R(T)¯≠X and T-1 exists but is discontinuous (see [3]). Figure 1 due to Wenger [4] may be useful for the readers.Figure 1
State diagram forB(X) and B(X*) for a non-reflective Banach space X.Ifα is a complex number such that Tα∈A1 or Tα∈B1, then α∈ρ(T,X). All scalar values of α 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 A2σ(T,X)=∅, A3σ(T,X), B2σ(T,X), B3σ(T,X), C1σ(T,X), C2σ(T,X), and C3σ(T,X). For example, if Tα is in a given state, C2 (say), then we write α∈C2σ(T,X).By the definitions given above, we can illustrate subdivision (3) in Table 1.Table 1
Subdivision of spectrum of a linear operator.
1
2
3
T
α
-
1 exists
T
α
-
1 exists
T
α
-
1
and is bounded
and is unbounded
does not exist
A
R
(
α
I
-
T
)
=
X
α
∈
ρ
(
T
,
X
)
—
α
∈
σ
p
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
c
(
T
,
X
)
α
∈
σ
p
(
T
,
X
)
B
R
(
α
I
-
T
)
¯
=
X
α
∈
ρ
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
r
(
T
,
X
)
α
∈
σ
r
(
T
,
X
)
α
∈
σ
p
(
T
,
X
)
C
R
(
α
I
-
T
)
¯
≠
X
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
ap
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
δ
(
T
,
X
)
α
∈
σ
co
(
T
,
X
)
α
∈
σ
co
(
T
,
X
)
α
∈
σ
co
(
T
,
X
)Observe that the case in the first row and the second column cannot occur in a Banach spaceX, by the closed graph theorem. If we are not in the third column, that is, if α is not an eigenvalue of T, we may always consider the resolvent operator Tα-1 (on a possibly “thin” domain of definition) as “algebraic inverse” of αI-T.By asequence space, we understand a linear subspace of the space ω=ℂℕ of all complex sequences which contain ϕ, the set of all finitely nonzero sequences, where ℕ={0,1,2,…}. We write ℓ∞, c, c0, and bv for the spaces of all bounded, convergent, null, and bounded variation sequences which are the Banach spaces with the sup-norm ∥x∥∞=supk∈ℕ|xk| and ∥x∥bv=∑k=0∞|xk-xk+1|, respectively, while ϕ is not a Banach space with respect to any norm. Also by ℓp, we denote the space of all p-absolutely summable sequences which is a Banach space with the norm ∥x∥p=(∑k=0∞|xk|p)1/p, where 1≤p<∞.Letμ and ν be two sequence spaces, and let A=(ank) be an infinite matrix of complex numbers ank, where k,n∈ℕ. Then, we say that A defines a matrix transformation from μ into ν, and we denote it by writing A:μ→ν if for every sequence x=(xk)∈μ, the sequence Ax={(Ax)n}, the A-transform of x, is in ν, where
(10)(Ax)n=∑k=0∞‍ankxkforeachn∈ℕ.
By (μ:ν), we denote the class of all matrices A such that A:μ→ν. Thus, A∈(μ:ν) if and only if the series on the right side of (10) converges for each n∈ℕ and each x∈μ, and we have Ax={(Ax)n}n∈ℕ∈ν for all x∈μ.Throughout this paper, letλ=(λk) be a strictly increasing sequence of positive reals tending to infinity; that is,
(11)0<λ0<λ1<λ2<⋯,limk→∞λk=∞.
Following Mursaleen and Noman [20], we define the matrix Λ=(λnk) of weighted mean relative to the sequence λ by
(12)λnk={λk-λk-1λn,0≤k≤n0,k>n
for all k,n∈ℕ. It is easy to show that the matrix Λ is regular and is reduced, in the special case λk=k+1 for all k∈ℕ, to the matrix C1 of Cesàro mean of order one. Introducing the concept of Λ-strong convergence, several results on Λ-strong convergence of numerical sequences and Fourier series were given by Móricz [21]. Since we have
(13)Qn=∑‍k=0nqk=λn,rnk=qkQn=λk-λk-1λn=λnk
in the special case qk=λk-λk-1 for all k∈ℕ, the matrix Λ is also reduced to the Riesz means Rq=(rnk) with respect to the sequence q=(qk). We say that a sequence x=(xk)∈ω is λ-convergent if Λx∈c. In particular, we say that x is a λ-null sequence if Λx∈c0 and we say that x is λ-bounded if Λx∈ℓ∞.Lemma 2 (see [22, Theorem 1.3.6, page 6]).
The matrixA = (ank) gives rise to a bounded linear operator T∈B(c) from c to itself if and only if (1)
the rows ofA are in ℓ1 and their ℓ1 norms are bounded;(2)
the columns ofA are in c;(3)
the sequence of row sums ofA is in c.The operator norm ofT is the supremum of the ℓ1 norms of the rows.Corollary 3.
Λ
:
c
→
c is a bounded linear operator with the norm ∥Λ∥(c:c)=1.Lemma 4 (see [22, Example 8.4.5.A, page 129]).
The matrixA=(ank) gives rise to a bounded linear operator T∈B(c0) from c0 to itself if and only if  (1)
the rows ofA are in ℓ1 and their ℓ1 norms are bounded,(2)
the columns ofA are in c0.The operator norm ofT is the supremum of the ℓ1 norms of the rows.Corollary 5.
Λ
:
c
0
→
c
0 is a bounded linear operator with the norm ∥Λ∥(c0:c0)=1.We give a short survey concerned with the spectrum of the linear operators defined by some triangle matrices over certain sequence spaces. Wenger [4] examined the fine spectrum of the integer power of the Cesàro operator in c and Rhoades [5] generalized this result to the weighted mean methods. The fine spectrum of the operator on the sequence space ℓp was studied by González [23], where 1<p<∞. The  spectrum of the Cesàro operator on the sequence spaces c0 and bv were also investigated by Reade [6], Akhmedov and Başar [7], and Okutoyi [8], respectively. The fine  spectrum of the Rhaly operators on the sequence spaces c0 and c were examined by Yıldırım [9]. Furthermore, Coşkun [10] has studied the spectrum and fine spectrum for p-Cesàro operator acting on the space c0. Besides, de Malafosse [11] and Altay and Başar [12], respectively, studied the spectrum and the fine spectrum of the difference operator on the sequence spaces sr and c0, c, where sr denotes the Banach space of all sequences x=(xk) normed by ∥x∥sr=supk∈ℕ|xk|/rk, (r>0). Altay and Karakuş [24] determined the fine spectrum of the Zweier matrix which is a band matrix as an operator over the sequence spaces ℓ1 and bv. In 2010, Srivastava and Kumar [16] determined the spectra and the fine spectra of the double sequential band matrix Δν on ℓ1, where Δν is defined by (Δν)nn=νn and (Δν)n+1,n=-νn for all n∈ℕ, under certain conditions on the sequence ν=(νk) and they have just generalized these results by the double sequential band matrix Δuv defined by Δuvx=(unxn+vn-1xn-1)n∈ℕ for all n∈ℕ (see [18]). Altun [25] studied the fine spectra of the Toeplitz operators, which are represented by upper and lower triangular n-band infinite matrices, over the sequence spaces c0 and c. Later, Karakaya and Altun determined the fine spectra of upper triangular double-band matrices over the sequence spaces c0 and c, in [26]. Quite recently, Akhmedov and El-Shabrawy [15] obtained the fine spectrum of the double sequential band matrix Δa,b, defined as a double-band matrix with the convergent sequences a~=(ak) and b~=(bk) having certain properties, over the sequence space c. The fine spectrum with respect to Goldberg’s classification of the operator B(r,s,t) defined by a triple band matrix over the sequence spaces ℓp and bvp with 1<p<∞ has recently been studied by Furkan et al. [14]. Quite recently, Karaisa and Başar [19] have determined the fine spectrum of the upper triangular triple band matrix B′(r,s,t) over the sequence space ℓp, where 0<p<∞. At this stage, Table 2 may be useful.Table 2
Spectrum and fine spectrum of some triangle matrices in certain sequence spaces.
σ
(
A
,
λ
)
σ
p
(
A
,
λ
)
σ
c
(
A
,
λ
)
σ
r
(
A
,
λ
)
Refer to
σ
(
C
1
p
,
c
)
—
—
—
[4]
σ
(
W
,
c
)
—
—
—
[5]
σ
(
C
1
,
c
0
)
—
—
—
[6]
σ
(
C
1
,
c
0
)
σ
p
(
C
1
,
c
0
)
σ
c
(
C
1
,
c
0
)
σ
r
(
C
1
,
c
0
)
[7]
σ
(
C
1
,
b
v
)
—
—
—
[8]
σ
(
R
,
c
0
)
σ
p
(
R
,
c
0
)
σ
c
(
R
,
c
0
)
σ
r
(
R
,
c
0
)
[9]
σ
(
R
,
c
)
σ
p
(
R
,
c
)
σ
c
(
R
,
c
)
σ
r
(
R
,
c
)
[9]
σ
(
C
1
p
,
c
0
)
—
—
—
[10]
σ
(
Δ
,
s
r
)
—
—
—
[11]
σ
(
Δ
,
c
0
)
—
—
—
[11]
σ
(
Δ
,
c
)
—
—
—
[11]
σ
(
Δ
(
1
)
,
c
)
σ
p
(
Δ
(
1
)
,
c
)
σ
c
(
Δ
(
1
)
,
c
)
σ
r
(
Δ
(
1
)
,
c
)
[12]
σ
(
Δ
(
1
)
,
c
0
)
σ
p
(
Δ
(
1
)
,
c
0
)
σ
c
(
Δ
(
1
)
,
c
0
)
σ
r
(
Δ
(
1
)
,
c
0
)
[12]
σ
(
B
(
r
,
s
)
,
ℓ
p
)
σ
p
(
B
(
r
,
s
)
,
ℓ
p
)
σ
c
(
B
(
r
,
s
)
,
ℓ
p
)
σ
r
(
B
(
r
,
s
)
,
ℓ
p
)
[13]
σ
(
B
(
r
,
s
)
,
b
v
p
)
σ
p
(
B
(
r
,
s
)
,
b
v
p
)
σ
c
(
B
(
r
,
s
)
,
b
v
p
)
σ
r
(
B
(
r
,
s
)
,
b
v
p
)
[13]
σ
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
σ
p
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
σ
c
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
σ
r
(
B
(
r
,
s
,
t
)
,
ℓ
p
)
[14]
σ
(
B
(
r
,
s
,
t
)
,
b
v
p
)
σ
p
(
B
(
r
,
s
,
t
)
,
b
v
p
)
σ
c
(
B
(
r
,
s
,
t
)
,
b
v
p
)
σ
r
(
B
(
r
,
s
,
t
)
,
b
v
p
)
[14]
σ
(
Δ
a
,
b
,
c
)
σ
p
(
Δ
a
,
b
,
c
)
σ
c
(
Δ
a
,
b
,
c
)
σ
r
(
Δ
a
,
b
,
c
)
[15]
σ
(
Δ
ν
,
ℓ
1
)
σ
p
(
Δ
ν
,
ℓ
1
)
σ
c
(
Δ
ν
,
ℓ
1
)
σ
r
(
Δ
ν
,
ℓ
1
)
[16]
σ
(
Δ
u
v
2
,
c
0
)
σ
p
(
Δ
u
v
2
,
c
0
)
σ
c
(
Δ
u
v
2
,
c
0
)
σ
r
(
Δ
u
v
2
,
c
0
)
[17]
σ
(
Δ
u
v
,
ℓ
1
)
σ
p
(
Δ
u
v
,
ℓ
1
)
σ
c
(
Δ
u
v
,
ℓ
1
)
σ
r
(
Δ
u
v
,
ℓ
1
)
[18]
σ
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
σ
p
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
σ
c
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
σ
r
(
B
′
(
r
,
s
,
t
)
,
ℓ
p
)
[19]In this work, our purpose is to determine the fine spectrum of the operatorΛ over the sequence spaces c0 and c with respect to Goldberg’s classification. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator Λ over the spaces c0 and c. Finally, we state and prove a Mercerian theorem.


## 3. The Fine Spectrum of the OperatorΛ on the Sequence Space c0

In this section, we examine the spectrum, the point spectrum, the continuous spectrum, the residual spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum of the operatorΛ on the sequence space c0. For simplicity in the notation, we write throughout that cn=(λn-λn-1)/λn for all n∈ℕ and we use this abbreviation with other letters.Theorem 6.
σ
(
Λ
,
c
0
)
⊆
{
α
∈
ℂ
:
|
2
α
-
1
|
≤
1
}.Proof.
Let|2α-1|>1. Since Λ-αI is triangle, (Λ-αI)-1 exists, and solving the matrix equation (Λ-αI)x=y for x in terms of y gives the matrix (Λ-αI)-1=B=(bnk), where
(14)bnk={(-1)n-k(λk-λk-1)λnα2∏j=kn(cj/α-1),0≤k≤n-1,λnλn-λn-1-αλn,k=n,0,k>n,
for all k,n∈ℕ. Thus, we observe that
(15)∥(Λ-αI)-1∥(c0:c0)=supn∈ℕ∑k=0∞‍|bnk|.
The inequality |2α-1|>1 is equivalent to γ>-1, where -(1/α)=γ+iβ. For all α∈ℂ,
(16)|1-cjα|=|1+(γ+iβ)cj|≥1+γcj
holds for all j∈ℕ. So, 1/|1-(cj/α)|≤1/(1+γcj).
Firstly we take-1<γ<0. Since 0<cj≤1, we have 1+γ≤1+γcj<1. Therefore 1/(1+γcj)<1/(1+γ) and 1<1/(1+γ)<∞ for 0<1+γ<1. 							(17)∑k=0∞‍|bnk|=∑k=0n‍|bnk|<λn-1λn|α|2(1+γ)n+1+1|α|(1+γ)<∞.
Secondly we get0≤γ. Since 1<1+γcj≤1+γ, 1/(1+γcj)<1. So,
(18)∑k=0∞‍|bnk|=∑k=0n‍|bnk|<λn-1λn|α|2+1|α|<∞.
Therefore, we have
(19)∥(Λ-αI)-1∥(c0:c0)=supn∈ℕ∑k=0n‍|bnk|<∞,
that is, (Λ-αI)-1∈(c0:c0). But for |2α-1|≤1,
(20)∥(Λ-αI)-1∥(c0:c0)=∞,
that is, (Λ-αI)-1 is not in B(c0). This completes the proof.Theorem 7.
Defineμ and η by μ=limsupj→∞cj and η=liminfj→∞cj. Then,
(21){α∈ℂ:|α-12-μ|≤1-μ2-μ}∪S⊆σ(Λ,c0),whereS={cj:j∈ℕ}¯.Proof.
Let|α-1/(2-μ)|<(1-μ)/(2-μ) and α≠cj for any j∈ℕ. Then,
(22)1-cjα=λjλj-1λjλj-1-cjα=λj-1λj[λjλj-1-λj-λj-1λj-1+(1-1α)λj-λj-1λj-1]=λj-1λj[1+(1-1α)λj-λj-1λj-1].
So we have,
(23)|bnk|=λk-λk-1λk-1|α|2∏j=kn|1+(1-1/α)((λj-λj-1)/λj-1)|.
Note that |1+(1-α-1)(λj-λj-1)/λj-1|≤1 if and only if
(24)[1+(1+γ)λj-λj-1λj-1]2+(βλj-λj-1λj-1)2≤1,
where -α-1=γ+iβ. So, one can see that
(25)2(1+γ)λj-λj-1λj-1+[(1+γ)2+β2](λj-λj-1λj-1)2≤0,
which is equivalent to the inequality
(26)2(1+γ)+[(1+γ)2+β2](λj-λj-1λj-1)≤0.
For inequality (26) to be true for all sufficiently large j, it is sufficient to have
(27)limsupj→∞[2(1+γ)+[(1+γ)2+β2]λj-λj-1λj-1]<0.
We can write (λj-λj-1)/λj-1=λj(λj-λj-1)/(λjλj-1) and λj/λj-1=1/(1-cj). Therefore,
(28)λj-λj-1λj-1=cj1-cj,(29)limsupj→∞λj-λj-1λj-1=μ1-μ,
since the function g defined by g(x)=x/(1-x) is monotone increasing in x for 0<x<1.
For (27) to be true for all sufficiently large j, it is sufficient to have μ satisfying
(30)2(1+γ)+[(1+γ)2+β2]μ1-μ<0
which is equivalent to
(31)|α-12-μ|<1-μ2-μ.
Therefore, for all n≥N, for some fixed N,
(32)∑k=Nn-1‍|bnk|=∑k=Nn-1‍λk-λk-1λk-1|α|2∏j=kn|1+(1-1/α)((λj-λj-1)/λj-1)|≥1|α|2∑k=Nn-1‍λk-λk-1λk-1
which diverges in the light of (29).
Ifα=cj for any j∈ℕ, then clearly α lies in the spectrum of Λ. This completes the proof.Theorem 8.
σ
(
Λ
,
c
0
)
⊆
{
α
∈
ℂ
:
|
α
-
1
/
(
2
-
η
)
|
≤
(
1
-
η
)
/
(
2
-
η
)
}
∪
S.Proof.
Letα be fixed and satisfy the inequality
(33)|α-12-η|>1-η2-η,
and α≠cj for any j∈ℕ. We will show that α∈ρ(Λ,c0). From Theorem 6, we need consider only those values of α satisfying |2α-1|>1; that is, γ>-1. Under the assumption on α, we wish to verify that
(34)|1+(1-1α)λj-λj-1λj-1|>1
for all sufficiently large j. It will be sufficient to show that
(35)liminfj→∞{2(1+γ)+[(1+γ)2+β2]λj-λj-1λj-1}>0,
that is,
(36)2(1+γ)+[(1+γ)2+β2]η1-η>0
which is equivalent to (33).
Define the functionf by f(t)=1+2(1+γ)t+[(1+γ)2+β2]t2. f has a minumum at t0=-(1+γ)/[(1+γ)2+β2]. The above inequality is equivalent to η(γ2+β2)+2γ>η-2 and is also equivalent to
(37)η2(1-η)>-1+γ(1+γ)2+β2=t0.
Therefore, for those values of η satisfying (37), f is monotone increasing. Let ϵ>0 and small. Then f(η/(1-η)-ϵ) = f(η/(1-η))-(2ϵ)g(ϵ), where g(ϵ) = 1+γ+[(1+γ)2+β2][η/(1-η)-ϵ/2]. Note that g(ϵ)>0 for small ϵ, since f is monotone increasing for t>η/[2(1-η)], we will now show that f(η/(1-η))>1. From (37),
(38)γ2+β2+2γη>η-2η
which is equivalent to
(39)|11-η-ηα(1-η)|>1.
But 1/(1-η) = 1+η/(1-η), so we have f(η/(1-η)) = |1+(1-α-1)η/(1-η)|2>1. Now choose ϵ>0 and so small that f(η/(1-η)-ϵ) = f(η/(1-η))-2ϵg(ϵ) = m2>1. Then, by the definition of η, there exists an N such that n>N implies (λn+1-λn)/λn>η/(1-η)-ϵ, so that f((λn+1-λn)/λn)>f(η/(1-η)-ϵ) = m2. Using (23),
(40)|bnk||bn+1,k|=(λk-λk-1)/λk-1|α|2∏j=kn|1+(1-1/α)((λj-λj-1)/λj-1)|(λk-λk-1)/λk-1|α|2∏j=kn+1|1+(1-1/α)((λj-λj-1)/λj-1)|=|1+(1-1α)λn+1-λnλn|=f(λn+1-λnλn)>m2>1,
for all n≥N. Therefore {|bnk|} is monotone decreasing in n for each k,n>N, so that B has bounded columns. It remains to show that B has finite norm.
Forϵ being used, from (29), we can enlarge N, if necessary, to ensure that (λn-λn-1)/λn-1<μ/(1-μ)+1 for n≥N. From (23),
(41)∑k=Nn-1‍|bnk|=∑k=Nn-1‍λk-λk-1λk-1|α|2∏j=kn|1+(1-1/α)((λj-λj-1)/λj-1)|≤1|α|2(μ1-μ+1)∑k=Nn-1‍1∏j=kn|1+(1-1/α)((λj-λj-1)/λj-1)|≤1|α|2(μ1-μ+1)∑k=Nn-1‍m-n+k-1<H,
where H is a constant independent of n. Further
(42)|bnn|=λn|α||λn-((λn-λn-1)/α)|=λn|α||λn-1+(1-1/α)(λn-λn-1)|=λn/λn-1|α||1+(1-1/α)((λn-λn-1)/λn-1)|=1+(λn-λn-1)/λn-1|α||1+(1-1/α)((λn-λn-1)/λn-1)|<1+μ/(1-μ)+1|α|m.
Hence, B has a finite norm.Corollary 9.
Letδ=limj→∞cj exist. Then,
(43)σ(Λ,c0)={α∈ℂ:|α-12-δ|≤1-δ2-δ}∪S.IfT∈B(c0) with the matrix A, then it is known that the adjoint operator T*:c0*→c0* is dened by the transpose At of the matrix A. It should be noted that the dual space c0*  of c0  is isometrically isomorphic to the Banach space ℓ1 of absolutely summable sequences normed by ∥x∥=∑k=0∞|xk|.Theorem 10.
Letδ be defined as in Corollary 9. Then, σp(Λ*,c0*)={α∈ℂ:|α-1/(2-δ)|<(1-δ)/(2-δ)}∪S.Proof.
Suppose thatΛ*x=αx for x≠θ in c0*≅ℓ1. Then, by solving the system of linear equations
(44)x1=λ1-λ0λ0(1-1α)x0,x2=λ2-λ1λ0(1-c1α)(1-1α)x0⋮xn=λn-λn-1λ0(1-1α)x0∏‍j=1n-1(1-cjα)⋮
we can write xn = (λn-λn-1)/λn-1(1-α-1)x0∏j=1n-1[1+(1-α-1)(λj-λj-1)/λj-1]. Let |α-1/(2-δ)|<(1-δ)/(2-δ) or α∈S and un=∏j=1n-1[1+(1-α-1)(λj-λj-1)/λj-1]. One can see that |1+(1-α-1)(λj-λj-1)/λj-1|<1 for all sufficiently large j if and only if
(45)[1+(1+γ)λj-λj-1λj-1]2+(βλj-λj-1λj-1)2<1,where-1α=γ+iβ.
Then, we have, from the discussion in Theorem 7 and the hypothesis on α,
(46)|un+1un|=|1+(1-1α)λn-λn-1λn-1|<1
for all sufficiently large n, so ∑n=0∞|un| is convergent. Since |(1-α-1)(λn-λn-1)/λn-1x0| is bounded, it follows that ∑n=0∞|xn| is convergent, so that Λ*x=αx has nonzero solutions. Therefore, the proof is completed.Theorem 11.
Letδ be defined as in Corollary 9. Then
(47)σp(Λ,c0)={α=cn∈ℂ:0≤α≤δ2-δ}∪{1}.Proof.
Letck be any diagonal entry satisfying 0<ck≤δ/(2-δ). Let j be the smallest integer such that cj=ck. By setting xn=0 for n>j+1, x0=0, the system (Λ*-cjI)x=θ reduces to a homogeneous linear system of j equations in j+1 unknowns, so that nontrivial solutions exist. Therefore Λ-cjI∈3.
Λ
-
α
I is not one to one for α=0,1 and so Λ-αI∈3. This step concludes the proof.Lemma 12 (see [3, page 59]).
T has a dense range if and only if T* is one to one.Theorem 13.
σ
r
(
Λ
,
c
0
)
=
σ
p
(
Λ
*
,
c
0
*
)
∖
σ
p
(
Λ
,
c
0
).Proof.
Forα∈σp(Λ*,c0*)∖σp(Λ,c0), the operator Λ-αI is triangle, so has an inverse. But Λ*-αI is not one to one by Theorem 10. Therefore by Lemma 12, R(Λ-αI)¯≠c0, and this step concludes the proof.Theorem 14.
Letδ be defined as in Corollary 9 and cn≥δ for all sufficiently large n. Then,
(48)σc(Λ,c0)={α∈ℂ:|α-12-δ|=1-δ2-δ,α≠1,δ2-δ}.Proof.
Fixα≠1, δ/(2-δ), and satisfying |α-1/(2-δ)|=(1-δ)/(2-δ). Since the operator Λ-αI is triangle, it has an inverse. Consider the adjoint operator Λ*-αI. As in Theorem 11, x0 is arbitrary and
(49)xn=λn-λn-1λn-1(1-1α)x0∏‍j=kn[1+(1-1α)λj-λj-1λj-1]
for all positive n. From the hypothesis, there exists a positive integer N such that n≥N implies cn≥δ. This fact, together with the condition on α, implies that |1+(1-α-1)(λn-λn-1)/λn-1|≥1 for n≥N. Thus, |xn|=C(λn-λn-1)/λn-1 for n≥N, where C is a positive constant independent of n. We can write
(50)λn-λn-1λn-1=cn(1+λn-λn-1λn-1)≥cn.
Therefore (xn)∈ℓ1⇔x0=0; that is, Λ*-αI is one to one. From Lemma 12, the range of Λ-αI is dense in c0. This completes the proof.Lemma 15 (see [3, page 60]).
T has a bounded inverse if and only if T* is onto.Theorem 16.
Letδ be defined as in Corollary 9 and less than 1. If α satisfies |α-1/(2-δ)|<(1-δ)/(2-δ) and α∉S, then α∈C1σ(Λ,c0).Proof .
First of allΛ-αI is a triangle; hence 1-1. Therefore Λ-αI∈1∪2. To verify that Λ-αI∈C1σ(Λ,c0) it is sufficient to show that Λ*-αI is onto by Lemma 15.
Supposey=(Λ*-αI)x, where x,y∈ℓ1. Then, x0=1/(1-α)y0-λ0/[(λ1-λ0)(1-α)]y1 and
(51)(cn-α)xn+(λn-λn-1)∑k=n+1∞‍xkλk=yn,n>0.
Choose x1=0 and solve (51) for x in terms of y to get
(52)(λ1-λ0)∑k=2∞‍xkλk=y1,(53)(cn-α)xn=yn-(λn-λn-1)∑k=n+1∞‍xkλk.
For example, substituting (52) into (53), with n=2, yields
(54)(c2-α)x2=y2-(λ2-λ1)∑k=3∞‍xkλk,
so that x2=(λ2-λ1)/[α(λ1-λ0)]y1-(1/α)y2. For n=3,
(55)x3=-(λ3-λ2λ1-λ0)(c2-α)1α2y1+(λ3-λ2λ2)1α2y2-1αy3.
Continuing this process, the entries of the matrixB=(bnk) such that By=x are calculated as
(56)b00=11-α,b01=-λ0(λ1-λ0)(1-α)b21=λ2-λ1(λ1-λ0)α,bnn=-1α,n>1,bn,n-1=λn-λn-1λn-1α2,n>2bn1=λn-λn-1(λ1-λ0)α∏j=2n-1‍(1-cjα),n>2,bnk=λn-λn-1λkα2∏j=k+1n-1‍(1-cjα),1<k<n-1,
and bnk=0 otherwise.
To show thatB∈B(ℓ1), it is sufficient to establish that ∑n=0∞|bnk| is finite independent of k. ∑n=0∞|bn0|=1/|1-α|. We may write 1-(cj/α)=(λj-1/λj)[1+(1-α-1)(λj-λj-1)/λj-1]. Also, supn∈ℕ|(λn-λn-1)/λn-1|≤M<∞. Therefore,
(57)∑n=0∞‍|bn1|≤1|α|[M+M∑n=3∞‍∏j=2n-1|1+(1-1α)λj-λj-1λj-1|]
and, for k>1,
(58)∑n=0∞‍|bnk|≤1|α|+M|α|2+M|α|2∑n=k+2∞∏j=k+1n-1|1+(1-1α)λj-λj-1λj-1|.
Sincek>1, the series in inequality (24) is absolutely convergent from Theorem 7. Therefore, ∥B∥(ℓ1:ℓ1) is finite.
Because of(Λ-αI)-1 is bounded, it is continuous, and α∈C1σ(Λ,c0). This completes the proof.Theorem 17.
Letδ be defined as in Corollary 9 and δ<1. If α=δ or α=cn for all n∈ℕ and δ/(2-δ)<α<1, then α∈C1σ(Λ,c0).Proof.
First assume thatΛ has distinct diagonal entries and fix j≥1. Then the system (Λ-cjI)x=θ implies that xn=0 for n=0,1,…,j-1, and for n≥j(59)(cj-cn)xn-∑k=0n-1‍λnkxk=0.
The system (59) yields the following recursion relation:
(60)xn+1=λncjxnλn+1(cj-cn+1)
which can be solved for xn to yield
(61)xj+m=λjxjcjmλj+m∏i=1m‍(cj-cj+i)=xj∏i=1mλj+i-1λj+i(1-cj+i/cj)=xj{∏i=1m‍λj+iλj+i-1[1-λj(λj+i-λj+i-1)λj+i(λj-λj-1)]}-1=xj{∏i=1m‍[λj+iλj+i-1-λj(λj+i-λj+i-1)λj+i-1(λj-λj-1)]}-1=xj{∏i=1m‍[λj+iλj+i-1-λj+i-λj+i-1λj+i-1==xj∏i=1m‍+(1-1cj)λj+i-λj+i-1λj+i-1]∏i=1m‍}-1=xj{∏i=1m‍[1+(1-1cj)λj+i-λj+i-1λj+i-1]}-1.
Since0<cj<1, the argument of Theorem 7 applies and (24) is true. Therefore x∈c0 implies x=θ and Λ-cjI is 1-1, so that Λ-cjI∈1∪2.
ClearlyΛ-cjI∈C. It remains to show that Λ*-cjI is onto.
Suppose that(Λ*-cjI)x=y, where x,y∈ℓ1. By choosing xj+1=0 we can solve for x0,x1,…,xj in terms of y0,y1,…,yj+1. As in Theorem 16, the remaining equations can be written in the form x=By, where the nonzero entries of B=(bnk) are as follows
(62)bj+m,j+m=-1cj;bj+2,j+1=λj+2-λj+1cj(λj+1-λj);bj+m,j+m-1=λj+m-λj+m-1cj2λj+m-1,m>2;bj+m,j+k=λj+m-λj+m-1cj2λj+k∏i=j+k+1j+m-1‍(1-cicj),1<k<m-1,m>3;bj+m,j+1=λj+m-λj+m-1cj(λj+1-λj)∏i=j+2j+m-1‍(1-cicj),m>2.
From (62), we have
(63)∑n=j+1∞‍|bn,j+1|=λj+2-λj+1cj(λj+1-λj)+1cj(λj+1-λj)×∑n=j+3∞‍(λn-λn-1)∏i=j+2n-1‍|1-cicj|.
For m>1,
(64)∑n=m+j∞‍|bn,m+j|=1cj+λj+m+1-λj+mcj2λj+m+1cj2∑n=j+m+2∞‍λn-λn-1λj+m∏i=j+m+1n-1‍|1-cicj|.
Using 1-(cj/α)=(λj-1/λj)[1+(1-α-1)(λj-λj-1)/λj-1], one can convert (63) and (64) the similar expressions to (57) and (58), and therefore ∥B∥(ℓ1:ℓ1) is finite.
Suppose thatΛ does not have distinct diagonal entries. The restriction on α guarantees that no zero diagonal entries are being considered. Let cj≠0 be any diagonal entry which occurs more than once, and let k,r denote, respectively, the smallest and largest integers for which cj=ck=cr. From (61) it follows that xn=0 for n≥r. Also, xn=0 for 0≤n<k. Therefore the system (Λ-cjI)x=θ becomes
(65)(cj-cn)xn-∑i=jn-1‍λnixi=0,k<n≤r.Case  1. Let r=k+1. Then (65) reduces to the single equation
(66)(cj-ck+1)xj+1-(λk-λk-1λk+1)xk=0
which implies that xk=0, since cj=cr=ck+1 and cj≠0. Therefore x=θ.
Case  2. Let r>k+1. From (65) one can obtain the recursion formula xn=λn+1(cj-cn+1)xn+1/(λncj) with k<n<r. Since xr=0 it then follows that xn=0 for k<n<r. Using (65) with n=k+1 yields xk=0 and so again x=θ.
To show thatΛ*-cjI is onto, suppose (Λ*-cjI)x=y, where x,y∈ℓ1. By choosing xj+1=0 one can solve for x0,x1,…,xj in terms of y0,y1,…,yj+1. As in Theorem 16, the remaining equations can be written in the form x=By, where the nonzero entries of B are as in (62) with the other entries of B clearly zero.
Sincek≤j≤r, there are two cases to consider.
Case  1. If j=r, then the proof proceeds exactly as in the argument following (62).
Case  2. If j<r, then from (62), bj+m,j+k=bj+m,j+1=0 at least for m≥r-j+2. If there are other values of n with j<n<r for which cn-cj, then additional entries of B will be zero. These zero entries do not affect the validity of the argument showing that (63) converges.
Ifδ=0, then 0 does not lie inside the disc, and so it is not considered in this theorem.
Letα=δ>0. If λnn≤δ for each n≥1, all i sufficiently large, then the argument of Theorem 16 applies and Λ-δI∈C1. If λnn=δ for some n, then the proof of Theorem 17 applies with cj replaced by δ and again, Λ-δI∈C1.
Therefore, in all cases,Λ-cjI∈1∪2.Theorem 18.
Ifα∈σp(Λ,c0),α∈C3σ(Λ,c0).Proof .
Forα∈σp(Λ,c0),Λ-αI∈3 and Λ*-αI is not one to one. Therefore R(Λ-αI)¯≠c0 by Lemma 12. This concludes the proof.Theorem 19.
The statementA3σ(Λ,c0)=C2σ(Λ,c0)=∅ holds.Proof.
Letδ be defined as in Corollary 9 and cn≥δ for all sufficiently large n, then A3σ(Λ,c0)=∅ and C2σ(Λ,c0)=∅ follow from Corollary 9, Theorems 14, and 16–18.Define the setE by
(67)E={cj:cj≤η2-η}¯,
where η is as in Theorem 7.We will considerδ=η, that is, for which the main diagonal entries converge, where δ as in Corollary 9.Theorem 20.
The following results hold:(a)
σap(Λ,c0) = {α:|α-(2-δ)-1|=(1-δ)/(2-δ)}∪E,(b)
σδ(Λ,c0) = σ(Λ,c0),(c)
σco(Λ,c0) = {α∈ℂ:|α-(2-δ)-1|<(1-δ)/(2-δ)}∪S.Proof .
(a) Since the relation(68)C1σ(Λ,c0)={{α:|α-12-δ|<1-δ2-δ}∖S}⋃‍{α=λnn:δ2-δ<α<1}
holds by Theorems 16 and 17 and from Table 1, σap(Λ,c0)=σ(Λ,c0)∖C1σ(Λ,c0). Therefore, we have σap(Λ,c0)={α:|α-(2-δ)-1|=(1-δ)/(2-δ)}∪E.
(b) Sinceσδ(Λ,c0)=σ(Λ,c0)∖A3σ(Λ,c0) from Table 1 and A3σ(Λ,c0)=∅ by Theorem 19, we have σδ(Λ,c0)=σ(Λ,c0).
(c) Since the equalityσco(Λ,c0)=C1σ(Λ,c0)∪C2σ(Λ,c0)∪C3σ(Λ,c0) holds from Table 1, we have σco(Λ,c0)={α∈ℂ:|α-(2-δ)-1|<(1-δ)/(2-δ)}∪S by Theorems 16–19.The next corollary can be obtained from Proposition1.Corollary 21.
The following results hold:(a)
σap(Λ*,ℓ1)=σ(Λ,c0),(b)
σδ(Λ*,ℓ1)={α:|α-(2-δ)-1|=(1-δ)(2-δ)}∪E,(c)
σp(Λ*,ℓ1)={α∈ℂ:|α-(2-δ)-1|<(1-δ)/(2-δ)}∪S.


## 4. The Fine Spectrum of the OperatorΛ on the Sequence Space c

In this section, we investigate the fine spectrum of the operatorΛ over the sequence space c.Theorem 22.
σ
(
Λ
,
c
)
⊆
{
α
∈
ℂ
:
|
2
α
-
1
|
≤
1
}.Proof.
This is obtained in a similar way to that used in the proof of Theorem6.Theorem 23.
Suppose thatμ, η and S be defined as in Theorem 7. Then,
(69){α∈ℂ:|α-12-μ|≤1-μ2-μ}∪S⊆σ(Λ,c)⊆{α∈ℂ:|α-12-η|≤1-η2-η}∪S.Proof.
This is similar to the proof of Theorems7 and 8. To avoid the repetition of the similar statements, we omit the detail.Corollary 24.
Letδ be defined as in Corollary 9. Then,
(70)σ(Λ,c)={α∈ℂ:|α-12-δ|≤1-δ2-δ}∪S.IfT:c→c is a bounded linear operator with the matrix A, then T*:c*→c* acting on ℂ⊕ℓ1 has a matrix representation of the form
(71)[χ0bAt],
where χ is the limit of the sequence of row sums of A minus the sum of the limit of the columns of A and b is the column vector whose kth entry is the limit of the kth column of A for each k∈ℕ. For Λ:c→c, the matrix Λ*∈B(ℓ1) is of the form
(72)Λ*=[100Λt].Theorem 25.
Letδ be defined as in Corollary 9. Then,
(73)σp(Λ*,c*)={α∈ℂ:|α-12-δ|<1-δ2-δ}∪S.Proof.
Suppose thatΛ*x=αx for x≠θ in c*≅ℓ1. Then, by solving the system of linear equations
(74)(1-α)x0=0,x2=λ1-λ0λ0(1-1α)x1,x3=λ2-λ1λ0(1-c1α)(1-1α)x1⋮xn=λn-1-λn-2λ0(1-1α)x1∏‍j=1n-1(1-cjα)⋮
we get by assumption (1-α)x0=0 with α=1 that x=(x0,x1,0,0,…)∈c. If α≠1, we have x0=0 and (xn)∈ℓ1 if and only if |1+(1-α-1)(λj-λj-1)/λj-1|<1, by Theorem 11. This completes the proof.Theorem 26.
Letδ be defined as in Corollary 9. Then,
(75)σp(Λ,c)={α=cn∈ℂ:0≤α≤δ2-δ}∪{1}.Proof.
The proof is identical to the proof of Theorem11.Theorem 27.
σ
r
(
Λ
,
c
)
=
σ
p
(
Λ
*
,
c
*
)
∖
σ
p
(
Λ
,
c
).Proof.
Forα∈σp(Λ*,c*)∖σp(Λ,c), the operator Λ-αI is triangle, so has an inverse. But Λ*-αI is not one to one by Theorem 26. Therefore by Lemma 12,  R(Λ-αI)¯≠c and this concludes the proof.Since Theorems28–31 can be proved in a similar way to that used in the proof of Theorems 14 and 16–18; respectively, to avoid the repetition of the similar statements we omit the detailed proof and give them without proof.Theorem 28.
Letδ be defined as in Corollary 9 and cn≥δ for all sufficiently large n. Then,
(76)σc(Λ,c)={α∈ℂ:|α-12-δ|=1-δ2-δ,α≠1,δ2-δ}.Theorem 29.
Letδ be defined as in Corollary 9 and less than 1. If α satisfies |α-1/(2-δ)|<(1-δ)/(2-δ) and α∉S, then α∈C1σ(Λ,c).Theorem 30.
Letδ be defined as in Corollary 9 and δ<1. If α=δ or α=cn for all n∈ℕ and δ/(2-δ)<α<1, then α∈C1σ(Λ,c).Theorem 31.
Ifα∈σp(Λ,c), α∈C3σ(Λ,c).Theorem 32.
The following statement holds:A3σ(Λ,c)=C2σ(Λ,c)=∅.Proof.
Letδ be defined as in Corollary 9 and cn≥δ for all sufficiently large n, then A3σ(Λ,c)=∅ and C2σ(Λ,c)=∅ follow from Corollary 24 and Theorems 28–31.Theorem 33.
The following results hold:(a)
σap(Λ,c)={α:|α-(2-δ)-1|=(1-δ)/(2-δ)}∪E,(b)
σδ(Λ,c)=σ(Λ,c),(c)
σco(Λ,c)={α∈ℂ:|α-(2-δ)-1|<(1-δ)/(2-δ)}∪S.Proof.
(a) Since the relationC1σ(Λ,c)={{α:|α-(2-δ)-1|<(1-δ)/(2-δ)}∖S}⋃‍{α=λnn:δ/(2-δ)<α<1} holds by Theorems 29 and 30 and from Table 1, σap(Λ,c)=σ(Λ,c)∖C1σ(Λ,c). Therefore, we have σap(Λ,c)={α:|α-(2-δ)-1|=(1-δ)/(2-δ)}∪E.
(b) Sinceσδ(Λ,c)=σ(Λ,c)∖A3σ(Λ,c) from Table 1 and A3σ(Λ,c)=∅ by Theorem 32, we have σδ(Λ,c)=σ(Λ,c).
(c) Since the equalityσco(Λ,c)=C1σ(Λ,c)∪C2σ(Λ,c)∪C3σ(Λ,c) holds from Table 1, we have σco(Λ,c)={α∈ℂ:|α-(2-δ)-1|<(1-δ)/(2-δ)}∪S by Theorems 29–32.The next corollary can be obtained from Proposition1.Corollary 34.
The following results hold:(a)
σap(Λ*,ℓ1)=σ(Λ,c),(b)
σδ(Λ*,ℓ1)={α:|α-(2-δ)-1|=(1-δ)/(2-δ)}∪E,(c)
σp(Λ*,ℓ1)={α∈ℂ:|α-(2-δ)-1|<(1-δ)/(2-δ)}∪S.LetA be an infinite matrix and let the set cA denote the convergence domain of that matrix A, a theorem which proves that cA=c is called a Mercerian theorem, after Mercer, who proved a significant theorem of this type [28, page 186].Now, we may give our final theorem.Theorem 35.
Suppose that|α+1|>|α-1|. Then the convergence field of A=αI+(1-α)Λ is c.Proof.
By Theorem22, Λ-[α/(α-1)]I has an inverse in B(c). That is to say that
(77)A-1=11-α(Λ-αα-1I)-1∈B(c).
Since A is a triangle and is in B(c), A-1 is also conservative which implies that cA=c [22, page 12].


## 5. Conclusion

Although the matrixΛ is used for obtaining some new sequence spaces by its domain from the classical sequence spaces, it is not considered for determining the spectrum or fine spectrum acting as a linear operator on any of the classical sequence spaces c0, c, or ℓp. Following Altay and Başar [12] and Karakaya and Altun [26], we determine the fine spectrum with respect to Goldberg’s classification of the operator defined by the triangle matrix Λ over the sequence spaces c0 and c which reduces to a new regular triangle matrix depending on choosing the strictly increasing sequence λ=(λk) of positive real numbers tending to infinity. Additionally, we give the approximate point spectrum, the defect spectrum, and the compression spectrum of the matrix operator Λ over the spaces c0 and c. Since the present paper is devoted to the fine spectrum of the operator defined by the lambda matrix over the sequence spaces c0 and c with new subdivision of spectrum, this makes it significant. We should note that the main results of the present paper are given as an extended abstract without proof by Yeşilkayagil and Başar [29].The generalized weighted meansG(u,v) = (gnk) is defined by
(78)gnk={unvk,0≤k≤n,0,k>n,
for all k,n∈ℕ, where un depends only on n and vk only on k such that un,vk≠0. It is immediate that in the case un=1/λn and vk=λk-λk-1, the generalized weighted means G(u,v) corresponds to the matrix Λ. Although the Riesz means Rq, the generalized weighted means G(u,v), and the matrix Λ were used for different purposes, their fine spectrum over the classical sequence spaces was not studied. As a beginning, the present work has an advantage.Finally, we record from now on that our next paper will be devoted to the investigation of the fine spectrum of the matrix operatorΛ on the spaces ℓp and bvp in the cases 0<p<1 and 1≤p<∞, where bvp denotes the space of all sequences whose Δ-transforms are in the space ℓp and was studied in the case 0<p<1 by Altay and Başar [30] and in the case 1≤p≤∞ by Başar and Altay [31].



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