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

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 spaces 𝑐₀ 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 (𝜆_𝑘) having certain properties and its spectrum is firstly investigated, our work is new and the results are comprehensive.

1. Introduction

Let 𝑋 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, 𝑅(𝑇)=𝑦∈𝑌∶𝑦=𝑇𝑥,𝑥∈𝑋. (1) 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 adjoint 𝑇^∗ 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.

Let 𝑋≠{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 “𝑇 is 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 𝑇_𝛼=𝛼𝐼−𝑇 (2) defined on the same domain 𝐷(𝑇) as 𝑇. The spectrum 𝜎(𝑇,𝑋) consists of those 𝛼∈ℂ, the complex field, for which 𝑇_𝛼 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↦𝑇_𝛼⁻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 since 𝑇_𝛼⁻¹ helps to solve the equation 𝑇_𝛼𝑥=𝑦. Thus, 𝑥=𝑇_𝛼⁻¹𝑦 provided that 𝑇_𝛼⁻¹ exists. More importantly, the investigation of properties of 𝑇_𝛼⁻¹ will be basic for an understanding of the operator 𝑇 itself. Naturally, many properties of 𝑇_𝛼 and 𝑇_𝛼⁻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 𝑇_𝛼⁻¹ exists. Boundedness of 𝑇_𝛼⁻¹ is another property that will be essential. We will also ask for what 𝛼's the domain of 𝑇_𝛼⁻¹ is dense in 𝑋, to name just a few aspects. A regular value 𝛼 of 𝑇 is a complex number such that 𝑇_𝛼⁻¹ exists and is bounded and whose domain is dense in 𝑋. For our investigation of 𝑇, 𝑇_𝛼, and 𝑇_𝛼⁻¹, we need some basic concepts in the spectral theory which are given, as follows (see [1, pages 370-371]).

The resolvent set 𝜌(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.

The point (discrete) spectrum 𝜎_𝑝(𝑇,𝑋) is the set such that 𝑇_𝛼⁻¹ does not exist. An 𝛼∈𝜎_𝑝(𝑇,𝑋) is called an eigenvalue of 𝑇.

The continuous spectrum 𝜎_𝑐(𝑇,𝑋) is the set such that 𝑇_𝛼⁻¹ exists and is unbounded, and the domain of 𝑇_𝛼⁻¹ is dense in 𝑋.

The residual spectrum 𝜎_𝑟(𝑇,𝑋) is the set such that 𝑇_𝛼⁻¹ exists (and may be bounded or not) but the domain of 𝑇_𝛼⁻¹ is not dense in 𝑋.

Therefore, these three subspectra form a disjoint subdivision 𝜎(𝑇,𝑋)=𝜎_𝑝(𝑇,𝑋)∪𝜎_𝑐(𝑇,𝑋)∪𝜎_𝑟(𝑇,𝑋). (3) 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 𝜎_𝑐(𝑇,𝑋)=𝜎_𝑟(𝑇,𝑋)=∅ and the spectrum 𝜎(𝑇,𝑋) consists of only the set 𝜎_𝑝(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 operator 𝑇 in a Banach space 𝑋, we call a sequence (𝑥_𝑘) in 𝑋 a Weyl sequence for 𝑇 if ‖𝑥_𝑘‖=1 and ‖𝑇𝑥_𝑘‖→0, as 𝑘→∞. Then, the approximate point spectrum 𝜎_ap(𝑇,𝑋) of 𝑇 is defined by 𝜎_ap().𝑇,𝑋∶=𝛼∈ℂ∶thereexistsaWeylsequencefor𝛼𝐼−𝑇 (4) Moreover, the subspectrum 𝜎_𝛿(𝑇,𝑋)∶=𝛼∈ℂ∶𝛼𝐼−𝑇isnotsurjective (5) is called the defect spectrum of 𝑇.

The two subspectra given by (4) and (5) form a (not necessarily disjoint) subdivision 𝜎(𝑇,𝑋)=𝜎_ap(𝑇,𝑋)∪𝜎_𝛿(𝑇,𝑋) (6) of the spectrum. There is another subspectrum, 𝜎_co(𝑇,𝑋)=𝛼∈ℂ∶𝑅(𝛼𝐼−𝑇)≠𝑋 (7) which is often called compression spectrum in the literature. The compression spectrum gives rise to another (not necessarily disjoint) decomposition 𝜎(𝑇,𝑋)=𝜎_ap(𝑇,𝑋)∪𝜎_co(𝑇,𝑋) (8) of the spectrum. Clearly, 𝜎_𝑝(𝑇,𝑋)⊆𝜎_ap(𝑇,𝑋) and 𝜎_co(𝑇,𝑋)⊆𝜎_𝛿(𝑇,𝑋). Moreover, comparing these subspectra with those in (3), we note that 𝜎_𝑟(𝑇,𝑋)=𝜎_co(𝑇,𝑋)⧵𝜎_𝑝(),𝜎𝑇,𝑋_𝑐(𝑇,𝑋)=𝜎(𝑇,𝑋)⧵𝜎_𝑝(𝑇,𝑋)∪𝜎_co(𝑇,𝑋). (9)

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 operator 𝑇∈𝐵(𝑋) and its adjoint 𝑇^∗∈𝐵(𝑋^∗) are related by the following relations: (a)𝜎(𝑇^∗,𝑋^∗)=𝜎(𝑇,𝑋), (b)𝜎_𝑐(𝑇^∗,𝑋^∗)⊆𝜎_𝑎𝑝(𝑇,𝑋), (c)𝜎_ap(𝑇^∗,𝑋^∗)=𝜎_𝛿(𝑇,𝑋), (d)𝜎_𝛿(𝑇^∗,𝑋^∗)=𝜎_𝑎𝑝(𝑇,𝑋), (e)𝜎_𝑝(𝑇^∗,𝑋^∗)=𝜎_𝑐𝑜(𝑇,𝑋), (f)𝜎_co(𝑇^∗,𝑋^∗)⊇𝜎_𝑝(𝑇,𝑋), (g)𝜎(𝑇,𝑋) = 𝜎_𝑎𝑝(𝑇,𝑋) ∪𝜎_𝑝(𝑇^∗,𝑋^∗) = 𝜎_𝑝(𝑇,𝑋)∪𝜎_𝑎𝑝(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, that 𝜎(𝑇,𝑋)=𝜎_ap(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

If 𝑋 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 for 𝐵(𝑋) and 𝐵(𝑋∗) for a non-reflective Banach space 𝑋.

If 𝛼 is a complex number such that 𝑇_𝛼∈𝐴₁ or 𝑇_𝛼∈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 𝑇_𝛼 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.

Observe that the case in the first row and the second column cannot occur in a Banach space 𝑋, 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 𝑇_𝛼⁻¹ (on a possibly “thin” domain of definition) as “algebraic inverse” of 𝛼𝐼−𝑇.

By a sequence space, we understand a linear subspace of the space 𝜔=ℂ^ℕ 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‖_∞=sup_𝑘∈ℕ|𝑥_𝑘| and ‖𝑥‖_𝑏𝑣=∑^∞_𝑘=0|𝑥_𝑘−𝑥_𝑘+1|, respectively, while 𝜙 is not a Banach space with respect to any norm. Also by ℓ_𝑝, we denote the space of all 𝑝-absolutely summable sequences which is a Banach space with the norm ‖𝑥‖_𝑝=(∑^∞_𝑘=0|𝑥_𝑘|^𝑝)^1/𝑝, where 1≤𝑝<∞.

Let 𝜇 and 𝜈 be two sequence spaces, and let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of complex numbers 𝑎_𝑛𝑘, 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=(𝑥_𝑘)∈𝜇, the sequence 𝐴𝑥={(𝐴𝑥)_𝑛}, the 𝐴-transform of 𝑥, is in 𝜈, where (𝐴𝑥)_𝑛=^∞_𝑘=0𝑎_𝑛𝑘𝑥_𝑘foreach𝑛∈ℕ. (10) 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, let 𝜆=(𝜆_𝑘) be a strictly increasing sequence of positive reals tending to infinity; that is, 0<𝜆₀<𝜆₁<𝜆₂<⋯,lim_𝑘→∞𝜆_𝑘=∞. (11) Following Mursaleen and Noman [20], we define the matrix Λ=(𝜆_𝑛𝑘) of weighted mean relative to the sequence 𝜆 by 𝜆_𝑛𝑘=𝜆_𝑘−𝜆_𝑘−1𝜆_𝑛,0≤𝑘≤𝑛0,𝑘>𝑛 (12) for all 𝑘,𝑛∈ℕ. It is easy to show that the matrix Λ is regular and is reduced, in the special case 𝜆_𝑘=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 𝑄_𝑛=^𝑛_𝑘=0𝑞_𝑘=𝜆_𝑛,𝑟_𝑛𝑘=𝑞_𝑘𝑄_𝑛=𝜆_𝑘−𝜆_𝑘−1𝜆_𝑛=𝜆_𝑛𝑘 (13) in the special case 𝑞_𝑘=𝜆_𝑘−𝜆_𝑘−1 for all 𝑘∈ℕ, the matrix Λ is also reduced to the Riesz means 𝑅^𝑞=(𝑟_𝑛𝑘) with respect to the sequence 𝑞=(𝑞_𝑘). We say that a sequence 𝑥=(𝑥_𝑘)∈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 matrix 𝐴 = (𝑎_𝑛𝑘) gives rise to a bounded linear operator 𝑇∈𝐵(𝑐) from 𝑐 to itself if and only if (1)the rows of 𝐴 are in ℓ₁ and their ℓ₁ norms are bounded; (2)the columns of 𝐴 are in 𝑐; (3)the sequence of row sums of 𝐴 is in 𝑐.

The operator norm of 𝑇 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 matrix 𝐴=(𝑎_𝑛𝑘) gives rise to a bounded linear operator 𝑇∈𝐵(𝑐₀) from 𝑐₀ to itself if and only if (1)the rows of 𝐴 are in ℓ₁ and their ℓ₁ norms are bounded, (2)the columns of 𝐴 are in 𝑐₀.

The operator norm of 𝑇 is the supremum of the ℓ₁ norms of the rows.

Corollary 5. Λ∶𝑐₀→𝑐₀ 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 ℓ_𝑝 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 𝑠_𝑟 and 𝑐₀, 𝑐, where 𝑠_𝑟 denotes the Banach space of all sequences 𝑥=(𝑥_𝑘) normed by ‖𝑥‖_𝑠𝑟=sup_𝑘∈ℕ|𝑥_𝑘|/𝑟^𝑘, (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 Δ_𝜈 on ℓ₁, where Δ_𝜈 is defined by (Δ_𝜈)_𝑛𝑛=𝜈_𝑛 and (Δ_𝜈)_𝑛+1,𝑛=−𝜈_𝑛 for all 𝑛∈ℕ, under certain conditions on the sequence 𝜈=(𝜈_𝑘) and they have just generalized these results by the double sequential band matrix Δ_𝑢𝑣 defined by Δ_𝑢𝑣𝑥=(𝑢_𝑛𝑥_𝑛+𝑣_𝑛−1𝑥_𝑛−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 Δ_𝑎,𝑏, defined as a double-band matrix with the convergent sequences 𝑎^∼=(𝑎_𝑘) and 𝑏^∼=(𝑏_𝑘) 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 ℓ_𝑝 and 𝑏𝑣_𝑝 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 ℓ_𝑝, 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.

In this work, our purpose is to determine the fine spectrum of the operator Λ 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 Operator Λ 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 operator Λ on the sequence space 𝑐₀. For simplicity in the notation, we write throughout that 𝑐_𝑛=(𝜆_𝑛−𝜆_𝑛−1)/𝜆_𝑛 for all 𝑛∈ℕ and we use this abbreviation with other letters.

Theorem 6. 𝜎(Λ,𝑐₀)⊆{𝛼∈ℂ∶|2𝛼−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=(𝑏_𝑛𝑘), where 𝑏_𝑛𝑘=(−1)^𝑛−𝑘𝜆_𝑘−𝜆_𝑘−1𝜆_𝑛𝛼²∏^𝑛_𝑗=𝑘𝑐_𝑗𝜆/𝛼−1,0≤𝑘≤𝑛−1,_𝑛𝜆_𝑛−𝜆_𝑛−1−𝛼𝜆_𝑛,𝑘=𝑛,0,𝑘>𝑛, (14) for all 𝑘,𝑛∈ℕ. Thus, we observe that (Λ−𝛼𝐼)⁻¹_{(𝑐0∶𝑐0)}=sup^∞_{𝑛∈ℕ𝑘=0}𝑏_𝑛𝑘. (15) The inequality |2𝛼−1|>1 is equivalent to 𝛾>−1, where −(1/𝛼)=𝛾+𝑖𝛽. For all 𝛼∈ℂ, 𝑐1−^𝑗𝛼=1+𝛾+𝑖𝛽𝑐_𝑗≥1+𝛾𝑐_𝑗 (16) holds for all 𝑗∈ℕ. So, 1/|1−(𝑐_𝑗/𝛼)|≤1/(1+𝛾𝑐_𝑗).
Firstly we take −1<𝛾<0. Since 0<𝑐_𝑗≤1, we have 1+𝛾≤1+𝛾𝑐_𝑗<1. Therefore 1/(1+𝛾𝑐_𝑗)<1/(1+𝛾) and 1<1/(1+𝛾)<∞ for 0<1+𝛾<1. ^∞_𝑘=0𝑏_𝑛𝑘=^𝑛_𝑘=0𝑏_𝑛𝑘<𝜆_𝑛−1𝜆_𝑛|𝛼|²1+𝛾^𝑛+1+1|𝛼|1+𝛾<∞. (17)
Secondly we get 0≤𝛾. Since 1<1+𝛾𝑐_𝑗≤1+𝛾, 1/(1+𝛾𝑐_𝑗)<1. So, ^∞_𝑘=0𝑏_𝑛𝑘=^𝑛_𝑘=0𝑏_𝑛𝑘<𝜆_𝑛−1𝜆_𝑛|𝛼|²+1|𝛼|<∞. (18) Therefore, we have (Λ−𝛼𝐼)⁻¹_{(𝑐0∶𝑐0)}=sup^𝑛_{𝑛∈ℕ𝑘=0}𝑏_𝑛𝑘<∞, (19) that is, (Λ−𝛼𝐼)⁻¹∈(𝑐₀∶𝑐₀). But for |2𝛼−1|≤1, (Λ−𝛼𝐼)⁻¹_{(𝑐0∶𝑐0)}=∞, (20) that is, (Λ−𝛼𝐼)⁻¹ is not in 𝐵(𝑐₀). This completes the proof.

Theorem 7. Define 𝜇 and 𝜂 by 𝜇=limsup_𝑗→∞𝑐_𝑗 and 𝜂=liminf_𝑗→∞𝑐_𝑗. Then, 1𝛼∈ℂ∶𝛼−≤2−𝜇1−𝜇2−𝜇∪𝑆⊆𝜎Λ,𝑐₀,where𝑆=𝑐_𝑗∶𝑗∈ℕ. (21)

Proof. Let |𝛼−1/(2−𝜇)|<(1−𝜇)/(2−𝜇) and 𝛼≠𝑐_𝑗 for any 𝑗∈ℕ. Then, 𝑐1−^𝑗𝛼=𝜆_𝑗𝜆_𝑗−1𝜆_𝑗𝜆_𝑗−1−𝑐_𝑗𝛼=𝜆_𝑗−1𝜆_𝑗𝜆_𝑗𝜆_𝑗−1−𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1+11−𝛼𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1=𝜆_𝑗−1𝜆_𝑗11+1−𝛼𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1. (22) So we have, 𝑏_𝑛𝑘=𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−1|𝛼|²∏^𝑛_𝑗=𝑘1+(1−1/𝛼)𝜆_𝑗−𝜆_𝑗−1/𝜆_𝑗−1. (23) Note that |1+(1−𝛼⁻¹)(𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1|≤1 if and only if 𝜆1+1+𝛾^𝑗−𝜆_𝑗−1𝜆_𝑗−1²+𝛽𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1²≤1, (24) where −𝛼⁻¹=𝛾+𝑖𝛽. So, one can see that 2𝜆1+𝛾^𝑗−𝜆_𝑗−1𝜆_𝑗−1+1+𝛾²+𝛽²𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1²≤0, (25) which is equivalent to the inequality 21+𝛾+1+𝛾²+𝛽²𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1≤0. (26) For inequality (26) to be true for all sufficiently large 𝑗, it is sufficient to have limsup_𝑗→∞21+𝛾+1+𝛾²+𝛽²𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1<0. (27) We can write (𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1=𝜆_𝑗(𝜆_𝑗−𝜆_𝑗−1)/(𝜆_𝑗𝜆_𝑗−1) and 𝜆_𝑗/𝜆_𝑗−1=1/(1−𝑐_𝑗). Therefore, 𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1=𝑐_𝑗1−𝑐_𝑗,limsup_𝑗→∞𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1=𝜇,1−𝜇 (28)(29) 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 21+𝛾+1+𝛾²+𝛽²𝜇1−𝜇<0 (30) which is equivalent to 1𝛼−<2−𝜇1−𝜇.2−𝜇 (31) Therefore, for all 𝑛≥𝑁, for some fixed 𝑁, ^𝑛−1_𝑘=𝑁𝑏_𝑛𝑘=^𝑛−1_𝑘=𝑁𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−1|𝛼|²∏^𝑛_𝑗=𝑘1+(1−1/𝛼)𝜆_𝑗−𝜆_𝑗−1/𝜆_𝑗−1≥1|𝛼|^2𝑛−1_𝑘=𝑁𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−1 (32) which diverges in the light of (29).
If 𝛼=𝑐_𝑗 for any 𝑗∈ℕ, then clearly 𝛼 lies in the spectrum of Λ. This completes the proof.

Theorem 8. 𝜎(Λ,𝑐₀)⊆{𝛼∈ℂ∶|𝛼−1/(2−𝜂)|≤(1−𝜂)/(2−𝜂)}∪𝑆.

Proof. Let 𝛼 be fixed and satisfy the inequality 1𝛼−>2−𝜂1−𝜂,2−𝜂 (33) and 𝛼≠𝑐_𝑗 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 11+1−𝛼𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1>1 (34) for all sufficiently large 𝑗. It will be sufficient to show that liminf_𝑗→∞21+𝛾+1+𝛾²+𝛽²𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1>0, (35) that is, 21+𝛾+1+𝛾²+𝛽²𝜂1−𝜂>0 (36) which is equivalent to (33).
Define the function 𝑓 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 𝜂21−𝜂>−1+𝛾1+𝛾²+𝛽²=𝑡₀. (37) 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), 𝛾²+𝛽²+2𝛾𝜂>𝜂−2𝜂 (38) which is equivalent to 1−𝜂1−𝜂𝛼1−𝜂>1. (39) 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 (𝜆_𝑛+1−𝜆_𝑛)/𝜆_𝑛>𝜂/(1−𝜂)−𝜖, so that 𝑓((𝜆_𝑛+1−𝜆_𝑛)/𝜆_𝑛)>𝑓(𝜂/(1−𝜂)−𝜖) = 𝑚². Using (23), 𝑏_𝑛𝑘𝑏_𝑛+1,𝑘=𝜆_𝑘−𝜆_𝑘−1/𝜆_𝑘−1|𝛼|²∏^𝑛_𝑗=𝑘1+(1−1/𝛼)𝜆_𝑗−𝜆_𝑗−1/𝜆_𝑗−1𝜆_𝑘−𝜆_𝑘−1/𝜆_𝑘−1|𝛼|²∏^𝑛+1_𝑗=𝑘1+(1−1/𝛼)𝜆_𝑗−𝜆_𝑗−1/𝜆_𝑗−1=11+1−𝛼𝜆_𝑛+1−𝜆_𝑛𝜆_𝑛𝜆=𝑓^𝑛+1−𝜆_𝑛𝜆_𝑛>𝑚²>1, (40) for all 𝑛≥𝑁. Therefore {|𝑏_𝑛𝑘|} 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.
For 𝜖 being used, from (29), we can enlarge 𝑁, if necessary, to ensure that (𝜆_𝑛−𝜆_𝑛−1)/𝜆_𝑛−1<𝜇/(1−𝜇)+1 for 𝑛≥𝑁. From (23), ^𝑛−1_𝑘=𝑁𝑏_𝑛𝑘=^𝑛−1_𝑘=𝑁𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−1|𝛼|²∏^𝑛_𝑗=𝑘1+(1−1/𝛼)𝜆_𝑗−𝜆_𝑗−1/𝜆_𝑗−1≤1|𝛼|²𝜇1−𝜇+1^𝑛−1_𝑘=𝑁1∏^𝑛_𝑗=𝑘1+(1−1/𝛼)𝜆_𝑗−𝜆_𝑗−1/𝜆_𝑗−1≤1|𝛼|²𝜇1−𝜇+1^𝑛−1_𝑘=𝑁𝑚^{−𝑛+𝑘−1}<𝐻, (41) where 𝐻 is a constant independent of 𝑛. Further 𝑏_𝑛𝑛=𝜆_𝑛|𝛼|𝜆_𝑛−𝜆_𝑛−𝜆_𝑛−1=𝜆/𝛼_𝑛|𝛼|𝜆_𝑛−1+(1−1/𝛼)𝜆_𝑛−𝜆_𝑛−1=𝜆_𝑛/𝜆_𝑛−1|𝛼|1+(1−1/𝛼)𝜆_𝑛−𝜆_𝑛−1/𝜆_𝑛−1=1+𝜆_𝑛−𝜆_𝑛−1/𝜆_𝑛−1|𝛼|1+(1−1/𝛼)𝜆_𝑛−𝜆_𝑛−1/𝜆_𝑛−1<1+𝜇/1−𝜇+1|𝛼|𝑚. (42) Hence, 𝐵 has a finite norm.

Corollary 9. Let 𝛿=lim_𝑗→∞𝑐_𝑗 exist. Then, 𝜎Λ,𝑐₀=1𝛼∈ℂ∶𝛼−≤2−𝛿1−𝛿2−𝛿∪𝑆. (43)

If 𝑇∈𝐵(𝑐₀) with the matrix 𝐴, then it is known that the adjoint operator 𝑇^∗∶𝑐^∗₀→𝑐^∗₀ is dened by the transpose 𝐴^𝑡 of the matrix 𝐴. It should be noted that the dual space 𝑐^∗₀ of 𝑐₀ is isometrically isomorphic to the Banach space ℓ₁ of absolutely summable sequences normed by ‖𝑥‖=∑^∞_𝑘=0|𝑥_𝑘|.

Theorem 10. Let 𝛿 be defined as in Corollary 9. Then, 𝜎_𝑝(Λ^∗,𝑐^∗₀)={𝛼∈ℂ∶|𝛼−1/(2−𝛿)|<(1−𝛿)/(2−𝛿)}∪𝑆.

Proof. Suppose that Λ^∗𝑥=𝛼𝑥 for 𝑥≠𝜃 in 𝑐^∗₀≅ℓ₁. Then, by solving the system of linear equations 𝑥₁=𝜆₁−𝜆₀𝜆₀11−𝛼𝑥₀,𝑥₂=𝜆₂−𝜆₁𝜆₀𝑐1−¹𝛼11−𝛼𝑥₀⋮𝑥_𝑛=𝜆_𝑛−𝜆_𝑛−1𝜆₀11−𝛼𝑥₀^𝑛−1_𝑗=1𝑐1−^𝑗𝛼⋮ (44) we can write 𝑥_𝑛 = (𝜆_𝑛−𝜆_𝑛−1)/𝜆_𝑛−1(1−𝛼⁻¹)𝑥₀∏^𝑛−1_𝑗=1[1+(1−𝛼⁻¹)(𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1]. Let |𝛼−1/(2−𝛿)|<(1−𝛿)/(2−𝛿) or 𝛼∈𝑆 and 𝑢_𝑛=∏^𝑛−1_𝑗=1[1+(1−𝛼⁻¹)(𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1]. One can see that |1+(1−𝛼⁻¹)(𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1|<1 for all sufficiently large 𝑗 if and only if 𝜆1+(1+𝛾)^𝑗−𝜆_𝑗−1𝜆_𝑗−1²+𝛽𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1²1<1,where−𝛼=𝛾+𝑖𝛽. (45) Then, we have, from the discussion in Theorem 7 and the hypothesis on 𝛼, 𝑢_𝑛+1𝑢_𝑛=11+1−𝛼𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛−1<1 (46) for all sufficiently large 𝑛, so ∑^∞_𝑛=0|𝑢_𝑛| is convergent. Since |(1−𝛼⁻¹)(𝜆_𝑛−𝜆_𝑛−1)/𝜆_𝑛−1𝑥₀| is bounded, it follows that ∑^∞_𝑛=0|𝑥_𝑛| is convergent, so that Λ^∗𝑥=𝛼𝑥 has nonzero solutions. Therefore, the proof is completed.

Theorem 11. Let 𝛿 be defined as in Corollary 9. Then 𝜎_𝑝Λ,𝑐₀=𝛼=c_𝑛𝛿∈ℂ∶0≤𝛼≤∪{1}2−𝛿. (47)

Proof. Let 𝑐_𝑘 be any diagonal entry satisfying 0<𝑐_𝑘≤𝛿/(2−𝛿). Let 𝑗 be the smallest integer such that 𝑐_𝑗=𝑐_𝑘. By setting 𝑥_𝑛=0 for 𝑛>𝑗+1, 𝑥₀=0, the system (Λ^∗−𝑐_𝑗𝐼)𝑥=𝜃 reduces to a homogeneous linear system of 𝑗 equations in 𝑗+1 unknowns, so that nontrivial solutions exist. Therefore Λ−𝑐_𝑗𝐼∈3.
Λ−𝛼𝐼 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,𝑐₀)=𝜎_𝑝(Λ^∗,𝑐^∗₀)⧵𝜎_𝑝(Λ,𝑐₀).

Proof. For 𝛼∈𝜎_𝑝(Λ^∗,𝑐^∗₀)⧵𝜎_𝑝(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. Let 𝛿 be defined as in Corollary 9 and 𝑐_𝑛≥𝛿 for all sufficiently large 𝑛. Then, 𝜎_𝑐Λ,𝑐₀=1𝛼∈ℂ∶𝛼−=2−𝛿1−𝛿𝛿2−𝛿,𝛼≠1,2−𝛿. (48)

Proof. Fix 𝛼≠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 𝑥_𝑛=𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛−111−𝛼𝑥₀^𝑛_𝑗=𝑘11+1−𝛼𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1 (49) for all positive 𝑛. From the hypothesis, there exists a positive integer 𝑁 such that 𝑛≥𝑁 implies 𝑐_𝑛≥𝛿. This fact, together with the condition on 𝛼, implies that |1+(1−𝛼⁻¹)(𝜆_𝑛−𝜆_𝑛−1)/𝜆_𝑛−1|≥1 for 𝑛≥𝑁. Thus, |𝑥_𝑛|=𝐶(𝜆_𝑛−𝜆_𝑛−1)/𝜆_𝑛−1 for 𝑛≥𝑁, where 𝐶 is a positive constant independent of 𝑛. We can write 𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛−1=𝑐_𝑛𝜆1+^𝑛−𝜆_𝑛−1𝜆_𝑛−1≥𝑐_𝑛. (50) Therefore (𝑥_𝑛)∈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. Let 𝛿 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 all Λ−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.
Suppose 𝑦=(Λ^∗−𝛼𝐼)𝑥, where 𝑥,𝑦∈ℓ₁. Then, 𝑥₀=1/(1−𝛼)𝑦₀−𝜆₀/[(𝜆₁−𝜆₀)(1−𝛼)]𝑦₁ and 𝑐_𝑛−𝛼𝑥_𝑛+𝜆_𝑛−𝜆_𝑛−1^∞_𝑘=𝑛+1𝑥_𝑘𝜆_𝑘=𝑦_𝑛,𝑛>0. (51) Choose 𝑥₁=0 and solve (51) for 𝑥 in terms of 𝑦 to get 𝜆₁−𝜆₀^∞_𝑘=2𝑥_𝑘𝜆_𝑘=𝑦₁,𝑐_𝑛−𝛼𝑥_𝑛=𝑦_𝑛−𝜆_𝑛−𝜆_𝑛−1^∞_𝑘=𝑛+1𝑥_𝑘𝜆_𝑘. (52)(53) For example, substituting (52) into (53), with 𝑛=2, yields 𝑐₂−𝛼𝑥₂=𝑦₂−𝜆₂−𝜆₁^∞_𝑘=3𝑥_𝑘𝜆_𝑘, (54) so that 𝑥₂=(𝜆₂−𝜆₁)/[𝛼(𝜆₁−𝜆₀)]𝑦₁−(1/𝛼)𝑦₂. For 𝑛=3, 𝑥₃𝜆=−³−𝜆₂𝜆₁−𝜆₀𝑐₂1−𝛼𝛼²𝑦₁+𝜆₃−𝜆₂𝜆₂1𝛼²𝑦₂−1𝛼𝑦₃. (55)
Continuing this process, the entries of the matrix 𝐵=(𝑏_𝑛𝑘) such that 𝐵𝑦=𝑥 are calculated as 𝑏₀₀=11−𝛼,𝑏₀₁𝜆=−⁰𝜆₁−𝜆₀()𝑏1−𝛼₂₁=𝜆₂−𝜆₁𝜆₁−𝜆₀𝛼,𝑏_𝑛𝑛1=−𝛼𝑏,𝑛>1,_{𝑛,𝑛−1}=𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛−1𝛼²𝑏,𝑛>2_𝑛1=𝜆_𝑛−𝜆_𝑛−1𝜆₁−𝜆₀𝛼^𝑛−1_𝑗=2𝑐1−^𝑗𝛼𝑏,𝑛>2,_𝑛𝑘=𝜆_𝑛−𝜆_𝑛−1𝜆_𝑘𝛼^2𝑛−1_𝑗=𝑘+1𝑐1−^𝑗𝛼,1<𝑘<𝑛−1, (56) and 𝑏_𝑛𝑘=0 otherwise.
To show that 𝐵∈𝐵(ℓ₁), it is sufficient to establish that ∑^∞_𝑛=0|𝑏_𝑛𝑘| is finite independent of 𝑘. ∑^∞_𝑛=0|𝑏_𝑛0|=1/|1−𝛼|. We may write 1−(𝑐_𝑗/𝛼)=(𝜆_𝑗−1/𝜆_𝑗)[1+(1−𝛼⁻¹)(𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1]. Also, sup_𝑛∈ℕ|(𝜆_𝑛−𝜆_𝑛−1)/𝜆_𝑛−1|≤𝑀<∞. Therefore, ^∞_𝑛=0𝑏_𝑛1≤1|𝛼|𝑀+𝑀^∞_𝑛=3^𝑛−1_𝑗=211+1−𝛼𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1 (57) and, for 𝑘>1, ^∞_𝑛=0𝑏_𝑛𝑘≤1|𝛼|+𝑀|𝛼|²+𝑀|𝛼|^2∞_𝑛=𝑘+2^𝑛−1_𝑗=𝑘+111+1−𝛼𝜆_𝑗−𝜆_𝑗−1𝜆_𝑗−1. (58)
Since 𝑘>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. Let 𝛿 be defined as in Corollary 9 and 𝛿<1. If 𝛼=𝛿 or 𝛼=𝑐_𝑛 for all 𝑛∈ℕ and 𝛿/(2−𝛿)<𝛼<1, then 𝛼∈𝐶₁𝜎(Λ,𝑐₀).

Proof. First assume that Λ has distinct diagonal entries and fix 𝑗≥1. Then the system (Λ−𝑐_𝑗𝐼)𝑥=𝜃 implies that 𝑥_𝑛=0 for 𝑛=0,1,…,𝑗−1, and for 𝑛≥𝑗𝑐_𝑗−𝑐_𝑛𝑥_𝑛−^𝑛−1_𝑘=0𝜆_𝑛𝑘𝑥_𝑘=0. (59) The system (59) yields the following recursion relation: 𝑥_𝑛+1=𝜆_𝑛𝑐_𝑗𝑥_𝑛𝜆_𝑛+1𝑐_𝑗−𝑐_𝑛+1 (60) which can be solved for 𝑥_𝑛 to yield 𝑥_𝑗+𝑚=𝜆_𝑗𝑥_𝑗𝑐^𝑚_𝑗𝜆_𝑗+𝑚∏^𝑚_𝑖=1𝑐_𝑗−𝑐_𝑗+𝑖=𝑥_𝑗^𝑚_𝑖=1𝜆_{𝑗+𝑖−1}𝜆_𝑗+𝑖1−𝑐_𝑗+𝑖/𝑐_𝑗=𝑥_𝑗^𝑚_𝑖=1𝜆_𝑗+𝑖𝜆_{𝑗+𝑖−1}𝜆1−^𝑗𝜆_𝑗+𝑖−𝜆_{𝑗+𝑖−1}𝜆_𝑗+𝑖𝜆_𝑗−𝜆_𝑗−1⁻¹=𝑥_𝑗^𝑚_𝑖=1𝜆_𝑗+𝑖𝜆_{𝑗+𝑖−1}−𝜆_𝑗𝜆_𝑗+𝑖−𝜆_{𝑗+𝑖−1}𝜆_{𝑗+𝑖−1}𝜆_𝑗−𝜆_𝑗−1⁻¹=𝑥_𝑗^𝑚_𝑖=1𝜆_𝑗+𝑖𝜆_{𝑗+𝑖−1}−𝜆_𝑗+𝑖−𝜆_{𝑗+𝑖−1}𝜆_{𝑗+𝑖−1}+11−𝑐_𝑗𝜆_𝑗+𝑖−𝜆_{𝑗+𝑖−1}𝜆_{𝑗+𝑖−1}⁻¹=𝑥_𝑗^𝑚_𝑖=111+1−𝑐_𝑗𝜆_𝑗+𝑖−𝜆_{𝑗+𝑖−1}𝜆_{𝑗+𝑖−1}⁻¹. (61)
Since 0<𝑐_𝑗<1, the argument of Theorem 7 applies and (24) is true. Therefore 𝑥∈𝑐₀ implies 𝑥=𝜃 and Λ−𝑐_𝑗𝐼 is 1−1, so that Λ−𝑐_𝑗𝐼∈1∪2.
Clearly Λ−𝑐_𝑗𝐼∈𝐶. It remains to show that Λ^∗−𝑐_𝑗𝐼 is onto.
Suppose that (Λ^∗−𝑐_𝑗𝐼)𝑥=𝑦, where 𝑥,𝑦∈ℓ₁. By choosing 𝑥_𝑗+1=0 we can solve for 𝑥₀,𝑥₁,…,𝑥_𝑗 in terms of 𝑦₀,𝑦₁,…,𝑦_𝑗+1. As in Theorem 16, the remaining equations can be written in the form 𝑥=𝐵𝑦, where the nonzero entries of 𝐵=(𝑏_𝑛𝑘) are as follows 𝑏_{𝑗+𝑚,𝑗+𝑚}1=−𝑐_𝑗;𝑏_{𝑗+2,𝑗+1}=𝜆_𝑗+2−𝜆_𝑗+1𝑐_𝑗𝜆_𝑗+1−𝜆_𝑗;𝑏_{𝑗+𝑚,𝑗+𝑚−1}=𝜆_𝑗+𝑚−𝜆_{𝑗+𝑚−1}𝑐²_𝑗𝜆_{𝑗+𝑚−1}𝑏,𝑚>2;_{𝑗+𝑚,𝑗+𝑘}=𝜆_𝑗+𝑚−𝜆_{𝑗+𝑚−1}𝑐²_𝑗𝜆_𝑗+𝑘^{𝑗+𝑚−1}_{𝑖=𝑗+𝑘+1}𝑐1−^𝑖𝑐_𝑗,𝑏1<𝑘<𝑚−1,𝑚>3;_{𝑗+𝑚,𝑗+1}=𝜆_𝑗+𝑚−𝜆_{𝑗+𝑚−1}𝑐_𝑗𝜆_𝑗+1−𝜆_𝑗^{𝑗+𝑚−1}_𝑖=𝑗+2𝑐1−^𝑖𝑐_𝑗,𝑚>2. (62) From (62), we have ^∞_𝑛=𝑗+1𝑏_𝑛,𝑗+1=𝜆_𝑗+2−𝜆_𝑗+1𝑐_𝑗𝜆_𝑗+1−𝜆_𝑗+1𝑐_𝑗𝜆_𝑗+1−𝜆_𝑗×^∞_𝑛=𝑗+3𝜆_𝑛−𝜆_𝑛−1^𝑛−1_𝑖=𝑗+2𝑐1−^𝑖𝑐_𝑗. (63) For 𝑚>1, ^∞_{𝑛=𝑚+𝑗}𝑏_{𝑛,𝑚+𝑗}=1𝑐_𝑗+𝜆_𝑗+𝑚+1−𝜆_𝑗+𝑚𝑐²_𝑗𝜆_𝑗+𝑚+1𝑐²_𝑗^∞_{𝑛=𝑗+𝑚+2}𝜆_𝑛−𝜆_𝑛−1𝜆_𝑗+𝑚^𝑛−1_{𝑖=𝑗+𝑚+1}𝑐1−^𝑖𝑐_𝑗. (64) Using 1−(𝑐_𝑗/𝛼)=(𝜆_𝑗−1/𝜆_𝑗)[1+(1−𝛼⁻¹)(𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1], one can convert (63) and (64) the similar expressions to (57) and (58), and therefore ‖𝐵‖_{(ℓ1∶ℓ1)} is finite.
Suppose that Λ does not have distinct diagonal entries. The restriction on 𝛼 guarantees that no zero diagonal entries are being considered. Let 𝑐_𝑗≠0 be any diagonal entry which occurs more than once, and let 𝑘,𝑟 denote, respectively, the smallest and largest integers for which 𝑐_𝑗=𝑐_𝑘=𝑐_𝑟. From (61) it follows that 𝑥_𝑛=0 for 𝑛≥𝑟. Also, 𝑥_𝑛=0 for 0≤𝑛<𝑘. Therefore the system (Λ−𝑐_𝑗𝐼)𝑥=𝜃 becomes 𝑐_𝑗−𝑐_𝑛𝑥_𝑛−^𝑛−1_𝑖=𝑗𝜆_𝑛𝑖𝑥_𝑖=0,𝑘<𝑛≤𝑟. (65)Case 1. Let 𝑟=𝑘+1. Then (65) reduces to the single equation 𝑐_𝑗−𝑐_𝑘+1𝑥_𝑗+1−𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘+1𝑥_𝑘=0 (66) which implies that 𝑥_𝑘=0, since 𝑐_𝑗=𝑐_𝑟=𝑐_𝑘+1 and 𝑐_𝑗≠0. Therefore 𝑥=𝜃.
Case 2. Let 𝑟>𝑘+1. From (65) one can obtain the recursion formula 𝑥_𝑛=𝜆_𝑛+1(𝑐_𝑗−𝑐_𝑛+1)𝑥_𝑛+1/(𝜆_𝑛𝑐_𝑗) with 𝑘<𝑛<𝑟. Since 𝑥_𝑟=0 it then follows that 𝑥_𝑛=0 for 𝑘<𝑛<𝑟. Using (65) with 𝑛=𝑘+1 yields 𝑥_𝑘=0 and so again 𝑥=𝜃.
To show that Λ^∗−𝑐_𝑗𝐼 is onto, suppose (Λ^∗−𝑐_𝑗𝐼)𝑥=𝑦, where 𝑥,𝑦∈ℓ₁. By choosing 𝑥_𝑗+1=0 one can solve for 𝑥₀,𝑥₁,…,𝑥_𝑗 in terms of 𝑦₀,𝑦₁,…,𝑦_𝑗+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.
Since 𝑘≤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), 𝑏_{𝑗+𝑚,𝑗+𝑘}=𝑏_{𝑗+𝑚,𝑗+1}=0 at least for 𝑚≥𝑟−𝑗+2. If there are other values of 𝑛 with 𝑗<𝑛<𝑟 for which 𝑐_𝑛−𝑐_𝑗, then additional entries of 𝐵 will be zero. These zero entries do not affect the validity of the argument showing that (63) converges.
If 𝛿=0, then 0 does not lie inside the disc, and so it is not considered in this theorem.
Let 𝛼=𝛿>0. If 𝜆_𝑛𝑛≤𝛿 for each 𝑛≥1, all 𝑖 sufficiently large, then the argument of Theorem 16 applies and Λ−𝛿𝐼∈𝐶₁. If 𝜆_𝑛𝑛=𝛿 for some 𝑛, then the proof of Theorem 17 applies with 𝑐_𝑗 replaced by 𝛿 and again, Λ−𝛿𝐼∈𝐶₁.
Therefore, in all cases, Λ−𝑐_𝑗𝐼∈1∪2.

Theorem 18. If 𝛼∈𝜎_𝑝(Λ,𝑐₀),𝛼∈𝐶₃𝜎(Λ,𝑐₀).

Proof . For 𝛼∈𝜎_𝑝(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 statement 𝐴₃𝜎(Λ,𝑐₀)=𝐶₂𝜎(Λ,𝑐₀)=∅ holds.

Proof. Let 𝛿 be defined as in Corollary 9 and 𝑐_𝑛≥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 set 𝐸 by 𝐸=𝑐_𝑗∶𝑐_𝑗≤𝜂,2−𝜂 (67) where 𝜂 is as in Theorem 7.

We will consider 𝛿=𝜂, that is, for which the main diagonal entries converge, where 𝛿 as in Corollary 9.

Theorem 20. The following results hold:(a)𝜎_𝑎𝑝(Λ,𝑐₀) = {𝛼∶|𝛼−(2−𝛿)⁻¹|=(1−𝛿)/(2−𝛿)}∪𝐸,(b)𝜎_𝛿(Λ,𝑐₀) = 𝜎(Λ,𝑐₀),(c)𝜎_𝑐𝑜(Λ,𝑐₀) = {𝛼∈ℂ∶|𝛼−(2−𝛿)⁻¹|<(1−𝛿)/(2−𝛿)}∪𝑆.

Proof . (a) Since the relation 𝐶₁𝜎Λ,𝑐₀=1𝛼∶𝛼−<2−𝛿1−𝛿2−𝛿⧵𝑆𝛼=𝜆_𝑛𝑛∶𝛿2−𝛿<𝛼<1 (68) holds by Theorems 16 and 17 and from Table 1, 𝜎_ap(Λ,𝑐₀)=𝜎(Λ,𝑐₀)⧵𝐶₁𝜎(Λ,𝑐₀). Therefore, we have 𝜎_ap(Λ,𝑐₀)={𝛼∶|𝛼−(2−𝛿)⁻¹|=(1−𝛿)/(2−𝛿)}∪𝐸.
(b) Since 𝜎_𝛿(Λ,𝑐₀)=𝜎(Λ,𝑐₀)⧵𝐴₃𝜎(Λ,𝑐₀) from Table 1 and 𝐴₃𝜎(Λ,𝑐₀)=∅ by Theorem 19, we have 𝜎_𝛿(Λ,𝑐₀)=𝜎(Λ,𝑐₀).
(c) Since the equality 𝜎_co(Λ,𝑐₀)=𝐶₁𝜎(Λ,𝑐₀)∪𝐶₂𝜎(Λ,𝑐₀)∪𝐶₃𝜎(Λ,𝑐₀) holds from Table 1, we have 𝜎_co(Λ,𝑐₀)={𝛼∈ℂ∶|𝛼−(2−𝛿)⁻¹|<(1−𝛿)/(2−𝛿)}∪𝑆 by Theorems 16–19.

The next corollary can be obtained from Proposition 1.

Corollary 21. The following results hold: (a)𝜎_ap(Λ^∗,ℓ₁)=𝜎(Λ,𝑐₀), (b)𝜎_𝛿(Λ^∗,ℓ₁)={𝛼∶|𝛼−(2−𝛿)⁻¹|=(1−𝛿)(2−𝛿)}∪𝐸,(c)𝜎_𝑝(Λ^∗,ℓ₁)={𝛼∈ℂ∶|𝛼−(2−𝛿)⁻¹|<(1−𝛿)/(2−𝛿)}∪𝑆.

4. The Fine Spectrum of the Operator Λ on the Sequence Space 𝑐

In this section, we investigate the fine spectrum of the operator Λ over the sequence space 𝑐.

Theorem 22. 𝜎(Λ,𝑐)⊆{𝛼∈ℂ∶|2𝛼−1|≤1}.

Proof. This is obtained in a similar way to that used in the proof of Theorem 6.

Theorem 23. Suppose that 𝜇, 𝜂 and 𝑆 be defined as in Theorem 7. Then, 1𝛼∈ℂ∶𝛼−≤2−𝜇1−𝜇()⊆12−𝜇∪𝑆⊆𝜎Λ,𝑐𝛼∈ℂ∶𝛼−≤2−𝜂1−𝜂2−𝜂∪𝑆. (69)

Proof. This is similar to the proof of Theorems 7 and 8. To avoid the repetition of the similar statements, we omit the detail.

Corollary 24. Let 𝛿 be defined as in Corollary 9. Then, 𝜎()=1Λ,𝑐𝛼∈ℂ∶𝛼−≤2−𝛿1−𝛿2−𝛿∪𝑆. (70)

If 𝑇∶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 𝜒0𝑏𝐴^𝑡, (71) 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 Λ^∗=100Λ^𝑡. (72)

Theorem 25. Let 𝛿 be defined as in Corollary 9. Then, 𝜎_𝑝Λ^∗,𝑐^∗=1𝛼∈ℂ∶𝛼−<2−𝛿1−𝛿2−𝛿∪𝑆. (73)

Proof. Suppose that Λ^∗𝑥=𝛼𝑥 for 𝑥≠𝜃 in 𝑐^∗≅ℓ₁. Then, by solving the system of linear equations (1−𝛼)𝑥₀𝑥=0,₂=𝜆₁−𝜆₀𝜆₀11−𝛼𝑥₁,𝑥₃=𝜆₂−𝜆₁𝜆₀𝑐1−¹𝛼11−𝛼𝑥₁⋮𝑥_𝑛=𝜆_𝑛−1−𝜆_𝑛−2𝜆₀11−𝛼𝑥₁^𝑛−1_𝑗=1𝑐1−^𝑗𝛼⋮ (74) 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 (𝑥_𝑛)∈ℓ₁ if and only if |1+(1−𝛼⁻¹)(𝜆_𝑗−𝜆_𝑗−1)/𝜆_𝑗−1|<1, by Theorem 11. This completes the proof.

Theorem 26. Let 𝛿 be defined as in Corollary 9. Then, 𝜎_𝑝(Λ,𝑐)=𝛼=𝑐_𝑛𝛿∈ℂ∶0≤𝛼≤∪{1}2−𝛿. (75)

Proof. The proof is identical to the proof of Theorem 11.

Theorem 27. 𝜎_𝑟(Λ,𝑐)=𝜎_𝑝(Λ^∗,𝑐^∗)⧵𝜎_𝑝(Λ,𝑐).

Proof. For 𝛼∈𝜎_𝑝(Λ^∗,𝑐^∗)⧵𝜎_𝑝(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 Theorems 28–31 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. Let 𝛿 be defined as in Corollary 9 and 𝑐_𝑛≥𝛿 for all sufficiently large 𝑛. Then, 𝜎_𝑐()=1Λ,𝑐𝛼∈ℂ∶𝛼−=2−𝛿1−𝛿𝛿2−𝛿,𝛼≠1,2−𝛿. (76)

Theorem 29. Let 𝛿 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. Let 𝛿 be defined as in Corollary 9 and 𝛿<1. If 𝛼=𝛿 or 𝛼=𝑐_𝑛 for all 𝑛∈ℕ and 𝛿/(2−𝛿)<𝛼<1, then 𝛼∈𝐶₁𝜎(Λ,𝑐).

Theorem 31. If 𝛼∈𝜎_𝑝(Λ,𝑐), 𝛼∈𝐶₃𝜎(Λ,𝑐).

Theorem 32. The following statement holds:𝐴₃𝜎(Λ,𝑐)=𝐶₂𝜎(Λ,𝑐)=∅.

Proof. Let 𝛿 be defined as in Corollary 9 and 𝑐_𝑛≥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)𝜎_𝑎𝑝(Λ,𝑐)={𝛼∶|𝛼−(2−𝛿)⁻¹|=(1−𝛿)/(2−𝛿)}∪𝐸, (b)𝜎_𝛿(Λ,𝑐)=𝜎(Λ,𝑐),(c)𝜎_𝑐𝑜(Λ,𝑐)={𝛼∈ℂ∶|𝛼−(2−𝛿)⁻¹|<(1−𝛿)/(2−𝛿)}∪𝑆.

Proof. (a) Since the relation 𝐶₁𝜎(Λ,𝑐)={{𝛼∶|𝛼−(2−𝛿)⁻¹|<(1−𝛿)/(2−𝛿)}⧵𝑆}⋃{𝛼=𝜆_𝑛𝑛∶𝛿/(2−𝛿)<𝛼<1} holds by Theorems 29 and 30 and from Table 1, 𝜎_ap(Λ,𝑐)=𝜎(Λ,𝑐)⧵𝐶₁𝜎(Λ,𝑐). Therefore, we have 𝜎_ap(Λ,𝑐)={𝛼∶|𝛼−(2−𝛿)⁻¹|=(1−𝛿)/(2−𝛿)}∪𝐸.
(b) Since 𝜎_𝛿(Λ,𝑐)=𝜎(Λ,𝑐)⧵𝐴₃𝜎(Λ,𝑐) from Table 1 and 𝐴₃𝜎(Λ,𝑐)=∅ by Theorem 32, we have 𝜎_𝛿(Λ,𝑐)=𝜎(Λ,𝑐).
(c) Since the equality 𝜎_co(Λ,𝑐)=𝐶₁𝜎(Λ,𝑐)∪𝐶₂𝜎(Λ,𝑐)∪𝐶₃𝜎(Λ,𝑐) holds from Table 1, we have 𝜎_co(Λ,𝑐)={𝛼∈ℂ∶|𝛼−(2−𝛿)⁻¹|<(1−𝛿)/(2−𝛿)}∪𝑆 by Theorems 29–32.

The next corollary can be obtained from Proposition 1.

Corollary 34. The following results hold: (a)𝜎_𝑎𝑝(Λ^∗,ℓ₁)=𝜎(Λ,𝑐),(b)𝜎_𝛿(Λ^∗,ℓ₁)={𝛼∶|𝛼−(2−𝛿)⁻¹|=(1−𝛿)/(2−𝛿)}∪𝐸,(c)𝜎_𝑝(Λ^∗,ℓ₁)={𝛼∈ℂ∶|𝛼−(2−𝛿)⁻¹|<(1−𝛿)/(2−𝛿)}∪𝑆.

Let 𝐴 be an infinite matrix and let the set 𝑐_𝐴 denote the convergence domain of that matrix 𝐴, a theorem which proves that 𝑐_𝐴=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 Theorem 22, Λ−[𝛼/(𝛼−1)]𝐼 has an inverse in 𝐵(𝑐). That is to say that 𝐴⁻¹=1𝛼1−𝛼Λ−𝐼𝛼−1⁻¹∈𝐵(𝑐). (77) Since 𝐴 is a triangle and is in 𝐵(𝑐), 𝐴⁻¹ is also conservative which implies that 𝑐_𝐴=𝑐 [22, page 12].

5. Conclusion

Although the matrix Λ 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 ℓ_𝑝. 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=(𝜆_𝑘) 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 means 𝐺(𝑢,𝑣) = (𝑔_𝑛𝑘) is defined by 𝑔_𝑛𝑘=𝑢_𝑛𝑣_𝑘,0≤𝑘≤𝑛,0,𝑘>𝑛, (78) for all 𝑘,𝑛∈ℕ, where 𝑢_𝑛 depends only on 𝑛 and 𝑣_𝑘 only on 𝑘 such that 𝑢_𝑛,𝑣_𝑘≠0. It is immediate that in the case 𝑢_𝑛=1/𝜆_𝑛 and 𝑣_𝑘=𝜆_𝑘−𝜆_𝑘−1, the generalized weighted means 𝐺(𝑢,𝑣) corresponds to the matrix Λ. Although the Riesz means 𝑅^𝑞, 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 operator Λ on the spaces ℓ_𝑝 and 𝑏𝑣_𝑝 in the cases 0<𝑝<1 and 1≤𝑝<∞, where 𝑏𝑣_𝑝 denotes the space of all sequences whose Δ-transforms are in the space ℓ_𝑝 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].

Acknowledgments

The authors would like to express their pleasure to Professor Bilâl Altay, Department of Mathematical Education, Faculty of Education, İnönü University, Malatya, Turkey, for many helpful suggestions and interesting comments on the main results of the earlier version of the paper. Additionally, the authors are very grateful to the referee for making some useful remarks which improved the presentation of the paper. The main results of this paper has been presented in part at the conference First International Conference on Analysis and Applied Mathematics (ICAAM 2012) to be held on October 18–21, 2012, in Gümüşhane, Turkey, at the University of Gümüşhane and published in the conference proceedings with AIP, as an extended abstract.