Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space ℓ_𝑝, (1<𝑝<∞)

Abstract

The operator 𝐴(𝑟,̃𝑠)̃ on sequence space on ℓ_𝑝 is defined 𝐴(𝑟,̃𝑠)̃𝑥=(𝑟_𝑘𝑥_𝑘+𝑠_𝑘𝑥_𝑘+1)^∞_𝑘=0, where 𝑥=(𝑥_𝑘)∈ℓ_𝑝, and 𝑟̃ and 𝑠̃ are two convergent sequences of nonzero real numbers satisfying certain conditions, where (1<𝑝<∞). The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg's classification of the operator 𝐴(𝑟,̃𝑠)̃ defined by a double sequential band matrix over the sequence space ℓ_𝑝. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator 𝐴(𝑟,̃𝑠)̃ over the space ℓ_𝑝.

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. (1.1) 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.

Given an operator 𝑇∈𝐵(𝑋), the set 𝜌(𝑇)∶=𝜆∈ℂ∶𝑇_𝜆=𝜆𝐼−𝑇isabijection (1.2) is called the resolvent set of 𝑇 and its complement with respect to the complex plain 𝜎(𝑇)∶=ℂ⧵𝜌(𝑇) (1.3) is called the spectrum of 𝑇. By the closed graph theorem, the inverse operator 𝑅(𝜆;𝑇)∶=(𝜆𝐼−𝑇)⁻¹,𝜆∈𝜌(𝑇) (1.4) is always bounded and is usually called resolvent operator of 𝑇 at 𝜆.

2. Subdivisions of the Spectrum

In this section, we give the definitions of the parts point spectrum, continuous spectrum, residual spectrum, approximate point spectrum, defect spectrum, and compression spectrum of the 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_𝜆⁻¹ helps to solve the equation 𝑇_𝜆𝑥=𝑦. Thus, 𝑥=𝑇_𝜆⁻¹𝑦 provided 𝑇_𝜆⁻¹ exists. More important, the investigation of properties of 𝑇_𝜆⁻¹ will be basic for an understanding of the operator 𝑇 itself. Naturally, many properties of 𝑇_𝜆 and 𝑇_𝜆⁻¹ depend on 𝜆, and spectral theory is concerned with those properties. For instance, we will be interested in the set of all 𝜆's in the complex plane such that 𝑇_𝜆⁻¹ exists. Boundedness of 𝑇_𝜆⁻¹ is another property that will be essential. We will also ask for what 𝜆's the domain of 𝑇_𝜆⁻¹ is dense in 𝑋, to name just a few aspects. A regular value 𝜆 of 𝑇 is a complex number such that 𝑇_𝜆⁻¹ exists and bounded and whose domain is dense in 𝑋. For our investigation of 𝑇, 𝑇_𝜆, and 𝑇_𝜆⁻¹, we need some basic concepts in spectral theory, which are given as follows (see [1, pp. 370-371]).

The resolvent set 𝜌(𝑇,𝑋) of 𝑇 is the set of all regular values 𝜆 of 𝑇. Furthermore, the spectrum 𝜎(𝑇,𝑋) is partitioned into three disjoint sets as follows.

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

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

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

Therefore, these three subspectra form a disjoint subdivisions 𝜎(𝑇,𝑋)=𝜎_𝑝(𝑇,𝑋)∪𝜎_𝑐(𝑇,𝑋)∪𝜎_𝑟(𝑇,𝑋). (2.1) 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 the three more subdivisions of the spectrum called as the 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 as a Weyl sequence for 𝑇 if ‖𝑥_𝑘‖=1 and ‖𝑇𝑥_𝑘‖→0, as 𝑘→∞.

In what follows, we call the set 𝜎_ap(𝑇,𝑋)∶=𝜆∈ℂ∶thereexistsaWeylsequencefor𝜆𝐼−𝑇 (2.2) the approximate point spectrum of 𝑇. Moreover, the subspectrum 𝜎_𝛿(𝑇,𝑋)∶=𝜆∈ℂ∶𝜆𝐼−𝑇isnotsurjective (2.3) is called defect spectrum of 𝑇.

The two subspectra given by (2.2) and (2.3) form a (not necessarily disjoint) subdivision 𝜎(𝑇,𝑋)=𝜎_ap(𝑇,𝑋)∪𝜎_𝛿(𝑇,𝑋) (2.4) of the spectrum. There is another subspectrum 𝜎_co(𝑇,𝑋)=𝜆∈ℂ∶𝑅(𝜆𝐼−𝑇)≠𝑋, (2.5) 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) (2.6) 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 (2.1) 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). (2.7)

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 is also useful.

Proposition 2.1 (see [2, Proposition 1.3, p. 28]). Spectra and subspectra 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 defect spectrum, and the point spectrum 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 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, 𝐶₃. If an operator is in state 𝐶₂, for example, then 𝑅(𝑇)≠𝑋 and 𝑇⁻¹ exist but is discontinuous (see [3] and Figure 1).

Figure 1 

State diagram for 𝐵(𝑋) and 𝐵(𝑋∗) for a nonreflective Banach space 𝑋.

If 𝜆 is a complex number such that 𝑇_𝜆=𝜆𝐼−𝑇∈𝐴₁ or 𝑇_𝜆=𝜆𝐼−𝑇∈𝐵₁, 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 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_𝜆=𝜆𝐼−𝑇 is in a given state, 𝐶₂ (say), then we write 𝜆∈𝐶₂𝜎(𝑇,𝑋).

By the definitions given above, we can illustrate the subdivisions (2.1) in Table 1.

Table 1 

Subdivisions of spectrum of a linear operator.

Observe that the case in the first row and 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 𝜆 is not an eigenvalue of 𝑇, we may always consider the resolvent operator 𝑇_𝜆⁻¹ (on a possibly “thin” domain of definition) as “algebraic” inverse of 𝜆𝐼−𝑇.

By a sequence space, we understand a linear subspace of the space 𝜔=ℂ^ℕ1 of all complex sequences which contains 𝜙, the set of all finitely nonzero sequences, where ℕ₁ denotes the set of positive integers. 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|, while 𝜙 is not a Banach space with respect to any norm, respectively, where ℕ={0,1,2,…}. 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 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of complex numbers 𝑎_𝑛𝑘, where 𝑛,𝑘∈ℕ, and write (𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘𝑛∈ℕ,𝑥∈𝐷₀₀(𝐴), (2.8) where 𝐷₀₀(𝐴) denotes the subspace of 𝑤 consisting of 𝑥∈𝑤 for which the sum exists as a finite sum. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞, and we will use the convention that any term with negative subscript is equal to naught. More generally if 𝜇 is a normed sequence space, we can write 𝐷_𝜇(𝐴) for the 𝑥∈𝑤 for which the sum in (2.8) converges in the norm of 𝜇. We write 𝜆∶𝜇=𝐴∶𝜆⊆𝐷_𝜇(𝐴) (2.9) for the space of those matrices which send the whole of the sequence space 𝜆 into 𝜇 in this sense.

We give a short survey concerning the spectrum and the fine spectrum of the linear operators defined by some particular triangle matrices over certain sequence spaces. The fine spectrum of the Cesàro operator of order one on the sequence space ℓ_𝑝 studied by González [19], where 1<𝑝<∞. Also, weighted mean matrices of operators on ℓ_𝑝 have been investigated by Cartlidge [20]. The spectrum of the Cesàro operator of order one on the sequence spaces 𝑏𝑣₀ and 𝑏𝑣 investigated by Okutoyi [8, 21]. The spectrum and fine spectrum of the Rhally operators on the sequence spaces 𝑐₀, 𝑐, ℓ_𝑝, 𝑏𝑣, and 𝑏𝑣₀ were examined by Yıldırım [9, 22–28]. The fine spectrum of the difference operator Δ over the sequence spaces 𝑐₀ and 𝑐 was studied by Altay and Başar [12]. The same authors also worked the fine spectrum of the generalized difference operator 𝐵(𝑟,𝑠) over 𝑐₀ and 𝑐, in [29]. The fine spectra of Δ over ℓ₁ and 𝑏𝑣 studied by Kayaduman and Furkan [30]. Recently, the fine spectra of the difference operator Δ over the sequence spaces ℓ_𝑝 and 𝑏𝑣_𝑝 studied by Akhmedov and Başar [31, 32], where 𝑏𝑣_𝑝 is the space of 𝑝-bounded variation sequences and introduced by Başar and Altay [33] with 1⩽𝑝<∞. Also, the fine spectrum of the generalized difference operator 𝐵(𝑟,𝑠) over the sequence spaces ℓ₁ and 𝑏𝑣 determined by Furkan et al. [34]. Recently, the fine spectrum of 𝐵(𝑟,𝑠,𝑡) over the sequence spaces 𝑐₀ and 𝑐 has been studied by Furkan et al. [35]. Quite recently, de Malafosse [11] and Altay and Başar [12] have, 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 [36] have determined the fine spectrum of the Zweier matrix, which is a band matrix as an operator over the sequence spaces ℓ₁ and 𝑏𝑣. Farés and de Malafosse [37] studied the spectra of the difference operator on the sequence spaces ℓ_𝑝(𝛼), where (𝛼_𝑛) denotes the sequence of positive reals and ℓ_𝑝(𝛼) is the Banach space of all sequences 𝑥=(𝑥_𝑛) normed by ‖𝑥‖_{ℓ𝑝(𝛼)}=[∑^∞_𝑛=1(|𝑥_𝑛|/𝛼_𝑛)^𝑝]1𝑝 with 𝑝⩾1. Also the fine spectrum of the same operator over ℓ₁ and 𝑏𝑣 has been studied by Bilgiç and Furkan [13]. More recently the fine spectrum of the operator 𝐵(𝑟,𝑠) over ℓ_𝑝 and 𝑏𝑣_𝑝 has been studied by Bilgiç and Furkan [38]. In 2010, Srivastava and Kumar [16] have determined the spectra and the fine spectra of generalized difference operator 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_𝑛), and they have just generalized these results by the generalized difference operator Δ_𝑢𝑣 defined by Δ_𝑢𝑣𝑥=(𝑢_𝑛𝑥_𝑛+𝑣_𝑛−1𝑥_𝑛−1)_𝑛∈ℕ for all 𝑛∈ℕ, (see [18]). Altun [39] has 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 have determined the fine spectra of upper triangular double-band matrices over the sequence spaces 𝑐₀ and 𝑐, in [40]. Quite recently, Akhmedov and El-Shabrawy [15] have obtained the fine spectrum of the generalized difference operator Δ_𝑎,𝑏, defined as a double band matrix with the convergent sequences 𝑎=(𝑎_𝑘) and 𝑏̃=(𝑏_𝑘) having certain properties, over the sequence space 𝑐. Finally, the fine spectrum with respect to the 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]. At this stage, Table 2 may be useful.

Table 2 

Spectrum and fine spectrum of some triangle matrices in certain sequence spaces. In this paper, we study the fine spectrum of the generalized difference operator 𝐴(𝑟,̃𝑠)̃ defined by an upper double sequential band matrix acting on the sequence spaces ℓ_𝑝 with respect to the Goldberg's classification. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator 𝐴(𝑟,̃𝑠)̃ over the spaces ℓ_𝑝. We quote some lemmas, which are needed in proving the theorems given in Section 3.

Lemma 2.2 (see [41, p. 253, Theorem 34.16]). The matrix 𝐴=(𝑎_𝑛𝑘) gives rise to a bounded linear operator 𝑇∈𝐵(ℓ₁) from ℓ₁ to itself if and only if the supremum of ℓ₁ norms of the columns of 𝐴 is bounded.

Lemma 2.3 (see [41, p. 245, Theorem 34.3]). The matrix 𝐴=(𝑎_𝑛𝑘) gives rise to a bounded linear operator 𝑇∈𝐵(ℓ_∞) from ℓ_∞ to itself if and only if the supremum of ℓ₁ norms of the rows of 𝐴 is bounded.

Lemma 2.4 (see [41, p. 254, Theorem 34.18]). Let 1<𝑝<∞ and 𝐴∈(ℓ_∞∶ℓ_∞)∩(ℓ₁∶ℓ₁). Then, 𝐴∈(ℓ_𝑝∶ℓ_𝑝).

Let 𝑟̃=(𝑟_𝑘) and 𝑠̃=(𝑠_𝑘) be sequences whose entries either constants or distinct real numbers satisfying the following conditions: lim_𝑘→∞𝑟_𝑘=𝑟>0,lim_𝑘→∞𝑠_𝑘=𝑠;|𝑠|=𝑟,sup_𝑘∈ℕ𝑟_𝑘⩽𝑟,𝑠²_𝑘⩽𝑟²_𝑘. (2.10)

Then, we define the sequential generalized difference matrix 𝐴(𝑟,̃𝑠)̃ by 𝐴(̃𝑠)̃=𝑟𝑟,⁰𝑠₀00⋯0𝑟₁𝑠₁0⋯00𝑟₂𝑠₂⋯000𝑟₃⋯.⋮⋮⋮⋮⋱ (2.11)

Therefore, we introduce the operator 𝐴(𝑟,̃𝑠)̃ from ℓ_𝑝 to itself by 𝐴(𝑟,̃𝑠)̃𝑥=𝑟_𝑘𝑥_𝑘+𝑠_𝑘𝑥_𝑘+1^∞_𝑘=0,where𝑥=𝑥_𝑘∈ℓ_𝑝. (2.12)

3. Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space ℓ_𝑝

Theorem 3.1. The operator 𝐴(𝑟,̃𝑠)̃∶ℓ_𝑝→ℓ_𝑝 is a bounded linear operator and sup_𝑘∈ℕ𝑟_𝑘^𝑝+𝑠_𝑘^𝑝1/𝑝⩽‖𝐴(𝑟,̃𝑠‖̃_ℓ𝑝⩽sup_𝑘∈ℕ𝑟_𝑘+sup_𝑘∈ℕ𝑠_𝑘. (3.1)

Proof. Since the linearity of the operator 𝐴(𝑟,̃𝑠)̃   is not difficult to prove, we omit the detail. Now we prove that (3.1) holds for the operator 𝐴(𝑟,̃𝑠)̃ on the space ℓ_𝑝. It is trivial that 𝐴(𝑟,̃𝑠)̃𝑒^(𝑘)=(0,0,…,𝑠_𝑘−1,𝑟_𝑘,0,…,0,…) for 𝑒^(𝑘)∈ℓ_𝑝. Therefore, we have ‖𝐴(𝑟,̃𝑠)̃‖_ℓ𝑝⩾𝐴(𝑟,̃𝑠)̃𝑒^(𝑘)_ℓ𝑝𝑒^(𝑘)_ℓ𝑝=𝑟_𝑘^𝑝+𝑠_𝑘−1^𝑝1/𝑝, (3.2) which implies that ‖𝐴(𝑟,̃𝑠)̃‖_ℓ𝑝⩾sup_𝑘∈ℕ𝑟_𝑘^𝑝+𝑠_𝑘^𝑝1/𝑝. (3.3) Let 𝑥=(𝑥_𝑘)∈ℓ_𝑝, where 𝑝>1. Then, since (𝑠_𝑘𝑥_𝑘+1),(𝑟_𝑘𝑥_𝑘)∈ℓ_𝑝 it is easy to see by Minkowski's inequality that ‖𝐴(𝑟,̃𝑠)̃𝑥‖_ℓ𝑝=^∞_𝑘=0𝑠_𝑘𝑥_𝑘+1+𝑟_𝑘𝑥_𝑘^𝑝1/𝑝⩽^∞_𝑘=0𝑠_𝑘𝑥_𝑘+1^𝑝1/𝑝+^∞_𝑘=0𝑟_𝑘𝑥_𝑘^𝑝1/𝑝⩽sup_𝑘∈ℕ𝑟_𝑘^∞_𝑘=0𝑥_𝑘^𝑝1/𝑝+sup_𝑘∈ℕ𝑠_𝑘^∞_𝑘=0𝑥_𝑘+1^𝑝1/𝑝=sup_𝑘∈ℕ𝑟_𝑘‖𝑥‖_ℓ𝑝+sup_𝑘∈ℕ𝑠_𝑘‖𝑥‖_ℓ𝑝=sup_𝑘∈ℕ𝑟_𝑘+sup_𝑘∈ℕ𝑠_𝑘‖𝑥‖_ℓ𝑝, (3.4) which leads us to the result that ‖𝐴(𝑟,̃𝑠)̃‖_ℓ𝑝⩽sup_𝑘∈ℕ𝑟_𝑘+sup_𝑘∈ℕ𝑠_𝑘. (3.5) Therefore, by combining the inequalities in (3.3) and (3.5) we have (3.1), as desired.

Lemma 3.2 (see [42, p. 115, Lemma 3.1]). Let 1<𝑝<∞. If 𝛼∈{𝛼∈ℂ∶|𝑟−𝛼|=|𝑠|}, (3.6) then the series ^∞_𝑘=1𝑟_𝑘−1−𝛼𝑟_𝑘−2−𝛼⋯𝑟₁−𝛼𝑟₀−𝛼𝑠_𝑘−1𝑠_𝑘−2⋯𝑠₁𝑠₀^𝑝 (3.7) is not convergent.

Throughout the paper, by 𝒞 and 𝒮𝒟, we denote the set of constant sequences and the set of sequences of distinct real numbers, respectively.

Theorem 3.3. 𝜎_𝑝𝐴(𝑟,̃𝑠)̃,ℓ_𝑝={||<|𝑠,̃{||<|𝑠∪𝑟𝛼∈ℂ∶𝑟−𝛼|}𝑟∈𝒞,𝛼∈ℂ∶𝑟−𝛼|}_{𝑘𝑘∈ℕ},𝑟,̃𝑠̃∈𝒮𝒟. (3.8)

Proof. Let 𝐴(𝑟,̃𝑠)̃𝑥=𝛼𝑥 for 𝜃≠𝑥∈ℓ_𝑝 Then, by solving linear equation 𝑟₀𝑥₀+𝑠₀𝑥₁=𝛼𝑥₀,𝑟₁𝑥₁+𝑠₁𝑥₂=𝛼𝑥₁,𝑟₂𝑥₂+𝑠₂𝑥₃=𝛼𝑥₂,⋮𝑟_𝑘−1𝑥_𝑘−1+𝑠_𝑘−1𝑥_𝑘=𝛼𝑥_𝑘,⋮ (3.9)𝑥_𝑘=((𝛼−𝑟_𝑘)/𝑠_𝑘−1)𝑥_𝑘−1 for all 𝑘⩾1 and 𝑥_𝑘=𝑟_𝑘−1−𝛼𝑟_𝑘−2−𝛼⋯𝑟₁−𝛼𝑟₀−𝛼𝑠_𝑘−1𝑠_𝑘−2⋯𝑠₁𝑠₀𝑥₀. (3.10)Part 1. Assume that 𝑟,̃𝑠̃∈𝒞. Let 𝑟_𝑘=𝑟 and 𝑠_𝑘=𝑠 For all 𝑘∈ℕ. We observe that 𝑥_𝑘=((𝛼−𝑟)/𝑠)^𝑘𝑥₀. This shows that 𝑥∈ℓ_𝑝 if and if only |𝛼−𝑟|<|𝑠|, as asserted.
Part 2. Assume that 𝑟,̃𝑠̃∈𝒮𝒟. We must take 𝑥₀≠0, since 𝑥≠0. It is clear that, for all 𝑘∈ℕ, the vector 𝑥=(𝑥₀,𝑥₁,…,𝑥_𝑘,0,0,…) is an eigenvector of the operator 𝐴(𝑟,̃𝑠)̃ corresponding to the eigenvalue 𝛼=𝑟_𝑘, where 𝑥₀≠0 and 𝑥_𝑛=((𝛼−𝑟_𝑛)/𝑠_𝑛−1)𝑥_𝑛−1, for 1⩽𝑛⩽𝑘. Thus {𝑟_𝑘∶𝑘∈ℕ}⊆𝜎_𝑝(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝). If 𝑟_𝑘≠𝛼, for all 𝑘∈ℕ, then 𝑥_𝑘≠0. If we take |𝛼−𝑟|<|𝑠|, since lim_𝑘→∞|𝑥_𝑘+1/𝑥_𝑘|^𝑝=lim_𝑘→∞|(𝑟_𝑘−𝛼)/𝑠_𝑘|^𝑝=|(𝑟−𝛼)/𝑠|^𝑝<1, 𝑥∈ℓ_𝑝. Hence {𝛼∈ℂ∶|𝑟−𝛼|<|𝑠|}⊆𝜎_𝑝(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝). Conversely, let 𝛼∈𝜎_𝑝(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝). Then, there exists 𝑥=(𝑥₀,𝑥₁,𝑥₂,…) in ℓ_𝑝 and we have 𝑥_𝑘=((𝛼−𝑟_𝑘)/𝑠_𝑘−1)𝑥_𝑘−1, for all 𝑘⩾1. Since 𝑥∈ℓ_𝑝, we can use ratio test. And so lim_𝑘→∞|𝑥_𝑘+1/𝑥_𝑘|^𝑝=lim_𝑘→∞|(𝑟_𝑘−𝛼)/𝑠_𝑘|^𝑝=|(𝑟−𝛼)/𝑠|^𝑝<1 or 𝛼∈{𝑟_𝑘∶𝑘∈ℂ}. If |𝛼−𝑟|=|𝑠|, by Lemma 3.2  𝑥∉ℓ_𝑝. This completes the proof.

Theorem 3.4. 𝜎_𝑝𝐴(𝑟,̃𝑠)̃^∗,ℓ^∗_𝑝=̃̃𝑟∅,𝑟∈𝒞,ℬ,𝑟∈𝒮𝒟,𝑤ℎ𝑒𝑟𝑒ℬ=_𝑘∶𝑘∈ℕ,𝑟−𝑟_𝑘>|𝑠|. (3.11)

Proof. We prove the theorem by dividing into two parts.
Part 1. Assume that 𝑟,̃𝑠̃∈𝒞. Consider 𝐴(𝑟,̃𝑠)̃^∗𝑓=𝛼𝑓 for 𝑓≠𝜃=(0,0,0,…) in ℓ^∗_𝑝=ℓ_𝑞. Then, by solving the system of linear equations 𝑟₀𝑓₀=𝛼𝑓₀,𝑠₀𝑓₀+𝑟₁𝑓₁=𝛼𝑓₁,𝑠₁𝑓₁+𝑟₂𝑓₂=𝛼𝑓₂,⋮𝑠_𝑘−1𝑓_𝑘−1+𝑟_𝑘𝑓_𝑘=𝛼𝑓_𝑘,⋮ (3.12) we find that 𝑓₀=0 if 𝛼≠𝑟=𝑟_𝑘 and 𝑓₁=𝑓₂=⋯=0 if 𝑓₀=0, which contradicts 𝑓≠𝜃. If 𝑓_𝑛0 is the first nonzero entry of the sequence 𝑓=(𝑓_𝑛) and 𝛼=𝑟, then we get 𝑠_𝑛0𝑓_𝑛0+𝑟𝑓_𝑛0+1=𝛼𝑓_𝑛0+1 that implies 𝑓_𝑛0=0, which contradicts the assumption 𝑓_𝑛0≠0. Hence, the equation 𝐴(𝑟,̃𝑠)̃^∗𝑓=𝛼𝑓 has no solution 𝑓≠𝜃.
Part 2. Assume that 𝑟,̃𝑠̃∈𝒮𝒟. Then, by solving the equation 𝐴(𝑟,̃𝑠)̃^∗𝑓=𝛼𝑓 for 𝑓≠𝜃=(0,0,0,…) in ℓ_𝑞, we obtain (𝑟₀−𝛼)𝑓₀=0 and (𝑟_𝑘+1−𝛼)𝑓_𝑘+1+𝑠_𝑘𝑓_𝑘=0 for all 𝑘∈ℕ. Hence, for all 𝛼∉{𝑟_𝑘∶𝑘∈ℕ}, we have 𝑓_𝑘=0 for all 𝑘∈ℕ, which contradicts our assumption. So, 𝛼∉𝜎_𝑝(𝐴(𝑟,̃𝑠)̃^∗,ℓ_𝑞). This shows that 𝜎_𝑝(𝐴(𝑟,̃𝑠)̃^∗,ℓ_𝑞)⊆{𝑟_𝑘∶𝑘∈ℕ}⧵{𝑟}. Now, we prove that 𝛼∈𝜎_𝑝𝐴(𝑟,̃𝑠)̃^∗,ℓ_𝑞iff𝛼∈ℬ. (3.13) If 𝛼∈𝜎_𝑝(𝐴(𝑟,̃𝑠)̃^∗,ℓ_𝑞), then, by solving the equation 𝐴(𝑟,̃𝑠)̃^∗𝑓=𝛼𝑓 for 𝑓≠𝜃=(0,0,0,…) in ℓ_𝑞 with 𝛼=𝑟₀, 𝑓_𝑘=𝑠₀𝑠₁𝑠₂⋯𝑠_𝑘−1𝑟₀−𝑟_𝑘𝑟₀−𝑟_𝑘−1𝑟₀−𝑟_𝑘−2⋯𝑟₀−𝑟₁𝑓₀∀𝑘⩾1, (3.14) which can expressed by the recursion relation 𝑓_𝑘=𝑠₀𝑠₁𝑠₂⋯𝑠_𝑘−1𝑟₀−𝑟₁𝑟₀−𝑟₂⋯𝑟₀−𝑟_𝑘𝑓₀. (3.15) Using ratio test, lim_𝑘→∞𝑓_𝑘𝑓_𝑘−1^𝑞=lim_𝑘→∞𝑠_𝑘−1𝑟_𝑘−𝑟₀^𝑞=𝑠𝑟−𝑟₀^𝑞⩽1. (3.16) But |𝑠/(𝑟−𝑟₀)|≠1. Hence, 𝛼=𝑟₀∈𝑟_𝑘∶𝑘∈ℕ,𝑟_𝑘−𝑟>|𝑠|=ℬ. (3.17) If we choose 𝛼=𝑟_𝑘≠𝑟 for all 𝑘∈ℕ₁, then we get 𝑓₀=𝑓₁=𝑓₂=⋯=𝑓_𝑘−1=0 and 𝑓_𝑛+1=𝑠_𝑛𝑠_𝑛−1𝑠_𝑛−2⋯𝑠_𝑘𝑟_𝑘−𝑟_𝑛+1𝑟_𝑘−𝑟_𝑛𝑟_𝑘−𝑟_𝑛−1⋯𝑟_𝑘−𝑟_𝑘+1𝑓_𝑘∀𝑛⩾𝑘, (3.18) which can expressed by the recursion relation 𝑓_𝑛+1=𝑠_𝑛−1𝑠_𝑛−2𝑠_𝑛−2⋯𝑠_𝑘𝑟_𝑘−𝑟_𝑛+1𝑟_𝑘−𝑟_𝑛−1𝑟_𝑘−𝑟_𝑛−2⋯𝑟_𝑘−𝑟_𝑘+1𝑓_𝑘. (3.19) Using ratio test, lim_𝑛→∞𝑓_𝑛+1𝑓_𝑛^𝑞=lim_𝑛→∞𝑠_𝑛𝑟_𝑛+1−𝑟_𝑘^𝑞=𝑠𝑟−𝑟_𝑘^𝑞⩽1. (3.20) But |𝑠/(𝑟−𝑟_𝑘)|≠1. So we have 𝛼=𝑟_𝑘∈𝑟_𝑘∶𝑘∈ℕ,𝑟_𝑘−𝑟>|𝑠|=ℬ.Hence,𝜎_𝑝𝐴(𝑟,̃𝑠)̃^∗,ℓ_𝑞⊆ℬ. (3.21) Conversely, let 𝛼∈ℬ. Then exist 𝑘∈ℕ, 𝛼=𝑟_𝑘≠𝑟, and lim_𝑛→∞𝑓_𝑛𝑓_𝑛−1^𝑞=lim_𝑛→∞𝑠_𝑛𝑟_𝑛+1−𝑟_𝑘^𝑞=𝑠𝑟−𝑟_𝑘^𝑞<1. (3.22) That is, 𝑓∈ℓ_𝑞. So we have ℬ⊆𝜎_𝑝(𝐴(𝑟,̃𝑠)̃^∗,ℓ_𝑞). This completes the proof.

Lemma 3.5 (see [3, p. 60]). The adjoint operator 𝑇^∗ of 𝑇 is onto if and only if 𝑇 is a bounded operator.

Theorem 3.6. 𝜎_𝑟(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)=𝜎_𝑝(𝐴(𝑟,̃𝑠)̃^∗,ℓ^∗_𝑝)⧵𝜎_𝑝(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)

Proof. The proof is obvious so is omitted.

Theorem 3.7. Let (𝑟_𝑘),(𝑠_𝑘) in 𝒮𝒟 and 𝒞. 𝜎_𝑟(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)=∅.

Proof. By Theorems 3.4 and 3.6, 𝜎_𝑟(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)=∅.

Theorem 3.8. Let 𝒜={𝛼∈ℂ∶|𝑟−𝛼|⩽|𝑠|} and ℬ={𝑟_𝑘∶𝑘∈ℕ,|𝑟−𝑟_𝑘|>|𝑠|}. Then, the set ℬ is finite and 𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)=𝒜∪ℬ.

Proof. We will show that 𝐴_𝛼(𝑟,̃𝑠)̃^∗ is onto, for |𝑟−𝛼|>|𝑠|. Thus, for every 𝑦∈ℓ_𝑞, we find 𝑥∈ℓ_𝑞. 𝐴_𝛼(𝑟,̃𝑠)̃^∗ is triangle so it has an inverse. Also equation 𝐴_𝛼(𝑟,̃𝑠)̃^∗𝑥=𝑦 gives [𝐴_𝛼(𝑟,̃𝑠)̃^∗]⁻¹𝑦=𝑥. It is sufficient to show that [𝐴_𝛼(𝑟,̃𝑠)̃^∗]⁻¹∈(ℓ_𝑞∶ℓ_𝑞). We can calculate that 𝐴=(𝑎_𝑛𝑘)=[𝐴_𝛼(𝑟,̃𝑠)̃^∗]⁻¹ as follows: 𝑎_𝑛𝑘=1𝑟₀−𝛼00⋯−𝑠₀𝑟₁−𝛼𝑟₀1−𝛼𝑟₁𝑠−𝛼0⋯₀𝑠₁𝑟₀−𝛼𝑟₁−𝛼𝑟₂−𝛼−𝑠₁𝑟₂−𝛼𝑟₁1−𝛼𝑟₂⋯.−𝛼⋮⋮⋮⋱ (3.23) Therefore, the supremum of the ℓ₁ norms of the rows of [𝐴_𝛼(𝑟,̃𝑠)̃^∗]⁻¹ is 𝑆_𝑘, where 𝑆_𝑘=1𝑟_𝑘+𝑠−𝛼𝑟_𝑘−1−𝛼^𝑘−1𝑟_𝑘+𝑠−𝛼^𝑘−1𝑠_𝑘−2𝑟_𝑘−2−𝛼𝑟_𝑘−1−𝛼𝑟_𝑘𝑠−𝛼+⋯+⁰𝑠₁⋯𝑠_𝑘−1𝑟₀−𝛼𝑟₁−𝛼⋯𝑟_𝑘.−𝛼 (3.24) Now, we prove that (𝑆_𝑘)∈ℓ_∞. Since lim_𝑘→∞|𝑠_𝑘/(𝑟_𝑘−𝛼)|=|𝑠/(𝑟−𝛼)|=𝑝<1, then there exists 𝑘₀∈ℕ such that |𝑠_𝑘/(𝑟_𝑘−𝛼)|<𝑝₀ with 𝑝₀<1, for all 𝑘⩾𝑘₀+1, 𝑆_𝑘=1𝑟_𝑘𝑠−𝛼1+^𝑘−1𝑟_𝑘−1+𝑠−𝛼^𝑘−1𝑠_𝑘−2𝑟_𝑘−1−𝛼𝑟_𝑘−2𝑠−𝛼+⋯+^𝑘−1𝑠_𝑘−2⋯𝑠_𝑘0+1𝑠_𝑘0⋯𝑠₀𝑟_𝑘−1−𝛼𝑟_𝑘−2−𝛼⋯𝑟_𝑘0+1−𝛼𝑟_𝑘0−𝛼⋯𝑟₀⩽1−𝛼𝑟_𝑘−𝛼1+𝑝₀+𝑝²₀+⋯+𝑝^𝑘−𝑘0₀+𝑝^𝑘−𝑘0₀𝑠_𝑘0−1𝑟_𝑘0−1−𝛼+⋯+𝑝^𝑘−𝑘0₀𝑠_𝑘0−1𝑠_𝑘0−2⋯𝑠₀𝑟_𝑘0−1−𝛼𝑟_𝑘0−2−𝛼⋯𝑟₀.−𝛼 (3.25) Therefore, 𝑆_𝑘⩽1𝑟_𝑘−𝛼1+𝑝₀+𝑝²₀+⋯𝑝^𝑘−𝑘0₀+𝑝^𝑘−𝑘0₀𝑀𝑘₀, (3.26) where 𝑀𝑘₀𝑠=1+^𝑘0−1𝑟_𝑘0−1+𝑠−𝛼^𝑘0−1𝑠_𝑘0−2𝑟_𝑘0−1−𝛼𝑟_𝑘0−2𝑠−𝛼+⋯+^𝑘0−1𝑠_𝑘0−2⋯𝑠₀𝑟_𝑘0−1−𝛼𝑟_𝑘0−2−𝛼⋯𝑟₀.−𝛼 (3.27) Then, 𝑀𝑘₀⩾1 and so 𝑆_𝑘⩽𝑀𝑘₀𝑟_𝑘−𝛼1+𝑝₀+𝑝²₀+⋯+𝑝^𝑘−𝑘0₀. (3.28) But there exist 𝑘₁∈ℕ and a real number 𝑝₁ such that 1/|𝑟_𝑘−𝛼|<𝑝₁ for all 𝑘⩾𝑘₁. Then, 𝑆_𝑘⩽(𝑀𝑝₁𝑘₀)/(1−𝑝₀) for all 𝑘>max{𝑘₀,𝑘₁}. Hence, sup_𝑘∈ℕ𝑆_𝑘<∞. This shows that [𝐴^∗(𝑟,̃𝑠)̃−𝛼𝐼]⁻¹∈(ℓ_∞∶ℓ_∞). Similarly, we can show that [(𝐴(𝑟,̃𝑠)̃−𝛼𝐼)^∗]⁻¹∈(ℓ₁∶ℓ₁). By Lemma 2.4, we have (𝐴(𝑟,̃𝑠)̃−𝛼𝐼)^∗−1∈ℓ_𝑞∶ℓ_𝑞for𝛼∈ℂwith|𝑟−𝛼|>|𝑠|. (3.29) Hence, 𝐴_𝛼(𝑟,̃𝑠)̃^∗ is onto. By Lemma 3.5, 𝐴_𝛼(𝑟,̃𝑠)̃ is bounded inverse. This means that 𝜎_𝑐𝐴(𝑟,̃𝑠)̃,ℓ_𝑝⊆{𝛼∈ℂ∶|𝑟−𝛼|⩽|𝑠|}. (3.30) Combining this with Theorem 3.3 and Theorem 3.7, we get 𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝⊆{𝛼∈ℂ∶|𝑟−𝛼|⩽|𝑠|}∪ℬ (3.31) and again from Theorem 3.3  {𝛼∈ℂ∶|𝑟−𝛼|<|𝑠|}⊆𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝) and ℬ⊆𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝). Since the spectrum of any bounded operator is closed, we have {𝛼∈ℂ∶|𝑟−𝛼|⩽|𝑠|}∪ℬ⊆𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝. (3.32)
Combining (3.31) and (3.32), we get 𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=𝒜∪ℬ. (3.33)

Theorem 3.9. Let (𝑟_𝑘),(𝑠_𝑘) in 𝒮𝒟 or 𝒞. 𝜎_𝑐(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)={𝛼∈ℂ∶|𝑟−𝛼|=|𝑠|}.

Proof. The proof follows of immediately from Theorems 3.3, 3.7, and 3.8 because the parts 𝜎_𝑐(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝), 𝜎_𝑟(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝), and 𝜎_𝑝(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝) are pairwise disjoint sets and union of these sets is 𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝).

Theorem 3.10. Let (𝑟_𝑘),(𝑠_𝑘)∈𝒮𝒟 and 𝒞. If |𝛼−𝑟|<|𝑠|, 𝛼∈𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)𝐴₃.

Proof. From Theorem 3.3,  𝛼∈𝜎_𝑝(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝). Thus, (𝐴(𝑟,̃𝑠)̃−𝛼𝐼)⁻¹ does not exist. It is sufficient to show that the operator (𝐴(𝑟,̃𝑠)̃−𝛼𝐼) is onto, that is, for given 𝑦=(𝑦_𝑘)∈ℓ_𝑝, we have to find 𝑥=(𝑥_𝑘)∈ℓ_𝑝 such that (𝐴(𝑟,̃𝑠)̃−𝛼𝐼)𝑥=𝑦. Solving the linear equation (𝐴(𝑟,̃𝑠)̃−𝛼𝐼)𝑥=𝑦, [𝐴(̃𝑠)̃]𝑟𝑟,−𝛼𝐼𝑥=⁰−𝛼𝑠₀00⋯0𝑟₁−𝛼𝑠₁0⋯00𝑟₂−𝛼𝑠₂⋯000𝑟₃𝑥−𝛼⋯⋮⋮⋮⋮⋱⁰𝑥₁𝑥₂⋮=𝑦₀𝑦₁𝑦₂⋮, (3.34)
let 𝑥₀𝑥=0,₁=𝑦₀𝑠₀,𝑥₂=𝛼−𝑟₁𝑦₀𝑠₁𝑠₀+𝑦₁𝑠₁,⋮𝑥_𝑘=𝛼−𝑟₁𝛼−𝑟₂⋯𝛼−𝑟_𝑘−1𝑦₀𝑠₀𝑠₁⋯𝑠_𝑘−1𝑟+⋯+^𝑘−2−𝛼𝑦_𝑘−2𝑠_𝑘−1𝑠_𝑘−2+𝑦_𝑘−1𝑠_𝑘−1. (3.35) Then, ∑_𝑘|𝑥_𝑘|^𝑝⩽sup_𝑘(𝑅_𝑘)^𝑝∑_𝑘|𝑦_𝑘|^𝑝, where 𝑅_𝑘=1𝑠_𝑘+𝑟_𝑘+1−𝛼𝑠_𝑘𝑠_𝑘+1+𝑟_𝑘+1−𝛼𝑟_𝑘+2−𝛼𝑠_𝑘𝑠_𝑘+1𝑠_𝑘+2𝑅+⋯,_𝑛𝑘=1𝑠_𝑘+𝑟_𝑘+1−𝛼𝑠_𝑘𝑠_𝑘+1+𝑟_𝑘+1−𝛼𝑟_𝑘+2−𝛼𝑠_𝑘𝑠_𝑘+1𝑠_𝑘+2+𝑟+⋯,_𝑘+1−𝛼𝑟_𝑘+2−𝛼⋯𝑟_𝑘+𝑛−𝛼𝑠_𝑘𝑠_𝑘+1⋯𝑠_𝑛+𝑘 (3.36) for all 𝑘,𝑛∈ℕ. Then, since 𝑅^𝑛=lim_𝑘→∞𝑅^𝑘_𝑛=1𝑠+(𝑟−𝛼)𝑠²+(𝑟−𝛼)²𝑠³()+⋯+𝑟−𝛼^𝑛+1𝑠^𝑛+2, (3.37) we have 𝑅=lim_𝑛→∞𝑅^𝑛=1𝑠1+𝑟−𝛼𝑠¹+⋯<∞. (3.38) Since |𝑟−𝛼|<|𝑠|, (𝑅_𝑘) is a convergent sequence of positive real numbers with limit 𝑅. Hence, (𝑅_𝑘) bounded and we have sup_𝑘(𝑅_𝑘)^𝑝<∞. Therefore, _𝑘𝑥_𝑘^𝑝⩽sup_𝑘𝑅_𝑘^𝑝_𝑘𝑦_𝑘^𝑝<∞. (3.39) This shows that 𝑥=(𝑥_𝑘)∈ℓ_𝑝. Thus (𝐴(𝑟,̃𝑠)̃−𝛼𝐼) is onto. So we have 𝛼∈𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)𝐴₃.

Theorem 3.11. Let (𝑟_𝑘),(𝑠_𝑘)∈𝒞 with 𝑟_𝑘=𝑟, 𝑠_𝑘=𝑠 for all 𝑘∈ℕ. Then, the following statements hold: (i)𝜎_ap(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)=𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝), (ii)𝜎_𝛿(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)={𝛼∈ℂ∶|𝑟−𝛼|=|𝑠|}, (iii)𝜎_co(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)=∅.

Proof. (i) Since from Table 1, 𝜎_ap𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=𝜎𝐴(𝑟,̃𝑠)̃,ℓ₁⧵𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₁, (3.40) we have by Theorem 3.7𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₁=𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₂=∅. (3.41) Hence, 𝜎_ap𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=𝒜. (3.42)
(ii) Since the following equality: 𝜎_𝛿𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝⧵𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐴₃ (3.43) holds from Table 1, we derive by Theorems 3.8 and 3.10 that 𝜎_𝛿(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)={𝛼∈ℂ∶|𝑟−𝛼|=|𝑠|}.
(iii) From Table 1, we have 𝜎_co𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₁∪𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₂∪𝜎𝐴(𝑟,̃𝑠)̃,𝑐₀𝐶₃. (3.44)
By Theorem 3.4, it is immediate that 𝜎_co(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)=∅.

Theorem 3.12. Let (𝑟_𝑘)∈𝒮𝒟. Then 𝜎_𝑎𝑝𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝜎=𝒜∪ℬ,_𝛿𝐴(𝑟,̃𝑠)̃,ℓ_𝑝={||=|𝑠𝜎𝛼∈ℂ∶𝑟−𝛼|}∪ℬ,_𝑐𝑜𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=ℬ. (3.45)

Proof. We have by Theorem 3.4 and Part (e) of Proposition 2.1 that 𝜎_𝑝𝐴(𝑟,̃𝑠)̃^∗,ℓ^∗_𝑝=𝜎_co𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=ℬ. (3.46) By Theorems 3.7 and 3.4, we must have 𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₁=𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₂=∅. (3.47) Hence, 𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)𝐶₃={𝑟_𝑘}. Additionally, since 𝜎(𝐴(𝑟,̃𝑠)̃,ℓ_𝑝)𝐶₁=∅.
Therefore, we derive from Table 1, Theorems 3.8, and 3.10 that 𝜎_ap𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝⧵𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐶₁=𝜎𝐴(𝑟,̃𝑠)̃,ℓ₁,𝜎_𝛿𝐴(𝑟,̃𝑠)̃,ℓ_𝑝=𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝⧵𝜎𝐴(𝑟,̃𝑠)̃,ℓ_𝑝𝐴₃={𝛼∈ℂ∶|𝑟−𝛼|=|𝑠|}∪ℬ. (3.48)

4. Conclusion

In the present work, as a natural continuation of Akhmedov and El-Shabrawy [15] and Srivastava and Kumar [18], we have determined the spectrum and the fine spectrum of the double sequential band matrix 𝐴(𝑟,̃𝑠)̃ on the space ℓ_𝑝. Many researchers determine the spectrum and fine spectrum of a matrix operator in some sequence spaces. In addition to this, we add the definition of some new divisions of spectrum called as approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator and give the related results for the matrix operator 𝐴(𝑟,̃𝑠)̃ on the space ℓ_𝑝, which is a new development for this type works giving the fine spectrum of a matrix operator on a sequence space with respect to the Goldberg's classification.

Acknowledgment

The authors would like to express their gratitude to Professor Feyzi Basar, Fatih University, Faculty of Art and Sciences, Department of Mathematics, The Hadımköy Campus, Büyükçekmece, Turkey, for his careful reading and for making some useful corrections, which improved the presentation of the paper.