On the Spectral Properties of the Weighted Mean Difference Operator 𝐺(𝑢,𝑣;Δ) over the Sequence Space 1

Abstract

In the present work the generalized weighted mean difference operator 𝐺(𝑢,𝑣;Δ) has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences 𝑢=(𝑢_𝑘) and 𝑣=(𝑣_𝑘). For any two sequences 𝑢 and 𝑣 of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator 𝐺(𝑢,𝑣;Δ) is defined by (𝐺(𝑢,𝑣;Δ)𝑥)_𝑘=∑^𝑘_𝑖=0𝑢_𝑘𝑣_𝑖(𝑥_𝑖−𝑥_𝑖−1) with 𝑥_𝑘=0 for all 𝑘<0. Furthermore, we compute the spectrum and the fine spectrum of the operator 𝐺(𝑢,𝑣;Δ) over the sequence space 1. In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space 1.

1. Introduction, Preliminaries, and Definitions

Let 𝑢=(𝑢_𝑘) and 𝑣=(𝑣_𝑘) be two bounded sequences of either constant or strictly decreasing positive real numbers such that 𝑢_𝑘≠0 and 𝑣_𝑘≠0 for all 𝑘, and lim_𝑘→∞𝑢_𝑘=𝑢,lim_𝑘→∞𝑣_𝑘−𝑣_𝑘+1𝑣_𝑘−1−𝑣_𝑘=1forstrictlydecreasingsequence. (1)(2)

By 1 and 𝑏𝑣_𝑝, we denote the spaces of all absolutely summable and p-bounded variation series, respectively. Also, by ∞, 𝑐, and 𝑐₀, we denote the spaces of all bounded, convergent, and null sequences, respectively. The main perception of this paper is to introduce the weighted mean difference operator 𝐺(𝑢,𝑣;Δ) as follows.

Let 𝑥=(𝑥_𝑘) be any sequence in 1, and we define the weighted mean difference transform 𝐺(𝑢,𝑣;Δ)𝑥 of 𝑥 by (𝐺(𝑢,𝑣;Δ)𝑥)_𝑘=^𝑘_𝑖=0𝑢_𝑘𝑣_𝑖𝑥_𝑖−𝑥_𝑖−1𝑘∈ℕ₀, (3) where ℕ₀ denotes the set of nonnegative integers and we assume throughout that any term with negative subscript is zero. Instead of writing (3), the operator 𝐺(𝑢,𝑣;Δ) can be expressed as a lower triangular matrix (𝑔_𝑛𝑘), where 𝑔_𝑛𝑘=𝑢_𝑘𝑣_𝑘,(),𝑢𝑘=𝑛_𝑘𝑣_𝑘−𝑣_𝑘+1,(),().0≤𝑘≤𝑛−10,𝑘>0 (4) Equivalently, in componentwise the triangle (𝑔_𝑛𝑘) can be represented by𝑔_𝑛𝑘=𝑢₀𝑣₀𝑢0000⋯₁𝑣₀−𝑣₁𝑢₁𝑣₁𝑢000⋯₂𝑣₀−𝑣₁𝑢₂𝑣₁−𝑣₂𝑢₂𝑣₂𝑢00⋯₃𝑣₀−𝑣₁𝑢₃𝑣₁−𝑣₂𝑢₃𝑣₂−𝑣₃𝑢₃𝑣₃.0⋯⋮⋮⋮⋮⋮⋱ (5)

The main objective of this paper is to determine the spectrum of the operator 𝐺(𝑢,𝑣;Δ) over the basic sequence space 1. The operator 𝐺(𝑢,𝑣;Δ) has been studied by Polat et al. [1] in detail by introducing the difference sequence spaces _∞(𝑢,𝑣;Δ), 𝑐₀(𝑢,𝑣;Δ), and 𝑐(𝑢,𝑣;Δ). In the existing literature several researchers have been actively engaged in finding the spectrum and fine spectrum of different bounded linear operators over various sequence spaces. The spectrum of weighted mean operator has been studied by Rhoades [2], whereas that of the difference operator Δ over the sequence spaces 𝑝 for 0<𝑝<1 and 𝑐, 𝑐₀ has been studied by Altay and Başar [3, 4]. Kayaduman and Furkan [5] have determined the fine spectrum of the difference operator Δ over the sequence spaces 1 and 𝑏𝑣_𝑝 and on generalizing these results, Srivastava and Kumar [6, 7] have determined the fine spectrum of the operator Δ_𝜈 over the sequence spaces 1 and 𝑐₀, where (𝜈_𝑘) is a sequence of either constant or strictly deceasing sequence of reals satisfying certain conditions. Dutta and Baliarsingh [8–10] have computed the spectrum of the operator Δ^𝑟_𝜈 (𝑟∈ℕ₀) and Δ² over the sequence spaces 1, 𝑐₀, and 𝑐₀, respectively. The fine spectrum of the generalized difference operators 𝐵(𝑟,𝑠) and 𝐵(𝑟,𝑠,𝑡) over the sequence spaces 𝑝, 𝑏𝑣_𝑝 and 1, 𝑏𝑣 has been studied by Bilgiç and Furkan [11, 12], respectively. Recently, the spectrum of some particular limitation matrices in certain sequence spaces has been studied by [13–15] and many others.

Let 𝑋 and 𝑌 be Banach spaces and 𝑇∶𝑋→𝑌 be a bounded linear operator. By ℛ(𝑇), we denote the range of 𝑇; that is, ℛ(𝑇)=𝑦∈𝑌∶𝑦=𝑇𝑥;𝑥∈𝑋. (6)

By 𝐵(𝑋), we 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^∗𝜙)(𝑥)=𝜙(𝑇𝑥) for all 𝜙∈𝑋^∗ and 𝑥∈𝑋 with ‖𝑇‖=‖𝑇^∗‖.

Let 𝑋≠{𝟎} be a normed linear space over the complex field and 𝑇∶𝐷(𝑇)→𝑋 be a linear operator, where 𝐷(𝑇) denotes the domain of 𝑇. With 𝑇, for a complex number 𝜆, we associate an operator 𝑇_𝜆=(𝑇−𝜆𝐼), where 𝐼 is called identity operator on 𝐷(𝑇) and if 𝑇_𝜆 has an inverse, we denote it by 𝑇_𝜆⁻¹, that is, 𝑇_𝜆⁻¹=(𝑇−𝜆𝐼)⁻¹ (7) and is called the resolvent operator of 𝑇. Many properties of 𝑇_𝜆 and 𝑇_𝜆⁻¹ depend on 𝜆 and the spectral theory is concerned with those properties. We are interested in the set of all 𝜆’s in the complex plane such that 𝑇_𝜆⁻¹ exists/𝑇_𝜆⁻¹ is bounded/domain of 𝑇_𝜆⁻¹ is dense in 𝑋. Now, we state the following results which are essential for our investigation.

Definition 1 (see [16, page 371]). Let 𝑋≠{𝟎} be normed linear space over the complex field and 𝑇∶𝐷(𝑇)→𝑋 be a linear operator with domain 𝐷(𝑇)⊆𝑋. A regular value of 𝑇 is a complex number 𝜆 such that (R1)𝑇_𝜆⁻¹ exists;(R2)𝑇_𝜆⁻¹ is bounded;(R3)𝑇_𝜆⁻¹ is defined on a set which is dense in 𝑋.

The resolvent set 𝜌(𝑇,𝑋) of 𝑇 is the set of all regular values of 𝑇. Its complement 𝜎(𝑇,𝑋)=ℂ⧵𝜌(𝑇,𝑋) in the complex plane ℂ is called the spectrum of 𝑇. Furthermore, the spectrum 𝜌(𝑇,𝑋) is partitioned into three disjoint sets as follows.(I)Point Spectrum 𝜎_𝑝(𝑇,𝑋). It is the set of all 𝜆∈ℂ such that (R1) does not hold. The elements of 𝜎_𝑝(𝑇,𝑋) are called eigen values of 𝑇.(II)Continuous Spectrum 𝜎_𝑐(𝑇,𝑋). It is the set of all 𝜆∈ℂ such that (R1) holds and satisfies (R3) but does not satisfy (R2).(III)Residual Spectrum 𝜎_𝑟(𝑇,𝑋). It is the set of all 𝜆∈ℂ such that (R1) holds but does not satisfy (R3). Condition (R2) may or may not hold.

Goldberg’s Classification of the Operator 𝑇_𝜆 (see [17], page 58–71). Let 𝑋 be a Banach space and 𝑇_𝜆=(𝑇−𝜆𝐼)∈𝐵(𝑋), where 𝜆 is a complex number. Again, let 𝑅(𝑇_𝜆) and 𝑇_𝜆⁻¹ denote the range and inverse of the operator 𝑇_𝜆, respectively. Then the following possibilities may occur: (A)𝑅(𝑇_𝜆)=𝑋;(B)𝑅(𝑇_𝜆)≠𝑅(𝑇_𝜆)=𝑋; (C)𝑅(𝑇_𝜆)≠𝑋

and(1)𝑇_𝜆 is injective and 𝑇_𝜆⁻¹ is continuous;(2)𝑇_𝜆 is injective and 𝑇_𝜆⁻¹ is discontinuous;(3)𝑇_𝜆 is not injective.

Taking permutations (A), (B), (C) and (1), (2), (3), we get nine different states. These are labelled by 𝐴₁, 𝐴₂, 𝐴₃, 𝐵₁, 𝐵₂, 𝐵₃, 𝐶₁, 𝐶₂, and 𝐶₃. If 𝜆 is a complex number such that 𝑇_𝜆∈𝐴₁ or 𝑇_𝜆∈𝐵₁, then 𝜆 is in the resolvent set 𝜌(𝑇,𝑋) of 𝑇 on 𝑋. The other classifications give rise to the fine spectrum of 𝑇. We use 𝜆∈𝐵₂𝜎(𝑇,𝑋) which means the operator 𝑇_𝜆∈𝐵₂, that is, 𝑅(𝑇_𝜆)≠𝑅(𝑇_𝜆)=𝑋 and 𝑇_𝜆 is injective but 𝑇_𝜆⁻¹ is discontinuous; similarly are the others.

Lemma 2 (see [17, page 59]). A linear operator 𝑇 has a dense range if and only if the adjoint 𝑇^∗ is one to one.

Lemma 3 (see [17, page 60]). The adjoint operator 𝑇^∗ is onto if and and only if 𝑇 has a bounded inverse.

Let 𝑃, 𝑄 be two nonempty subsets of the space 𝑤 of all real or complex sequences and let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of complex numbers 𝑎_𝑛𝑘, where 𝑛,𝑘∈ℕ₀. For every 𝑥=(𝑥_𝑘)∈𝑃, we write 𝐴_𝑛(𝑥)=_𝑘𝑎_𝑛𝑘𝑥_𝑘(𝑛∈ℕ). (8) The sequence 𝐴𝑥=(𝐴_𝑛(𝑥)), if it exists, is called the transformation of 𝑥 by the matrix 𝐴. Infinite matrix 𝐴∈(𝑃,𝑄) if and only if 𝐴𝑥∈𝑄 whenever 𝑥∈𝑃.

Lemma 4 (see [18, page 126]). The matrix 𝐴=(𝑎_𝑛𝑘) gives rise to a bounded linear operator 𝑇∈𝐵(₁) from 1 to itself if and only if the supremum 1 norms of the columns of 𝐴 is bounded.

Lemma 5 (see [19], Theorem 2). If 𝑎_𝑘(0)≠0 for all 𝑘∈ℕ₀, then the inverse of the difference operator 𝐵(𝑎[𝑚]) is given by a lower triangular Toeplitz matrix 𝐶=(𝑐_𝑛𝑘) as follows:𝑐_𝑛𝑘=1𝑎_𝑛(0),(),()𝑘=𝑛−1^𝑛−𝑘_{𝑛𝑗=𝑘}𝑎_𝑗(0)×𝐷^(𝑘)_𝑛−𝑘(𝑎[𝑚,(),(),])0≤𝑘≤𝑛−10,𝑘>𝑛𝑛,𝑘∈ℕ₀, (9) where (𝐵(𝑎[𝑚])𝑥)_𝑘=𝑎_𝑘(0)𝑥_𝑘+𝑎_𝑘−1(1)𝑥_𝑘−1+⋯+𝑎_𝑘−𝑚(𝑚)𝑥_𝑘−𝑚, (𝑘∈ℕ₀) and𝐷_𝑛^(𝑘)(𝑎[𝑚=𝑎])^𝑘(1)𝑎_𝑘+1(0)𝑎0⋯00⋯0_𝑘(2)𝑎_𝑘+1(1)𝑎_𝑘+2(0)𝑎⋯00⋯0⋮⋮⋮⋮⋮⋮⋮⋮_𝑘(𝑚)𝑎_𝑘+1(𝑚−1)𝑎_𝑘+2(𝑚−2)⋯𝑎_𝑚−1(1)𝑎_𝑚(0)⋯00𝑎_𝑘+1(𝑚)𝑎_𝑘+2(𝑚−1)⋯𝑎_𝑚−1(2)𝑎_𝑚(1)⋯0⋮⋮⋮⋮⋮⋮⋱⋮00⋯𝑎_𝑛−𝑚(𝑚)⋯⋯⋯𝑎_𝑛−1(1). (10)

2. The Spectrum of the Operator 𝐺(𝑢,𝑣;Δ) over the Sequence 1

In this section, we compute the spectrum, the point spectrum, the continuous spectrum, and the residual spectrum of the difference matrix 𝐺(𝑢,𝑣;Δ) on the sequence space 1.

Theorem 6. The operator 𝐺(𝑢,𝑣;Δ)∶₁→₁ is a linear operator and ‖𝐺(𝑢,𝑣;Δ)‖_(1∶1)=sup_𝑛𝑢_𝑛^𝑛−1_𝑘=0𝑣_𝑘−𝑣_𝑘+1+𝑣_𝑛. (11)

Proof. Proof of this theorem follows from Lemma 4 and in particular cases. (i)If (𝑢_𝑘) is a strictly decreasing sequence and (𝑣_𝑘) is a constant sequence of positive reals (say 𝑣_𝑘=𝑣 for all 𝑘), then ‖𝐺(𝑢,𝑣;Δ)‖_(1∶1)=𝑢₀𝑣. (12)(ii)If (𝑢_𝑘) and (𝑣_𝑘) are strictly decreasing sequences of positive reals, then ‖𝐺(𝑢,𝑣;Δ)‖_(1∶1)=𝑢₀𝑣₀. (13)(iii)If (𝑢_𝑘) and (𝑣_𝑘) are constant sequences of positive reals, then ‖𝐺(𝑢,𝑣;Δ)‖_(1∶1)=𝑢𝑣. (14)(iv)If (𝑢_𝑘) is a constant sequence and (𝑣_𝑘) is a strictly decreasing sequence of positive reals, then ‖𝐺(𝑢,𝑣;Δ)‖_(1∶1)=𝑢𝑣₀. (15)

Theorem 7. The spectrum of 𝐺(𝑢,𝑣;Δ) over the sequence space 1 is given by 𝜎𝐺(𝑢,𝑣;Δ),₁=𝜆∈ℂ∶𝜆−𝑢_𝑘𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘≥1;𝑘∈ℕ₀. (16)

Proof. The proof is divided into four parts as follows.
Let (𝑢_𝑘) and (𝑣_𝑘) be two bounded sequences of positive reals satisfying (1) and (2) and 𝜆∈ℂ such that |(𝜆−𝑢_𝑘𝑣_𝑘+1)/(𝜆−𝑢_𝑘𝑣_𝑘)|<1 for all 𝑘∈ℕ₀; then (𝐺(𝑢,𝑣;Δ)−𝜆𝐼) is a triangle and hence has an inverse, that is, (𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹=1𝑢₀𝑣₀−𝑢−𝜆000⋯₁𝑣₀−𝑣_11𝑗=0𝑢_𝑗𝑣_𝑗1−𝜆𝑢₁𝑣₁𝑢−𝜆00⋯₂𝑣₀−𝑣₁𝜆−𝑢₁𝑣_22𝑗=0𝑢_𝑗𝑣_𝑗−𝑢−𝜆²𝑣₁−𝑣_22𝑗=1𝑢_𝑗𝑣_𝑗1−𝜆𝑢₂𝑣₂−𝑢−𝜆0⋯₃𝑣₀−𝑣₁𝜆−𝑢₁𝑣₂𝜆−𝑢₂𝑣_33𝑗=0𝑢_𝑗𝑣_𝑗𝑢−𝜆³𝑣₁−𝑣₂𝜆−𝑢₂𝑣_33𝑗=1𝑢_𝑗𝑣_𝑗−𝑢−𝜆³𝑣₂−𝑣_33𝑗=2𝑢_𝑗𝑣_𝑗1−𝜆𝑢₃𝑣₃⋯−𝜆⋮⋮⋮⋮⋱. (17)In general, by using Lemma 5, we observe that𝐷_𝑛^(𝑘)()=𝑢𝑢,𝑣,𝜆^𝑘+1𝑣_𝑘−𝑣_𝑘+1𝑢_𝑘+1𝑣_𝑘+1𝑢−𝜆0⋯0_𝑘+2𝑣_𝑘−𝑣_𝑘+1𝑢_𝑘+2𝑣_𝑘+1−𝑣_𝑘+2𝑢_𝑘+2𝑣_𝑘+2𝑢−𝜆⋯0_𝑘+3𝑣_𝑘−𝑣_𝑘+1𝑢_𝑘+3𝑣_𝑘+1−𝑣_𝑘+2𝑢_𝑘+3𝑣_𝑘+2−𝑣_𝑘+3𝑢⋯0⋮⋮⋮⋱⋮_𝑛𝑣_𝑘−𝑣_𝑘+1𝑢_𝑛𝑣_𝑘+1−𝑣_𝑘+2𝑢_𝑛𝑣_𝑘+2−𝑣_𝑘+3⋯𝑢_𝑛𝑣_𝑛−1−𝑣_𝑛=𝑣_𝑘−𝑣_𝑘+1𝑣_𝑘+1−𝑣_𝑘+2⋯𝑢_𝑛𝑣_𝑛−1−𝑣_𝑛×𝑢_𝑘+1−𝑢_𝑘+1𝑣_𝑘+1−𝜆𝑣_𝑘+1−𝑣_𝑘+2×𝑢_𝑘+2−𝑢_𝑘+2𝑣_𝑘+2−𝜆𝑣_𝑘+2−𝑣_𝑘+3×⋯×𝑢_𝑛−1−𝑢_𝑛−1𝑣_𝑛−1−𝜆𝑣_𝑛−1−𝑣_𝑛. (18)Therefore, for all 𝑛=𝑘, ((𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹)_𝑛,𝑘=1/(𝑢_𝑛𝑣_𝑛−𝜆), for all 𝑘>𝑛, ((𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹)_𝑛,𝑘=0, and for all 0≤𝑘<𝑛 one can deduce that (𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹_𝑛,𝑘=𝑣_𝑘−𝑣_𝑘+1𝑢_𝑛𝑢_𝑘𝑣_𝑘−𝜆𝑢_𝑛𝑣_𝑛−𝜆^𝑛−1_𝑗=𝑘+1𝜆−𝑢_𝑗𝑣_𝑗+1𝑢_𝑗𝑣_𝑗.−𝜆 (19) By using Lemma 4, we obtain that (𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹_(1∶1)=sup_𝑛^𝑛_𝑘=0(𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹_𝑛,𝑘≤sup_𝑛1𝑢_𝑛𝑣_𝑛+−𝜆^𝑛−1_𝑘=0𝑣_𝑘−𝑣_𝑘+1𝑢_𝑛𝑢_𝑘𝑣_𝑘−𝜆𝑢_𝑛𝑣_𝑛×−𝜆_{𝑛−1𝑗=𝑘+1}𝜆−𝑢_𝑗𝑣_𝑗+1𝑢_𝑗𝑣_𝑗−𝜆=sup_𝑛1𝑢_𝑛𝑣_𝑛−𝜆1+^𝑛−1_𝑘=0𝑣_𝑘−𝑣_𝑘+1𝑢_𝑛𝑢_𝑘𝑣_𝑘×−𝜆_{𝑛−1𝑗=𝑘+1}𝜆−𝑢_𝑗𝑣_𝑗+1𝑢_𝑗𝑣_𝑗−𝜆≤sup_𝑛𝑢_𝑛𝑢_𝑛𝑣_𝑛1−𝜆𝑢_𝑛+^𝑛−1_𝑘=0𝑣_𝑘−𝑣_𝑘+1𝑢_𝑘𝑣_𝑘−𝜆^𝑛−1_𝑗=𝑘+1𝑟_𝜆≤sup_𝑛𝑄_𝜆𝑢_𝑛𝑢_𝑛𝑣_𝑛1−𝜆𝑄_𝜆𝑢_𝑛+1𝑟_𝜆^𝑛−1_𝑘=0𝑟_𝜆^{𝑛−𝑘−1}=sup_𝑛𝑄_𝜆𝑢_𝑛𝑢_𝑛𝑣_𝑛1−𝜆𝑄_𝜆𝑢_𝑛+1𝑟_𝜆1−𝑟_𝜆^𝑛−11−𝑟_𝜆<∞. (20) For simplicity, we write 𝑄_𝜆=sup_𝑘((𝑣_𝑘−𝑣_𝑘+1)/(𝑢_𝑘𝑣_𝑘−𝜆)) and 𝑟_𝜆=sup_𝑗((𝜆−𝑢_𝑗𝑣_𝑗+1)/(𝑢_𝑗𝑣_𝑗−𝜆)). Since (𝑢_𝑘) and (𝑣_𝑘) are two bounded sequences of positive reals, the quantity 𝑄_𝜆 is finite. As per the assumption, 𝜆∈ℂ and |(𝜆−𝑢_𝑘𝑣_𝑘+1)/(𝜆−𝑢_𝑘𝑣_𝑘)|<1 for all 𝑘∈ℕ₀. Therefore, |𝑟_𝜆|<1. Now, we consider the four possible cases of the sequences (𝑢_𝑘) and (𝑣_𝑘).
Case  1. If (𝑢_𝑘) and (𝑣_𝑘) are strictly decreasing sequences of positive reals, then from the above relation we have (𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹∈(₁∶₁). Thus, 𝜎𝐺(𝑢,𝑣;Δ),₁⊆𝜆∈ℂ∶𝜆−𝑢_𝑘𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘≥1;𝑘∈ℕ₀. (21) Conversely, consider 𝜆≠𝑢_𝑘𝑣_𝑘 and |(𝜆−𝑢_𝑘𝑣_𝑘+1)/(𝜆−𝑢_𝑘𝑣_𝑘)|≥1 for all 𝑘∈ℕ₀, and clearly (𝐺(𝑢,𝑣;Δ)−𝜆𝐼) is a triangle and hence (𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹ exists, but ‖(𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹‖_(1∶1) is not finite. Again, for 𝜆=𝑢_𝑘𝑣_𝑘, the matrix (𝐺(𝑢,𝑣;Δ)−𝑢_𝑘𝑣_𝑘𝐼)(𝑘∈ℕ₀) is not invertible. Thus, 𝜆∈ℂ∶𝜆−𝑢_𝑘𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘≥1;𝑘∈ℕ₀⊆𝜎𝐺(𝑢,𝑣;Δ),₁. (22)
Combining (21) and (22) we complete the proof for this case.
Case  2. If (𝑢_𝑘) is a strictly decreasing sequence and (𝑣_𝑘) is a constant sequence of positive reals, then (𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹=1𝑢₀01𝑣−𝜆000⋯𝑢₁1𝑣−𝜆00⋯00𝑢₂1𝑣−𝜆0⋯000𝑢₃⋯.𝑣−𝜆⋮⋮⋮⋮⋱ (23) Since 𝑄=0, ‖(𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹‖_(1∶1)=sup_𝑛|1/(𝑢_𝑛𝑣−𝜆)|<∞ and the rest part of the proof is similar to that of Case 1.
Case  3. If (𝑢_𝑘) and (𝑣_𝑘) are constant sequences of positive reals, then the proof is similar to that of Case 2 and only the difference is ‖(𝐺(𝑢,𝑣;Δ)−𝜆𝐼)⁻¹‖_(1∶1)=|1/𝑢𝑣−𝜆|<∞.
Case  4. If (𝑢_𝑘) is constant sequence and (𝑣_𝑘) is strictly decreasing sequence of positive reals, then the proof is similar to that of Case 1.

Theorem 8. The point spectrum of the operator 𝐺(𝑢,𝑣;Δ) over 1 is given by 𝜎_𝑝𝐺(𝑢,𝑣;Δ),₁=𝑢_𝑘𝑣,𝑖𝑓𝑣_𝑘().𝑖𝑠𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒∅,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (24)

Proof. Suppose 𝑥∈₁ and consider the system of linear equations 𝐺(𝑢,𝑣;Δ)𝑥=𝜆𝑥 for 𝑥≠𝟎={0,0,0,…} in 1, 𝑢₀𝑣₀𝑥₀=𝜆𝑥₀𝑢₁𝑣₀−𝑣₁𝑥₀+𝑢₁𝑣₁𝑥₁=𝜆𝑥₁𝑢₂𝑣₀−𝑣₁𝑥₀+𝑢₂𝑣₁−𝑣₂𝑥₁+𝑢₂𝑣₂𝑥₂=𝜆𝑥₂⋮𝑢_𝑛𝑣₀−𝑣₁𝑥₀+𝑢_𝑛𝑣₁−𝑣₂𝑥₁+⋯+𝑢_𝑛𝑣_𝑛𝑥_𝑛=𝜆𝑥_𝑛⋮ (25) On solving the system of (25), we obtain that 𝜆=𝑢₀𝑣₀ whenever 𝑥₀≠0 and 𝑥₁=𝑢₁𝑣₀−𝑣₁𝑢₀𝑣₀−𝑢₁𝑣₁𝑥₀𝑥₂=𝑢₂𝑣₀−𝑣₁𝑢₀𝑣₀−𝑢₁𝑣₂𝑢₀𝑣₀−𝑢₁𝑣₁𝑢₀𝑣₀−𝑢₂𝑣₂𝑥₀andingeneral𝑥_𝑛=𝑢_𝑛𝑣₀−𝑣₁𝑢₀𝑣₀−𝑢_𝑛𝑣_𝑛^𝑛−1_𝑗=1𝑢₀𝑣₀−𝑢_𝑗𝑣_𝑗+1𝑢₀𝑣₀−𝑢_𝑗𝑣_𝑗𝑥₀. (26) Now, we have the following cases.
Case  1. If (𝑢_𝑘) and (𝑣_𝑘) are strictly decreasing sequences of positive reals, then we need to show that |(𝑢₀𝑣₀−𝑢_𝑗𝑣_𝑗+1)/(𝑢₀𝑣₀−𝑢_𝑗𝑣_𝑗)|≥1. Suppose, for the contrary, if |(𝑢₀𝑣₀−𝑢_𝑗𝑣_𝑗+1)/(𝑢₀𝑣₀−𝑢_𝑗𝑣_𝑗)|<1, then 𝑣_𝑗+1>𝑣_𝑗 which contradicts the fact that (𝑣_𝑘) is strictly decreasing sequence. Therefore, 𝑥∉₁ and hence 𝜎_𝑝(𝐺(𝑢,𝑣;Δ),₁)=∅.
Case  2. If (𝑢_𝑘) is a strictly decreasing and (𝑣_𝑘) is a constant sequence of positive reals, then for all 𝑘∈ℕ₀, 𝜆=𝑢_𝑘𝑣 is an eigen value corresponding to the eigen vector 𝑥={0,0,…,0,1,0,…} whose 𝑘th entry is 1. Thus, 𝜎_𝑝(𝐺(𝑢,𝑣;Δ),₁)=𝑢_𝑘𝑣.
Case  3. If (𝑢_𝑘) and (𝑣_𝑘) are constant sequences of positive reals, then the proof is similar to that of Case 2 and 𝜎_𝑝(𝐺(𝑢,𝑣;Δ),₁)=𝑢𝑣.
Case  4. If (𝑢_𝑘) is a constant and (𝑣_𝑘) is a strictly decreasing sequence of positive reals, then the proof is similar to that of Case 1.

Theorem 9. The point spectrum of the dual operator (𝐺(𝑢,𝑣;Δ))^∗ over the sequence space ^∗₁≊_∞ is given by𝜎_𝑝(𝐺(𝑢,𝑣;Δ))^∗,_∞=𝑢_𝑘𝑣,𝑖𝑓𝑣_𝑘𝑖𝑠𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝜆∈ℂ∶𝜆−𝑢_𝑘𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘>1;𝑘∈ℕ₀,(𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒). (27)

Proof. Consider (𝐺(𝑢,𝑣;Δ))^∗𝑓=𝜆𝑓 and 𝟎≠𝑓∈₁, and then the system of equations 𝑢₀𝑣₀𝑓₀+𝑣₀−𝑣₁𝑢₁𝑓₁+𝑣₀−𝑣₁𝑢₂𝑓₂+⋯=𝜆𝑓₀𝑢₁𝑣₁𝑓₁+𝑣₁−𝑣₂𝑢₂𝑓₂+𝑣₁−𝑣₂𝑢₃𝑓₃+⋯=𝜆𝑓₁𝑢₂𝑣₂𝑓₂+𝑣₂−𝑣₃𝑢₃𝑓₃+𝑣₂−𝑣₃𝑢₄𝑓₄+⋯=𝜆𝑓₂⋮𝑢_𝑘𝑣_𝑘𝑓_𝑘+𝑣_𝑘−𝑣_𝑘+1𝑢_𝑘+1𝑓_𝑘+1+𝑣_𝑘−𝑣_𝑘+1𝑢_𝑘+1𝑓_𝑘+1+⋯=𝜆𝑓_𝑘⋮ (28) From the system of linear equations (28), we obtain that 𝑓_𝑘−1=𝑣_𝑘−1−𝑣_𝑘𝜆−𝑢_𝑘−1𝑣_𝑘−1𝑢_𝑘𝑓_𝑘+𝑢_𝑘+1𝑓_𝑘+1+𝑢_𝑘+2𝑓_𝑘+2+⋯,𝑓_𝑘=𝑣_𝑘−𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘𝑢_𝑘+1𝑓_𝑘+1+𝑢_𝑘+2𝑓_𝑘+2+𝑢_𝑘+3𝑓_𝑘+3().+⋯𝑘≥1 (29)(30)
Combining (29) and (30), we have 𝑓_𝑘=𝑣_𝑘−𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘𝜆−𝑢_𝑘−1𝑣_𝑘−1𝑣_𝑘−1−𝑣_𝑘𝑓_𝑘−1−𝑢_𝑘𝑓_𝑘. (31) Therefore, by using ratio text, it is observed that lim_𝑘→∞𝑓_𝑘𝑓_𝑘−1=lim_𝑘→∞𝑣_𝑘−𝑣_𝑘+1𝜆−𝑢_𝑘−1𝑣_𝑘−1𝑣_𝑘−1−𝑣_𝑘𝜆−𝑢_𝑘𝑣_𝑘+1. (32) Now, we have the following cases.
Case  1. If (𝑢_𝑘) and (𝑣_𝑘) are strictly decreasing sequences of positive reals, then by using (2), we conclude that lim_𝑘→∞|𝑓_𝑘/𝑓_𝑘−1|<1 if and only if |(𝜆−𝑢_𝑘−1𝑣_𝑘−1)/(𝜆−𝑢_𝑘𝑣_𝑘+1)|<1 for all 𝑘≥1. Therefore, 𝜎_𝑝((𝐺(𝑢,𝑣;Δ))^∗,_∞)={𝜆∈ℂ∶|(𝜆−𝑢_𝑘𝑣_𝑘+1)/(𝜆−𝑢_𝑘𝑣_𝑘)|>1;(𝑘∈ℕ₀)}.
Case  2. If (𝑢_𝑘) is a strictly decreasing and (𝑣_𝑘) is a constant sequence of positive reals, then from the system of (28), we observe that for all 𝑘∈ℕ₀, 𝜆=𝑢_𝑘𝑣 is an eigen value corresponding to the eigen vector 𝑓={0,0,…,0,1,0,…} whose 𝑘th entry is 1. Thus, 𝜎_𝑝((𝐺(𝑢,𝑣;Δ))^∗,_∞)=𝑢_𝑘𝑣.
Case  3. If (𝑢_𝑘) and (𝑣_𝑘) are constant sequences of positive reals, then the proof is similar to that of Case 2.
Case  4. If (𝑢_𝑘) is a constant and (𝑣_𝑘) is a strictly decreasing sequence of positive reals, then the proof is similar to that of Case 1.

Theorem 10. The residual spectrum of the operator 𝐺(𝑢,𝑣;Δ) over the sequence space 1 is given by𝜎_𝑟𝐺(𝑢,𝑣;Δ),₁=∅,𝑖𝑓𝑣_𝑘𝑖𝑠𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝜆∈ℂ∶𝜆−𝑢_𝑘𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘>1;𝑘∈ℕ₀,(𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒). (33)

Proof. Proof follows from Lemma 2 and Theorems 8 and 9.

Theorem 11. The continuous spectrum of the operator 𝐺(𝑢,𝑣;Δ) over the sequence space 1 is given by𝜎_𝑐𝐺(𝑢,𝑣;Δ),₁=∅,𝑖𝑓𝑣_𝑘𝑖𝑠𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝜆∈ℂ∶𝜆−𝑢_𝑘𝑣_𝑘+1𝜆−𝑢_𝑘𝑣_𝑘=1;𝑘∈ℕ₀,(𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒). (34)

Proof. The proof of this theorem follows from Theorems 7, 8, and 10 and along with the fact that 𝜎𝐺(𝑢,𝑣;Δ),₁=𝜎_𝑝𝐺(𝑢,𝑣;Δ),₁∪𝜎_𝑟𝐺(𝑢,𝑣;Δ),₁∪𝜎_𝑐𝐺(𝑢,𝑣;Δ),₁. (35)

3. Conclusion

In this work, the authors have determined the spectrum of the generalized weighted mean difference operator 𝐺(𝑢,𝑣;Δ) over the Banach space 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.