On the Fine Spectrum of the Lower Triangular Matrix B(r; s) over the Hahn Sequence Space

Rituparna Das

doi:10.5666/KMJ.2017.57.3.441

Article

Kyungpook Mathematical Journal 2017; 57(3): 441-455

Published online September 23, 2017

On the Fine Spectrum of the Lower Triangular Matrix B(r; s) over the Hahn Sequence Space

Rituparna Das

Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim 737136, India

Received: January 6, 2017; Accepted: June 14, 2017

Abstract

Go to

Abstract
1. Introduction
2. Preliminaries and Background
3. Spectrum and Fine Spectrum of the Operator B(r, s) over the Sequence Space h
Tables
References

In this article we have determined the spectrum and fine spectrum of the lower triangular matrix B(r, s) on the Hahn sequence space h. We have also determined the approximate point spectrum, the defect spectrum and the compression spectrum of the operator B(r, s) on the sequence space h.

Keywords: spectrum of an operator, matrix mapping, sequence space

1. Introduction

Go to

Abstract
1. Introduction
2. Preliminaries and Background
3. Spectrum and Fine Spectrum of the Operator B(r, s) over the Sequence Space h
Tables
References

By w, we denote the space of all real or complex valued sequences. Throughout the paper c, c₀, bv, cs, bs, ℓ₁, ℓ_∞ represent the spaces of all convergent, null, bounded variation, convergent series, bounded series, absolutely summable and bounded sequences respectively. Also bv₀ denotes the sequence space bv ∩ c₀.

Fine spectra of various matrix operators on different sequence spaces have been examined by several authors. Fine spectrum of the operator Δ_a;b on the sequence space c was determined by Akhmedov and El-Shabrawy [1]. The fine spectra of the Cesàro operator C₁ over the sequence space bv_p, (1 ≤ p < ∞) was determined by Akhmedov and Başar [2]. Altay and Başar [3, 4] determined the fine spectrum of the difference operator Δ and the generalized difference operator B(r, s) on the sequence spaces c₀ and c. The spectrum and fine spectrum of the Zweier Matrix on the sequence spaces ℓ₁ and bv were studied by Altay and Karakuş [5]. Altun [6, 7] determined the fine spectra of triangular Toeplitz operators and tridiagonal symmetric matrices over some sequence spaces. Furkan, Bilgiç and Kayaduman [14] have determined the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces ℓ₁ and bv. Fine spectra of operator B(r, s, t) over the sequence spaces ℓ₁ and bv and generalized difference operator B(r, s) over the sequence spaces ℓ_p and bv_p, (1 ≤ p < ∞) were studied by Bilgiç and Furkan [11, 12]. Furkan, Bilgiç and Altay [15] have studied the fine spectrum of operator B(r, s, t) over the sequence spaces c₀ and c. Fine spectrum of the operator B(r, s, t) over the sequence spaces ℓ_p and bv_p, (1 ≤ p < ∞) were studied by Furkan, Bilgiç and Başar [16]. The spectrum of the operator D(r, 0, 0, s) over the sequence space bv₀ was investigated by Tripathy and Paul [30]. Tripathy and Paul [29, 31] also determined the spectrum of the operators D(r, 0, 0, s) and D(r, 0, s, 0, t) over the sequence spaces ℓ_p and bv_p, (1 ≤ p < ∞). Fine spectrum of the generalized difference operator Δ_v on the sequence space ℓ₁ was investigated by Srivastava and Kumar [26]. Panigrahi and Srivastava [23, 24] studied the spectrum and fine spectrum of the second order difference operator Δuv2 on the sequence space c₀ and generalized second order forward difference operator Δuvw2 on the sequence space ℓ₁. Fine spectra of upper triangular double-band matrix U(r, s) over the sequence spaces c₀ and c were studied by Karakaya and Altun [20]. Karaisa and Başar [19] have determined the spectrum and fine spectrum of the upper traiangular matrix A(r, s, t) over the sequence space ℓ_p, (0 < p < ∞). Dündar and Başar [13] have studied the fine spectrum of the linear operator Δ⁺ defined by an upper triangle double band matrix acting on the sequence space c₀ with respect to the Goldberg’s classification. Başar, Durna and Yildirim [9] subdivided the spectra for some generalized difference operators over certain sequence spaces. Başar [10] also determined the spectrum and fine spectrum of some particular limitation matrices over some sequence spaces. Tripathy and Das [27, 28] have studied the fine spectrum of the matrix operators B(r, 0, s) and U(r, s) over the sequence space cs. The fine spectrum of the forward difference operator on the Hahn sequence space h was determined by Yeşilkayagil and Kirişci [33].

The Hahn sequence space is defined as

h={x=(xn)∈w:∑k=1∞k∣Δxk∣<∞ and limk→∞xk=0},

where Δx_k = x_k −x_k₊₁, for all k ∈ ℕ. This space was defined and studied to some general properties by Hahn [18]. The norm ‖x‖h=∑k=1∞k∣Δxk∣+supk∣xk∣ on the space h was defined by Hahn [18]. Rao ( [25], Proposition 2.1) defined a new norm on h given by ‖x‖h=∑k=1∞k∣Δxk∣. Many other authors also investigated various properties of the Hahn sequence space.

In this paper, we shall determine the spectrum and fine spectrum of the lower triangular matrix B(r, s) on the Hahn sequence space h. Also we determine the approximate point spectrum, the defect spectrum and the compression spectrum of the operator B(r, s) on the sequence space h.

2. Preliminaries and Background

Go to

Abstract
1. Introduction
2. Preliminaries and Background
3. Spectrum and Fine Spectrum of the Operator B(r, s) over the Sequence Space h
Tables
References

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, i.e.

R(T)={y∈Y:y=Tx,x∈X}.

By B(X), we denote the set of all bounded linear operators on X into itself. If 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 complex normed linear space, where θ is the zero element and T : D(T) → X be a linear operator with domain D(T) ⊆ X. With T, we associate the operator

Tλ=T-λI,

where λ is a complex number and I is the identity operator on D(T). If T_λ has an inverse which is linear, we denote it by Tλ-1, that is Tλ-1=(T-λI)-1,

and call it the resolvent operator of T.

A regular valueλ of T is a complex number such that

(R1) Tλ-1 exists,
(R2) Tλ-1 is bounded
(R3) Tλ-1 is defined on a set which is dense in X i.e. R(Tλ)¯=X.

The resolvent set of T, denoted by ρ(T,X), is the set of all regular values λ of T. Its complement σ(T,X) = ℂρ(T,X) in the complex plane ℂ is called the spectrum of T. Furthermore, the spectrum σ(T,X) is partitioned into three disjoint sets as follows:

The point(discrete) spectrumσ_p(T,X) is the set of all λ ∈ ℂ such that Tλ-1 does not exist. Any such λ ∈ σ_p(T,X) is called an eigenvalue of T.

The continuous spectrumσ_c(T,X) is the set of all λ ∈ ℂ such that Tλ-1 exists and satisfies (R3), but not (R2), that is, Tλ-1 is unbounded.

The residual spectrumσ_r(T,X) is the set of all λ ∈ ℂ such that Tλ-1 exists (and may be bounded or not), but does not satisfy (R3), that is, the domain of Tλ-1 is not dense in X.

From Goldberg [17], if X is a Banach space and T ∈ B(X), then there are three possibilities for R(T) and T⁻¹:

R(T) = X,

R(T)≠R(T)¯=X

R(T)¯≠X

and

T⁻¹ exists and is continuous,
T⁻¹ exists but is discontinuous,
T⁻¹ does not exist.

If these possibilities are combined in all possible ways, nine different states are created which may be shown as in Table 1.

These are labeled by: I₁, I₂, I₃, II₁, II₂, II₃, III₁, III₂ and III₃. If λ is a complex number such that T_λ ∈ I₁ or T_λ ∈ I₂, then λ is in the resolvent set ρ(T,X) of T. The further classification gives rise to the fine spectrum of T. If an operator is in state II₂, then R(Tλ)≠R(Tλ)¯=X and Tλ-1 exists but is discontinuous and we write λ ∈ II₂σ(T,X). The state II₁ is impossible as if T_λis injective, then from Kreyszig [[22], Problem 6, p.290] Tλ-1 is bounded and hence continuous if and only if R(T_λ) is closed.

Again, following Appell et al. [8], 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 → ∞.

The approximate point spectrum of T, denoted by σ_ap(T,X), is defined as the set

(2.1)σap(T,X)={λ∈ℂ:there exists a Weyl sequence for T-λI}

The defect spectrum of T, denoted by σ_δ(T,X), is defined as the set

(2.2)σδ(T,X)={λ∈ℂ:T-λI is not surjective}

The two subspectra given by the relations (2.1) and (2.2) form a (not necessarily disjoint) subdivisions

(2.3)σ(T,X)=σap(T,X)∪σδ(T,X)

of the spectrum. There is another subspectrum

σco(T,X)={λ∈ℂ:R(T-λI)¯≠X}

which is often called the compression spectrum of T. The compression spectrum gives rise to another (not necessarily disjoint) decomposition

(2.4)σ(T,X)=σap(T,X)∪σco(T,X)

Clearly, σ_p(T,X) ⊆ σ_ap(T,X) and σ_co(T,X) ⊆ σ_δ(T,X). Moreover, it is easy to verify that

(2.5)σr(T,X)=σco(T,X)σp(T,X) and

(2.6)σc(T,X)=σ(T,X)[σp(T,X)∪σco(T,X)]

By the definitions given above, we can illustrate the subdivisions of spectrum of a bounded linear operator in Table 2.

Proposition 2.1. (Appell et al. [8], 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:

σ(T*,X*) = σ(T,X).
σ_c(T*,X*) ⊆ σ_ap(T,X).
σ_ap(T*,X*) = σ_δ(T,X).
σ_δ(T*,X*) = σ_ap(T,X).
σ_p(T*,X*) = σ_co(T,X).
σ_co(T*,X*) ⊇ σ_p(T,X).
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. Part (g) of Proposition 2.1 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 (Appell et al. [8]).

Let E and F be two sequence spaces and A = (a_nk) be an infinite matrix of real or complex numbers a_nk, where n, k ∈ ℕ. Then, we say that A defines a matrix mapping from E into F, and we denote it by A : E → F, if for every sequence x = (x_k) ∈ E, the sequence Ax = {(Ax)_n}, the A-transform of x, is in F, where

(2.7)(Ax)n=∑k=1∞ankxk, n∈ℕ.

By (E : F), we denote the class of all matrices such that A : E → F. Thus, A ∈ (E : F) if and only if the series on the right hand side of (2.7) converges for each n ∈ ℕ and every x ∈ E and we have Ax = {(Ax)_n}_n_∈ℕ ∈ F for all x ∈ E.

The matrix B(r, s) is an infinite lower triangular matrix of the form

B(r,s)=(r000⋯sr00⋯0sr0⋯00sr⋯⋮⋮⋮⋮⋱)

where s ≠ 0.

The following results will be used in order to establish the results of this article.

Lemma 2.1

Kirişci [21], Theorem 3.5) The matrix A = (a_nk) gives rise to a bounded linear operator T ∈ B(h) from h to itself if and only if:

∑n=1∞n∣(ank-an+1,k)∣converges, for each k,
supk1k∑n=1∞n|∑v=1k(anv-an+1,v)|<∞,
limn→∞ank=0, for each k.

Lemma 2.2. (Goldberg [17], Page 59)

T has a dense range if and only if T^*is one to one.

Lemma 2.3. (Goldberg [17], Page 60)

T has a bounded inverse if and only if T^*is onto.

3. Spectrum and Fine Spectrum of the Operator B(r, s) over the Sequence Space h

Go to

Abstract
1. Introduction
2. Preliminaries and Background
3. Spectrum and Fine Spectrum of the Operator B(r, s) over the Sequence Space h
Tables
References

Theorem 3.1

B(r, s) : h → h is a bounded linear operator and

‖B(r,s)‖(h:h)≤∣r∣+∣s∣.

Proof

From Lemma 2.1, B(r, s) : h → h is a bounded linear operator on h if

∑n=1∞n∣(ank-an+1,k)| converges, for each k,
supk1k∑n=1∞n|∑v=1k(anv-an+1,v)∣<∞,
limn→∞ank=0, for each k,

where

B(r,s)=(ank)=(r000⋯sr00⋯0sr0⋯00sr⋯⋮⋮⋮⋮⋱)

For each k, it is clear that limn→∞ank=0. Also for each k, ∑n=1∞n∣(ank-an+1,k)∣ is finite and so is convergent. It is easy to show that, for each k

1k∑n=1∞n|∑v=1k(anv-an+1,v)|≤∣r∣+(1+2k)∣s∣

and so

supk1k∑n=1∞n|∑v=1k(anv-an+1,v)|≤∣r∣+3∣s∣<∞.

Now,

‖B(r,s)(x)‖h=∑k=1∞k∣(sxk+rxk+1)-(sxk+1+rxk+2)∣=∑k=1∞k∣s(xk-xk+1)+r(xk+1-xk+2)∣≤∣s∣∑k=1∞k∣(xk-xk+1)∣+∣r∣∑k=1∞k∣(xk+1-xk+2)∣≤(∣s∣+∣r∣) ‖x‖h

and hence, || B(r, s) ||₍_h_:_h₎≤ |r| + |s|. Hence the result.

Theorem 3.2

The spectrum of the operator B(r, s) over h is given by

σ(B(r,s),h)={α∈ℂ:∣α-r∣≤∣s∣}.

Proof

We prove this theorem by showing that (B(r, s) − αI)⁻¹ exists and is in (h : h) for |α − r| > |s|, and then show that the operator B(r, s) − αI is not invertible for |α − r| ≤ |s|.

Let α be such that |α−r| > |s|. Since s ≠ 0 we have α ≠ r and so B(r, s)−αI is a triangle, therefore (B(r, s) − αI)⁻¹ exists. Let y = (y_n) ∈ h. On solving (B(r, s) − αI)x = y for x in terms of y we get

(B(r,s)-αI)-1=(bnk)=(1r-α000⋯-s(r-α)21r-α00⋯s2(r-α)3-s(r-α)21r-α0⋯-s3(r-α)4s2(r-α)3-s(r-α)21r-α⋯⋮⋮⋮⋮⋱)

From Lemma 2.1, (B(r, s) − αI)⁻¹ will be a bounded linear operator on h if

∑n=1∞n∣(bnk-bn+1,k)∣ converges, for each k,
supk1k∑n=1∞n|∑v=1k(bnv-bn+1,v)|<∞,
limn→∞bnk=0, for each k.

For each k, we get

bnk=(-s)n-k(r-α)n-k+1=1r-α(-sr-α)n-k

Since |α − r| > |s|, so for each k, limn→∞bnk=0. For each k, it is easy to show that

∑n=1∞n∣(bnk-bn+1,k)∣≤(2k-1)1∣r-α∣+(2k+1)∣s∣∣r-α∣2+(2k+3)∣s∣2∣r-α∣3+⋯

Now for a fixed k, considering 2k − 1 = a, from above we get

∑n=1∞n∣(bnk-bn+1,k)∣≤a1∣r-α∣+(a+2)∣s∣∣r-α∣2+(a+4)∣s∣2∣r-α∣3+⋯=a∣r-α∣(1+∣s∣∣r-α∣+∣s∣2∣r-α∣2+⋯)+2∣r-α∣(∣s∣∣r-α∣+2∣s∣2∣r-α∣2+3∣s∣3∣r-α∣3+⋯)

Since |α − r| > |s|, therefore the two series

1+∣s∣∣r-α∣+∣s∣2∣r-α∣2+⋯ and ∣s∣∣r-α∣+2∣s∣2∣r-α∣2+3∣s∣3∣r-α∣3+⋯

are convergent and converge to 11-∣s∣∣r-α∣ and ∣s∣∣r-α∣(1-∣s∣∣r-α∣)2 respectively. Therefore, ∑n=1∞n∣(bnk-bn+1,k)∣ converges, for each k. Also, for each k, it is easy to show that

1k∑n=1∞n|∑v=1k(bnv-bn+1,v)|≤1∣r-α∣+(1+2k)∣s∣∣r-α∣2+(1+4k)∣s∣2∣r-α∣3+⋯≤1∣r-α∣+3∣s∣∣r-α∣2+5∣s∣2∣r-α∣3+⋯

Since |α − r| > |s|, so by D’Alembert’s ratio test it is easy to show that the series 1∣r-α∣+3∣s∣∣r-α∣2+5∣s∣2∣r-α∣3+⋯ is convergent and therefore we have,

supk1k∑n=1∞n|∑v=1k(bnv-bn+1,v)|<∞.

So, by Lemma 2.1, (B(r, s) − αI)⁻¹ is in (h : h). This shows that σ(B(r, s), h) ⊆ {α ∈ ℂ : |α − r| ≤ |s|}.

Now, let α ∈ ℂ be such that |α − r| ≤ |s|. If α ≠ r, then B(r, s) − αI is a triangle and hence, (B(r, s) − αI)⁻¹ exists. Let y = (1, 0, 0, 0, …). Then y ∈ h. Now, (B(r, s) − αI)⁻¹y = x gives

xn=(-s)n-1(r-α)n.

Since |α−r| ≤ |s|, so the sequence (x_n) does not converge to 0 and so, x = (x_n)/∈ h. Therefore, (B(r, s)−αI)⁻¹ is not in (h : h) and so α ∈ σ(B(r, s), h). If α = r, then the operator B(r, s) − αI is represented by the matrix

B(r,s)-αI=(0000⋯s000⋯0s00⋯00s0⋯⋮⋮⋮⋮⋱)

Since, the range of B(r, s) − αI is not dense, so α ∈ σ(B(r, s), h). Hence,

{α∈ℂ:∣α-r∣≤∣s∣}⊆σ(B(r,s),h).

This completes the proof.

Theorem 3.3

The point spectrum of the operator B(r, s) over h is given by

σp(B(r,s),h)=∅.

Proof

Let α be an eigenvalue of the operator B(r, s). Then there exists x ≠ θ = (0, 0, 0, …) in h such that B(r, s)x = αx. Then, we have

rx1=αx1sx1+rx2=αx2sx2+rx3=αx3⋮sxn+rxn+1=αxn+1},

where n ≥ 1. If x_k is the first non-zero entry of the sequence (x_n), then α = r. Then from the relation sx_k + rx_k₊₁ = αx_k₊₁, we have sx_k = 0. But s ≠ 0 and hence, x_k = 0, a contradiction. Hence, σ_p(B(r, s), h) = ∅.

If T : h → h is a bounded linear operator represented by a matrix A, then it is known that the adjoint operator T*: h* → h* is defined by the transpose A^t of the matrix A. It should be noted that the dual space h* of h is isometrically isomorphic to the Banach space σ∞={x=(xk)∈w:supn1n|∑k=1nxk|<∞}.

Theorem 3.4

The point spectrum of the operator B(r, s)^*over h^*is given by

σp(B(r,s)*,h*≅σ∞)={α∈ℂ:∣α-r∣<∣s∣}.

Proof

Let α be an eigenvalue of the operator B(r, s)^*. Then there exists x ≠ θ = (0, 0, 0, …) in σ_∞ such that B(r, s)^*x = αx. Then, we have

B(r,s)tx=αx⇒rx1+sx2=αx1rx2+sx3=αx2⋮rxn+sxn+1=αxn},

where n ≥ 1. Solving, we get

xn=(α-rs)n-1x1, n≥1.

and so, sup supn1n|∑k=1nxk|<∞ if and only if |α − r| < |s|. Hence, σ_p(B(r, s)^*, h^*≅ σ_∞) = {α ∈ ℂ : |α − r| < |s|}.

Theorem 3.5

The residual spectrum of the operator B(r, s) over h is given by

σr(B(r,s),h)={α∈ℂ:∣α-r∣<∣s∣}.

Proof

From part (e) of Propostion 2.1 and relation (2.5), we get

σr(B(r,s),h)=σp(B(r,s)*,h*)σp(B(r,s),h).

So we get the required result by using Theorem 3.3 and Theorem 3.4.

Theorem 3.6

The continuous spectrum of the operator B(r, s) over h is given by

σc(B(r,s),h)={α∈ℂ:∣α-r∣=∣s∣}.

Proof

Since, σ(B(r, s), h) is the disjoint union of σ_p(B(r, s), h), σ_r(B(r, s), h) and σ_c(B(r, s), h), therefore, by Theorem 3.2, Theorem 3.3 and Theorem 3.5, we get σ_c(B(r, s), h) = {α ∈ ℂ : |α − r| = |s|}.

Theorem 3.7

Ifα = r, thenα ∈ III₁σ(B(r, s), h).

Proof

If α = r, the range of B(r, s) − αI is not dense. So, from Table 2 and Theorem 3.3, we have α ∈ σ_r(B(r, s), h). From Table 2,

σr(B(r,s),h)=III1σ(B(r,s),h)∪III2σ(B(r,s),h).

Therefore, α ∈ III₁σ(B(r, s), h) or α ∈ III₂σ(B(r, s), h). Also for α = r,

B(r,s)-αI=(0000⋯s000⋯0s00⋯00s0⋯⋮⋮⋮⋮⋱)

It is easy to show that the operator (B(r, s) − αI)^* : σ_∞ → σ_∞ is onto. So by Lemma 2.3 we get the operator B(r, s)−αI has a bounded inverse. This completes the theorem.

Theorem 3.8

Ifα ≠ r andα ∈ σ_r(B(r, s), h), thenα ∈ III₂σ(B(r, s), h).

Proof

Let α ≠ r. Since, α ∈ σ_r(B(r, s), h), therefore, from Table 2,

α∈III1σ(B(r,s),h) or α∈III2σ(B(r,s),h).

Now, α ∈ σ_r(B(r, s), h) implies that |α − r| < |s|. Therefore, for each k, the sequence (b_nk)_n in Theorem 3.2 does not converge to 0 as n → ∞ and hence, the operator B(r, s)−αI has no bounded inverse. Therefore, α ∈ III₂σ(B(r, s), h).

Theorem 3.9

Ifα ∈ σ_c(B(r, s), h), thenα ∈ II₂σ(B(r, s), h).

Proof

If α ∈ σ_c(B(r, s), h), then |α − r| = |s|. Therefore, for each k, the sequence (b_nk)_n in Theorem 3.2 does not converge to 0 as n → ∞ and hence, the operator B(r, s) − αI has no bounded inverse. So, α satisfies Goldberg’s condition 2.

Now we shall show that the operator B(r, s) − αI is not onto. Let y = (y_n) = (1, 0, 0, 0, …). Clearly, (y_n) ∈ h. Let x = (x_n) be a sequence such that (B(r, s) − αI)x = y. On solving, we get

xn=(-s)n-1(r-α)n.

Since, |α − r| = |s| so the sequence {x_n} does not converge to 0 as n → ∞ and so, x = (x_n)/∈ h. Hence the operator B(r, s) − αI is not onto. So, α satisfies Goldberg’s condition II. This completes the proof.

Theorem 3.10

The approximate point spectrum of the operator B(r, s) over h is given by

σap(B(r,s),h)={α∈ℂ:∣α-r∣≤∣s∣}{r}.

Proof

From Table 2,

σap(B(r,s),h)=σ(B(r,s),h)III1σ(B(r,s),h).

By Theorem 3.7, III₁σ(B(r, s), h) = {r}. This completes the proof.

Theorem 3.11

The compression spectrum of the operator B(r, s) over h is given by

σco(B(r,s),h)={α∈ℂ:∣α-r∣<∣s∣}.

Proof

From part (e) of Proposition 2.1, we get

σp(B(r,s)*,h*)=σco(B(r,s),h).

Using Theorem 3.4, we get the required result.

Theorem 3.12

The defect spectrum of the operator B(r, s) over h is given by

σδ(B(r,s),h)={α∈ℂ:∣α-r∣≤∣s∣}.

Proof

From Table 2, we have

σδ(B(r,s),h)=σ(B(r,s),h)I3σ(B(r,s),h).

Also,

σp(B(r,s),h)=I3σ(B(r,s),h)∪II3σ(B(r,s),h)∪III3σ(B(r,s),h).

By Theorem 3.3, we have σ_p(B(r, s), h) = ∅ and so I₃σ(B(r, s), h) = ∅. Hence, σ_δ(B(r, s), h) = {α ∈ ℂ : |α − r| ≤ |s|}.

Theorem 3.13

The following statements hold:

σ_ap(B(r, s)^*, h^* ≅ σ_∞) = {α ∈ ℂ : |α − r| ≤ |s|}.
σ_δ (B(r, s)^*, h^* ≅ σ_∞) = {α ∈ ℂ : |α − r| ≤ |s|}{r}.

Proof

From parts (c) and (d) of Proposition 2.1, we get

σap(B(r,s)*,h*≅σ∞)=σδ(B(r,s),h)

and

σδ(B(r,s)*,h*≅σ∞)=σap(B(r,s),h).

Using Theorem 3.10 and Theorem 3.12, we get the required results.

References

Go to

Abstract
1. Introduction
2. Preliminaries and Background
3. Spectrum and Fine Spectrum of the Operator B(r, s) over the Sequence Space h
Tables
References

Akhmedov, AM, and El-Shabrawy, SR (2011). On the fine spectrum of the operator Δ_a;b over the sequence space c. Comput Math Appl. 61, 2994-3002.
Akhmedov, AM, and Başar, F (2008). The fine spectra of the Cesàro operator C₁ over the sequence space bv_p (1 ≤ p < ∞). Math J Okayama Univ. 50, 135-147.
Altay, B, and Başar, F (2004). On the fine spectrum of the difference operator Δ on c₀ and c. Inform Sci. 168, 217-224.
Altay, B, and Başar, F (2005). On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c₀ and c. Int J Math Math Sci. 2005, 3005-3013.
Altay, B, and Karakuş, M (2005). On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces. Thai J Math. 3, 153-162.
Altun, M (2011). On the fine spectra of triangular Toeplitz operators. Appl Math Comput. 217, 8044-8051.
Altun, M (2011). Fine spectra of tridiagonal symmetric matrices. Int J Math Math Sci, 10.
Appell, J, Pascale, E, and Vignoli, A (2004). Nonlinear spectral theory. Berlin, New York: Walter de Gruyter
Başar, F, Durna, N, and Yildirim, M (2011). Subdivisions of the spectra for generalized difference operator over certain sequence spaces. Thai J Math. 9, 279-289.
Başar, F (2012). Summability theory and its applications. Istanbul: Bentham Science Publishers, e-books, Monographs
Bilgiç, H, and Furkan, H (2007). On the fine spectrum of operator B(r, s, t) over the sequence spaces ℓ₁ and bv. Math Comput Modelling. 45, 883-891.
Bilgiç, H, and Furkan, H (2008). On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces ℓ₁ and bv_p (1 ≤ p < ∞). Nonlinear Anal. 68, 499-506.
Dündar, E, and Başar, F (2013). On the fine spectrum of the upper triangle double band matrix Δ⁺ on the sequence space c₀. Math Commun. 18, 337-348.
Furkan, H, Bilgiç, H, and Kayaduman, K (2006). On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces ℓ₁ and bv. Hokkaido Math J. 35, 893-904.
Furkan, H, Bilgiç, H, and Altay, B (2007). On the fine spectrum of operator B(r, s, t) over c₀ and c. Comput Math Appl. 53, 989-998.
Furkan, H, Bilgiç, H, and Başar, F (2010). On the fine spectrum of operator B(r, s, t) over the sequence spaces ℓ_p and bv_p,(1 ≤ p < ∞). Comput Math Appl. 60, 2141-2152.
Goldberg, S (1966). Unbounded linear operators: theory and applications. New York-Tronto: McGraw-Hill Book co
Hahn, H (1922). Über folgen linearer operationen. Monatsh Math Phys. 32, 3-88.
Karaisa, A, and Başar, F (2013). Fine spectra of upper triangular triple-band matrices over the sequence space ℓ_p (0 < p < ∞). Abstr Appl Anal, 10.
Karakaya, V, and Altun, M (2010). Fine spectra of upper triangular double-band matrices. J Comput Appl Math. 234, 1387-1394.
Kirişci, M (2013). A survey on Hahn sequence space. Gen Math Notes. 19, 37-58.
Kreyszig, E (1989). Introductory functional analysis with application: John Wiley and Sons
Panigrahi, BL, and Srivastava, PD (2011). Spectrum and fine spectrum of generalised second order difference operator on sequence space c₀. Thai J Math. 9, 57-74.
Panigrahi, BL, and Srivastava, PD (2012). Spectrum and fine spectrum of generalized second order forward difference operator on sequence space ℓ₁. Demonstratio Math. 45, 593-609.
Rao, WC (1990). The Hahn sequence space. Bull Calcutta Math Soc. 82, 72-78.
Srivastava, PD, and Kumar, S (2010). Fine spectrum of the generalized difference operator Δ_v on sequence space ℓ₁. Thai J Math. 8, 221-233.
Tripathy, BC, and Das, R (2015). Spectrum and fine spectrum of the lower triangular matrix B(r, 0, s) over the sequence space cs. Appl Math Inf Sci. 9, 2139-2145.
Tripathy, BC, and Das, R (2015). Spectrum and fine spectrum of the upper triangular matrix U(r, s) over the sequence space cs. Proyecciones J Math. 34, 107-125.
Tripathy, BC, and Paul, A (2014). The spectrum of the operator D(r, 0, 0, s) over the sequence spaces ℓ_p and bv_p, Hacet. J Math Stat. 43, 425-434.
Tripathy, BC, and Paul, A (2015). The Spectrum of the operator D(r, 0, 0, s) over the sequence space bv₀. Georgian Math J. 22, 421-426.
Tripathy, BC, and Paul, A (2015). The spectrum of the operator D(r, 0, s, 0, t) over the sequence spaces ℓ_p and bv_p, (1 ≤ p < ∞). Afrika Mat. 26, 1137-1151.
Wilansky, A (1984). Summability through functional analysis: North Holland
Yeşilkayagil, M, and Kirişci, M (2016). On the fine spectrum of the forward difference operator on the Hahn space. Gen Math Notes. 33, 1-16.