The operator A(r̃,s̃) on sequence space on ℓp is defined A(r̃,s̃)x=(rkxk+skxk+1)k=0∞, where x=(xk)∈ℓp, 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(r̃,s̃) defined by a double sequential band matrix over the sequence space ℓp. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator A(r̃,s̃) over the space ℓp.
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,
(1.1)R(T)={y∈Y:y=Tx,x∈X}.
By B(X), we also denote the set of all bounded linear operators on X into itself. If X is any Banach space and T∈B(X), then the 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
(1.2)ρ(T):={λ∈ℂ:Tλ=λI-Tisabijection}
is called the resolvent set of T and its complement with respect to the complex plain
(1.3)σ(T):=ℂ∖ρ(T)
is called the spectrum of T. By the closed graph theorem, the inverse operator
(1.4)R(λ;T):=(λI-T)-1,(λ∈ρ(T))
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 Tλ-1 helps to solve the equation Tλx=y. Thus, x=Tλ-1y provided Tλ-1 exists. More important, the investigation of properties of Tλ-1 will be basic for an understanding of the operator T itself. Naturally, many properties of Tλ and Tλ-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 Tλ-1 exists. Boundedness of Tλ-1 is another property that will be essential. We will also ask for what λ's the domain of Tλ-1 is dense in X, to name just a few aspects. A regular value λ of T is a complex number such that Tλ-1 exists and bounded and whose domain is dense in X. For our investigation of T, Tλ, and Tλ-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 σp(T,X) is the set such that Tλ-1 does not exist. An λ∈σp(T,X) is called an eigenvalue of T.
The continuous spectrum σc(T,X) is the set such that Tλ-1 exists and is unbounded and the domain of Tλ-1 is dense in X.
The residual spectrum σr(T,X) is the set such that Tλ-1 exists (and may be bounded or not), but the domain of Tλ-1 is not dense in X.
Therefore, these three subspectra form a disjoint subdivisions
(2.1)σ(T,X)=σp(T,X)∪σc(T,X)∪σr(T,X).
To avoid trivial misunderstandings, let us say that some of the sets defined above, may be empty. This is an existence problem, which we will have to discuss. Indeed, it is well known that σc(T,X)=σr(T,X)=∅ and the spectrum σ(T,X) consists of only the set σp(T,X) in the finite-dimensional case.
2.2. The Approximate Point Spectrum, Defect Spectrum, and Compression Spectrum
In this subsection, following Appell et al. [2], we define 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 (xk) in X as a Weyl sequence for T if ∥xk∥=1 and ∥Txk∥→0, as k→∞.
In what follows, we call the set
(2.2)σap(T,X):={λ∈ℂ:thereexistsaWeylsequenceforλI-T}
the approximate point spectrum of T. Moreover, the subspectrum
(2.3)σδ(T,X):={λ∈ℂ:λI-Tisnotsurjective}
is called defect spectrum of T.
The two subspectra given by (2.2) and (2.3) form a (not necessarily disjoint) subdivision
(2.4)σ(T,X)=σap(T,X)∪σδ(T,X)
of the spectrum. There is another subspectrum
(2.5)σco(T,X)={λ∈ℂ:R(λI-T)¯≠X},
which is often called compression spectrum in the literature. The compression spectrum gives rise to another (not necessarily disjoint) decomposition
(2.6)σ(T,X)=σap(T,X)∪σco(T,X)
of the spectrum. Clearly, σp(T,X)⊆σap(T,X) and σco(T,X)⊆σδ(T,X). Moreover, comparing these subspectra with those in (2.1) we note that
(2.7)σr(T,X)=σco(T,X)∖σp(T,X),σc(T,X)=σ(T,X)∖[σp(T,X)∪σco(T,X)].
Sometimes it is useful to relate the spectrum of a bounded linear operator to that of its adjoint. Building on classical existence and uniqueness results for linear operator equations in Banach spaces and their adjoints is also useful.
Proposition 2.1 (see [<xref ref-type="bibr" rid="B9">2</xref>, Proposition <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M100"><mml:mrow><mml:mn mathvariant="normal">1.3</mml:mn></mml:mrow></mml:math></inline-formula>, 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),
σc(T*,X*)⊆σap(T,X),
σap(T*,X*)=σδ(T,X),
σδ(T*,X*)=σap(T,X),
σp(T*,X*)=σco(T,X),
σco(T*,X*)⊇σp(T,X),
σ(T,X)=σap(T,X)∪σp(T*,X*)=σp(T,X)∪σap(T*,X*).
The relations (c)–(f) show that the approximate point spectrum is in a certain sense dual to defect spectrum, and the point spectrum dual to the compression spectrum.
The equality (g) implies, in particular, that σ(T,X)=σap(T,X) if X is a Hilbert space and T is normal. Roughly speaking, this shows that normal (in particular, self-adjoint) operators on 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):
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. 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).
State diagram for B(X) and B(X*) for a nonreflective Banach space X.
If λ is a complex number such that Tλ=λI-T∈A1 or Tλ=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 Tλ=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.
Subdivisions of spectrum of a linear operator.
1
2
3
Tλ-1 exists and is bounded
Tλ-1 exists and is unbounded
Tλ-1 does not exist
A
R(λI-T)=X
λ∈ρ(T,X)
—
λ∈σp(T,X)
λ∈σap(T,X)
λ∈σc(T,X)
λ∈σp(T,X)
B
R(λI-T)¯=X
λ∈ρ(T,X)
λ∈σap(T,X)
λ∈σap(T,X)
λ∈σδ(T,X)
λ∈σδ(T,X)
λ∈σr(T,X)
λ∈σr(T,X)
λ∈σp(T,X)
C
R(λI-T)¯≠X
λ∈σδ(T,X)
λ∈σap(T,X)
λ∈σap(T,X)
λ∈σδ(T,X)
λ∈σδ(T,X)
λ∈σco(T,X)
λ∈σco(T,X)
λ∈σco(T,X)
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 Tλ-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=ℂℕ1 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∥∞=supk∈ℕ|xk| and ∥x∥bv=∑k=0∞|xk-xk+1|, while ϕ is not a Banach space with respect to any norm, respectively, where ℕ={0,1,2,…}. Also by ℓp, we denote the space of all p-absolutely summable sequences, which is a Banach space with the norm ∥x∥p=(∑k=0∞|xk|p)1/p, where 1⩽p<∞.
Let A=(ank) be an infinite matrix of complex numbers ank, where n,k∈ℕ, and write
(2.8)(Ax)n=∑kankxk(n∈ℕ,x∈D00(A)),
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 Dμ(A) for the x∈w for which the sum in (2.8) converges in the norm of μ. We write
(2.9)(λ:μ)={A:λ⊆Dμ(A)}
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 ℓp studied by González [19], where 1<p<∞. Also, weighted mean matrices of operators on ℓp 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, ℓp, 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 ℓp and bvp studied by Akhmedov and Başar [31, 32], where bvp 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 sr and c0, c, where sr denotes the Banach space of all sequences x=(xk) normed by ∥x∥sr=supk∈ℕ(|xk|/rk), (r>0). Altay and Karakuş [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 ℓp(α), where (αn) denotes the sequence of positive reals and ℓp(α) is the Banach space of all sequences x=(xn) normed by ∥x∥ℓp(α)=[∑n=1∞(|xn|/αn)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 ℓp and bvp 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 Δν on ℓ1, where Δν is defined by (Δν)nn=νn and (Δν)n+1,n=-νn for all n∈ℕ, under certain conditions on the sequence ν=(νn), and they have just generalized these results by the generalized difference operator Δuv defined by Δuvx=(unxn+vn-1xn-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 Δa,b, defined as a double band matrix with the convergent sequences a~=(ak) and b~=(bk) having certain properties, over the sequence space c. 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 ℓp and bvp with 1<p<∞ has recently been studied by Furkan et al. [14]. At this stage, Table 2 may be useful.
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 ℓp 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 ℓp. We quote some lemmas, which are needed in proving the theorems given in Section 3.
σ(A,λ)
σp(A,λ)
σc(A,λ)
σr(A,λ)
refer to
σ(C1p,c)
—
—
—
[4]
σ(W,c)
—
—
—
[5]
σ(C1,c0)
—
—
—
[6]
σ(C1,c0)
σp(C1,c0)
σc(C1,c0)
σr(C1,c0)
[7]
σ(C1,bv)
—
—
—
[8]
σ(R,c0)
σp(R,c0)
σc(R,c0)
σr(R,c0)
[9]
σ(R,c)
σp(R,c)
σc(R,c)
σr(R,c)
[9]
σ(C1p,c0)
—
—
—
[10]
σ(Δ,sr)
—
—
—
[11]
σ(Δ,c0)
—
—
—
[11]
σ(Δ,c)
—
—
—
[11]
σ(Δ(1),c)
σp(Δ(1),c)
σc(Δ(1),c)
σr(Δ(1),c)
[12]
σ(Δ(1),c0)
σp(Δ(1),c0)
σc(Δ(1),c0)
σr(Δ(1),c0)
[12]
σ(B(r,s),ℓp)
σp(B(r,s),ℓp)
σc(B(r,s),ℓp)
σr(B(r,s),ℓp)
[13]
σ(B(r,s),bvp)
σp(B(r,s),bvp)
σc(B(r,s),bvp)
σr(B(r,s),bvp)
[13]
σ(B(r,s,t),ℓp)
σp(B(r,s,t),ℓp)
σc(B(r,s,t),ℓp)
σr(B(r,s,t),ℓp)
[14]
σ(B(r,s,t),bvp)
σp(B(r,s,t),bvp)
σc(B(r,s,t),bvp)
σr(B(r,s,t),bvp)
[14]
σ(Δa,b,c)
σp(Δa,b,c)
σc(Δa,b,c)
σr(Δa,b,c)
[15]
σ(Δν,ℓ1)
σp(Δν,ℓ1)
σc(Δν,ℓ1)
σr(Δν,ℓ1)
[16]
σ(Δuv2,c0)
σp(Δuv2,c0)
σc(Δuv2,c0)
σr(Δuv2,c0)
[17]
σ(Δuv,ℓ1)
σp(Δuv,ℓ1)
σc(Δuv,ℓ1)
σr(Δuv,ℓ1)
[18]
Lemma 2.2 (see [<xref ref-type="bibr" rid="B15">41</xref>, p. 253, Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M365"><mml:mrow><mml:mn mathvariant="normal">34.16</mml:mn></mml:mrow></mml:math></inline-formula>]).
The matrix A=(ank) 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 [<xref ref-type="bibr" rid="B15">41</xref>, p. 245, Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M371"><mml:mrow><mml:mn mathvariant="normal">34.3</mml:mn></mml:mrow></mml:math></inline-formula>]).
The matrix A=(ank) 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 [<xref ref-type="bibr" rid="B15">41</xref>, p. 254, Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M377"><mml:mrow><mml:mn mathvariant="normal">34.18</mml:mn></mml:mrow></mml:math></inline-formula>]).
Let 1<p<∞ and A∈(ℓ∞:ℓ∞)∩(ℓ1:ℓ1). Then, A∈(ℓp:ℓp).
Let r~=(rk) and s~=(sk) be sequences whose entries either constants or distinct real numbers satisfying the following conditions:
(2.10)limk→∞rk=r>0,limk→∞sk=s;|s|=r,supk∈ℕ|rk|⩽r,sk2⩽rk2.
Then, we define the sequential generalized difference matrix A(r~,s~) by
(2.11)A(r~,s~)=[r0s000⋯0r1s10⋯00r2s2⋯000r3⋯⋮⋮⋮⋮⋱].
Therefore, we introduce the operator A(r~,s~) from ℓp to itself by
(2.12)A(r~,s~)x=(rkxk+skxk+1)k=0∞,wherex=(xk)∈ℓp.
3. Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M389"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>Theorem 3.1.
The operator A(r~,s~):ℓp→ℓp is a bounded linear operator and
(3.1)supk∈ℕ(|rk|p+|sk|p)1/p⩽∥A(r~,s~∥ℓp⩽supk∈ℕ|rk|+supk∈ℕ|sk|.
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 ℓp. It is trivial that A(r~,s~)e(k)=(0,0,…,sk-1,rk,0,…,0,…) for e(k)∈ℓp. Therefore, we have
(3.2)∥A(r~,s~)∥ℓp⩾∥A(r~,s~)e(k)∥ℓp∥e(k)∥ℓp=(|rk|p+|sk-1|p)1/p,
which implies that
(3.3)∥A(r~,s~)∥ℓp⩾supk∈ℕ(|rk|p+|sk|p)1/p.
Let x=(xk)∈ℓp, where p>1. Then, since (skxk+1),(rkxk)∈ℓp it is easy to see by Minkowski's inequality that
(3.4)∥A(r~,s~)x∥ℓp=(∑k=0∞|skxk+1+rkxk|p)1/p⩽(∑k=0∞|skxk+1|p)1/p+(∑k=0∞|rkxk|p)1/p⩽supk∈ℕ|rk|(∑k=0∞|xk|p)1/p+supk∈ℕ|sk|(∑k=0∞|xk+1|p)1/p=supk∈ℕ|rk|∥x∥ℓp+supk∈ℕ|sk|∥x∥ℓp=(supk∈ℕ|rk|+supk∈ℕ|sk|)∥x∥ℓp,
which leads us to the result that
(3.5)∥A(r~,s~)∥ℓp⩽supk∈ℕ|rk|+supk∈ℕ|sk|.
Therefore, by combining the inequalities in (3.3) and (3.5) we have (3.1), as desired.
Lemma 3.2 (see [<xref ref-type="bibr" rid="B43">42</xref>, p. 115, Lemma 3.1]).
Let 1<p<∞. If
(3.6)α∈{α∈ℂ:|r-α|=|s|},
then the series
(3.7)∑k=1∞|(rk-1-α)(rk-2-α)⋯(r1-α)(r0-α)sk-1sk-2⋯s1s0|p
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.
Let A(r~,s~)x=αx for θ≠x∈ℓp Then, by solving linear equation
(3.9)r0x0+s0x1=αx0,r1x1+s1x2=αx1,r2x2+s2x3=αx2,⋮rk-1xk-1+sk-1xk=αxk,⋮xk=((α-rk)/sk-1)xk-1 for all k⩾1 and
(3.10)xk=[(rk-1-α)(rk-2-α)⋯(r1-α)(r0-α)sk-1sk-2⋯s1s0]x0.Part 1. Assume that r~,s~∈𝒞. Let rk=r and sk=s For all k∈ℕ. We observe that xk=((α-r)/s)kx0. This shows that x∈ℓp if and if only |α-r|<|s|, as asserted.
Part 2. Assume that r~,s~∈𝒮𝒟. We must take x0≠0, since x≠0. It is clear that, for all k∈ℕ, the vector x=(x0,x1,…,xk,0,0,…) is an eigenvector of the operator A(r~,s~) corresponding to the eigenvalue α=rk, where x0≠0 and xn=((α-rn)/sn-1)xn-1, for 1⩽n⩽k. Thus {rk:k∈ℕ}⊆σp(A(r~,s~),ℓp). If rk≠α, for all k∈ℕ, then xk≠0. If we take |α-r|<|s|, since limk→∞|xk+1/xk|p=limk→∞|(rk-α)/sk|p=|(r-α)/s|p<1, x∈ℓp. Hence {α∈ℂ:|r-α|<|s|}⊆σp(A(r~,s~),ℓp). Conversely, let α∈σp(A(r~,s~),ℓp). Then, there exists x=(x0,x1,x2,…) in ℓp and we have xk=((α-rk)/sk-1)xk-1, for all k⩾1. Since x∈ℓp, we can use ratio test. And so limk→∞|xk+1/xk|p=limk→∞|(rk-α)/sk|p=|(r-α)/s|p<1 or α∈{rk:k∈ℂ}. If |α-r|=|s|, by Lemma 3.2x∉ℓp. This completes the proof.
Part 1. Assume that r~,s~∈𝒞. Consider A(r~,s~)*f=αf for f≠θ=(0,0,0,…) in ℓp*=ℓq. Then, by solving the system of linear equations
(3.12)r0f0=αf0,s0f0+r1f1=αf1,s1f1+r2f2=αf2,⋮sk-1fk-1+rkfk=αfk,⋮
we find that f0=0 if α≠r=rk and f1=f2=⋯=0 if f0=0, which contradicts f≠θ. If fn0 is the first nonzero entry of the sequence f=(fn) and α=r, then we get sn0fn0+rfn0+1=αfn0+1 that implies fn0=0, which contradicts the assumption fn0≠0. Hence, the equation A(r~,s~)*f=αf has no solution f≠θ.
Part 2. Assume that r~,s~∈𝒮𝒟. Then, by solving the equation A(r~,s~)*f=αf for f≠θ=(0,0,0,…) in ℓq, we obtain (r0-α)f0=0 and (rk+1-α)fk+1+skfk=0 for all k∈ℕ. Hence, for all α∉{rk:k∈ℕ}, we have fk=0 for all k∈ℕ, which contradicts our assumption. So, α∉σp(A(r~,s~)*,ℓq). This shows that σp(A(r~,s~)*,ℓq)⊆{rk:k∈ℕ}∖{r}. Now, we prove that
(3.13)α∈σp(A(r~,s~)*,ℓq)iffα∈ℬ.
If α∈σp(A(r~,s~)*,ℓq), then, by solving the equation A(r~,s~)*f=αf for f≠θ=(0,0,0,…) in ℓq with α=r0,
(3.14)fk=s0s1s2⋯sk-1(r0-rk)(r0-rk-1)(r0-rk-2)⋯(r0-r1)f0∀k⩾1,
which can expressed by the recursion relation
(3.15)|fk|=|s0s1s2⋯sk-1(r0-r1)(r0-r2)⋯(r0-rk)||f0|.
Using ratio test,
(3.16)limk→∞|fkfk-1|q=limk→∞|sk-1rk-r0|q=|sr-r0|q⩽1.
But |s/(r-r0)|≠1. Hence,
(3.17)α=r0∈{rk:k∈ℕ,|rk-r|>|s|}=ℬ.
If we choose α=rk≠r for all k∈ℕ1, then we get f0=f1=f2=⋯=fk-1=0 and
(3.18)fn+1=snsn-1sn-2⋯sk(rk-rn+1)(rk-rn)(rk-rn-1)⋯(rk-rk+1)fk∀n⩾k,
which can expressed by the recursion relation
(3.19)|fn+1|=|sn-1sn-2sn-2⋯sk(rk-rn+1)(rk-rn-1)(rk-rn-2)⋯(rk-rk+1)||fk|.
Using ratio test,
(3.20)limn→∞|fn+1fn|q=limn→∞|snrn+1-rk|q=|sr-rk|q⩽1.
But |s/(r-rk)|≠1. So we have
(3.21)α=rk∈{rk:k∈ℕ,|rk-r|>|s|}=ℬ.Hence,σp(A(r~,s~)*,ℓq)⊆ℬ.
Conversely, let α∈ℬ. Then exist k∈ℕ, α=rk≠r, and
(3.22)limn→∞|fnfn-1|q=limn→∞|snrn+1-rk|q=|sr-rk|q<1.
That is, f∈ℓq. So we have ℬ⊆σp(A(r~,s~)*,ℓq). This completes the proof.
Lemma 3.5 (see [<xref ref-type="bibr" rid="B21">3</xref>, p. 60]).
The adjoint operator T* of T is onto if and only if T is a bounded operator.
Theorem 3.6.
σr(A(r~,s~),ℓp)=σp(A(r~,s~)*,ℓp*)∖σp(A(r~,s~),ℓp)
Proof.
The proof is obvious so is omitted.
Theorem 3.7.
Let (rk),(sk) in 𝒮𝒟 and 𝒞. σr(A(r~,s~),ℓp)=∅.
Proof.
By Theorems 3.4 and 3.6, σr(A(r~,s~),ℓp)=∅.
Theorem 3.8.
Let 𝒜={α∈ℂ:|r-α|⩽|s|} and ℬ={rk:k∈ℕ,|r-rk|>|s|}. Then, the set ℬ is finite and σ(A(r~,s~),ℓp)=𝒜∪ℬ.
Proof.
We will show that Aα(r~,s~)* is onto, for |r-α|>|s|. Thus, for every y∈ℓq, we find x∈ℓq. Aα(r~,s~)* is triangle so it has an inverse. Also equation Aα(r~,s~)*x=y gives [Aα(r~,s~)*]-1y=x. It is sufficient to show that [Aα(r~,s~)*]-1∈(ℓq:ℓq). We can calculate that A=(ank)=[Aα(r~,s~)*]-1 as follows:
(3.23)(ank)=[1r0-α00⋯-s0(r1-α)(r0-α)1r1-α0⋯s0s1(r0-α)(r1-α)(r2-α)-s1(r2-α)(r1-α)1r2-α⋯⋮⋮⋮⋱].
Therefore, the supremum of the ℓ1 norms of the rows of [Aα(r~,s~)*]-1 is Sk, where
(3.24)Sk=|1rk-α|+|sk-1(rk-1-α)(rk-α)|+|sk-1sk-2(rk-2-α)(rk-1-α)(rk-α)|+⋯+|s0s1⋯sk-1(r0-α)(r1-α)⋯(rk-α)|.
Now, we prove that (Sk)∈ℓ∞. Since limk→∞|sk/(rk-α)|=|s/(r-α)|=p<1, then there exists k0∈ℕ such that |sk/(rk-α)|<p0 with p0<1, for all k⩾k0+1,
(3.25)Sk=1|rk-α|[1+|sk-1rk-1-α|+|sk-1sk-2(rk-1-α)(rk-2-α)|+⋯+|sk-1sk-2⋯sk0+1sk0⋯s0(rk-1-α)(rk-2-α)⋯(rk0+1-α)(rk0-α)⋯(r0-α)|]⩽1|rk-α|[1+p0+p02+⋯+p0k-k0+p0k-k0|sk0-1||rk0-1-α|+⋯+p0k-k0|sk0-1sk0-2⋯s0(rk0-1-α)(rk0-2-α)⋯(r0-α)|].
Therefore,
(3.26)Sk⩽1|rk-α|(1+p0+p02+⋯p0k-k0+p0k-k0Mk0),
where
(3.27)Mk0=1+|sk0-1rk0-1-α|+|sk0-1sk0-2(rk0-1-α)(rk0-2-α)|+⋯+|sk0-1sk0-2⋯s0(rk0-1-α)(rk0-2-α)⋯(r0-α)|.
Then, Mk0⩾1 and so
(3.28)Sk⩽Mk0|rk-α|(1+p0+p02+⋯+p0k-k0).
But there exist k1∈ℕ and a real number p1 such that 1/|rk-α|<p1 for all k⩾k1. Then, Sk⩽(Mp1k0)/(1-p0) for all k>max{k0,k1}. Hence, supk∈ℕSk<∞. 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
(3.29)[(A(r~,s~)-αI)*]-1∈(ℓq:ℓq)forα∈ℂwith|r-α|>|s|.
Hence, Aα(r~,s~)* is onto. By Lemma 3.5, Aα(r~,s~) is bounded inverse. This means that
(3.30)σc(A(r~,s~),ℓp)⊆{α∈ℂ:|r-α|⩽|s|}.
Combining this with Theorem 3.3 and Theorem 3.7, we get
(3.31)σ(A(r~,s~),ℓp)⊆{α∈ℂ:|r-α|⩽|s|}∪ℬ
and again from Theorem 3.3{α∈ℂ:|r-α|<|s|}⊆σ(A(r~,s~),ℓp) and ℬ⊆σ(A(r~,s~),ℓp). Since the spectrum of any bounded operator is closed, we have
(3.32){α∈ℂ:|r-α|⩽|s|}∪ℬ⊆σ(A(r~,s~),ℓp).
Combining (3.31) and (3.32), we get
(3.33)σ(A(r~,s~),ℓp)=𝒜∪ℬ.
Theorem 3.9.
Let (rk),(sk) in 𝒮𝒟 or 𝒞. σc(A(r~,s~),ℓp)={α∈ℂ:|r-α|=|s|}.
Proof.
The proof follows of immediately from Theorems 3.3, 3.7, and 3.8 because the parts σc(A(r~,s~),ℓp), σr(A(r~,s~),ℓp), and σp(A(r~,s~),ℓp) are pairwise disjoint sets and union of these sets is σ(A(r~,s~),ℓp).
Theorem 3.10.
Let (rk),(sk)∈𝒮𝒟 and 𝒞. If |α-r|<|s|, α∈σ(A(r~,s~),ℓp)A3.
Proof.
From Theorem 3.3, α∈σp(A(r~,s~),ℓp). 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=(yk)∈ℓp, we have to find x=(xk)∈ℓp such that (A(r~,s~)-αI)x=y. Solving the linear equation (A(r~,s~)-αI)x=y,
(3.34)[A(r~,s~)-αI]x=[r0-αs000⋯0r1-αs10⋯00r2-αs2⋯000r3-α⋯⋮⋮⋮⋮⋱][x0x1x2⋮]=[y0y1y2⋮],
let
(3.35)x0=0,x1=y0s0,x2=(α-r1)y0s1s0+y1s1,⋮xk=(α-r1)(α-r2)⋯(α-rk-1)y0s0s1⋯sk-1+⋯+(rk-2-α)yk-2sk-1sk-2+yk-1sk-1.
Then, ∑k|xk|p⩽supk(Rk)p∑k|yk|p, where
(3.36)Rk=|1sk|+|(rk+1-α)sksk+1|+|(rk+1-α)(rk+2-α)sksk+1sk+2|+⋯,Rkn=|1sk|+|(rk+1-α)sksk+1|+|(rk+1-α)(rk+2-α)sksk+1sk+2|+⋯,+|(rk+1-α)(rk+2-α)⋯(rk+n-α)sksk+1⋯sn+k|
for all k,n∈ℕ. Then, since
(3.37)Rn=limk→∞Rnk=|1s|+|(r-α)s2|+|(r-α)2s3|+⋯+|(r-α)n+1sn+2|,
we have
(3.38)R=limn→∞Rn=|1s|(1+|r-αs|1+⋯)<∞.
Since |r-α|<|s|, (Rk) is a convergent sequence of positive real numbers with limit R. Hence, (Rk) bounded and we have supk(Rk)p<∞. Therefore,
(3.39)∑k|xk|p⩽supk(Rk)p∑k|yk|p<∞.
This shows that x=(xk)∈ℓp. Thus (A(r~,s~)-αI) is onto. So we have α∈σ(A(r~,s~),ℓp)A3.
Theorem 3.11.
Let (rk),(sk)∈𝒞 with rk=r, sk=s for all k∈ℕ. Then, the following statements hold:
σap(A(r~,s~),ℓp)=σ(A(r~,s~),ℓp),
σδ(A(r~,s~),ℓp)={α∈ℂ:|r-α|=|s|},
σco(A(r~,s~),ℓp)=∅.
Proof.
(i) Since from Table 1,
(3.40)σap(A(r~,s~),ℓp)=σ(A(r~,s~),ℓ1)∖σ(A(r~,s~),ℓp)C1,
we have by Theorem 3.7(3.41)σ(A(r~,s~),ℓp)C1=σ(A(r~,s~),ℓp)C2=∅.
Hence,
(3.42)σap(A(r~,s~),ℓp)=𝒜.
(ii) Since the following equality:
(3.43)σδ(A(r~,s~),ℓp)=σ(A(r~,s~),ℓp)∖σ(A(r~,s~),ℓp)A3
holds from Table 1, we derive by Theorems 3.8 and 3.10 that σδ(A(r~,s~),ℓp)={α∈ℂ:|r-α|=|s|}.
(iii) From Table 1, we have
(3.44)σco(A(r~,s~),ℓp)=σ(A(r~,s~),ℓp)C1∪σ(A(r~,s~),ℓp)C2∪σ(A(r~,s~),c0)C3.
By Theorem 3.4, it is immediate that σco(A(r~,s~),ℓp)=∅.
Theorem 3.12.
Let (rk)∈𝒮𝒟. Then
(3.45)σap(A(r~,s~),ℓp)=𝒜∪ℬ,σδ(A(r~,s~),ℓp)={α∈ℂ:|r-α|=|s|}∪ℬ,σco(A(r~,s~),ℓp)=ℬ.
Proof.
We have by Theorem 3.4 and Part (e) of Proposition 2.1 that
(3.46)σp(A(r~,s~)*,ℓp*)=σco(A(r~,s~),ℓp)=ℬ.
By Theorems 3.7 and 3.4, we must have
(3.47)σ(A(r~,s~),ℓp)C1=σ(A(r~,s~),ℓp)C2=∅.
Hence, σ(A(r~,s~),ℓp)C3={rk}. Additionally, since σ(A(r~,s~),ℓp)C1=∅.
Therefore, we derive from Table 1, Theorems 3.8, and 3.10 that
(3.48)σap(A(r~,s~),ℓp)=σ(A(r~,s~),ℓp)∖σ(A(r~,s~),ℓp)C1=σ(A(r~,s~),ℓ1),σδ(A(r~,s~),ℓp)=σ(A(r~,s~),ℓp)∖σ(A(r~,s~),ℓp)A3={α∈ℂ:|r-α|=|s|}∪ℬ.
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 ℓp. 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 ℓp, 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.
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