Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space l p ( 0 < p < ∞ )

and Applied Analysis 3 mapping from μ into γ and we denote it by writing A : μ → γ if for every sequence x = (xk) ∈ μ the sequence Ax = {(Ax)n}, the A-transform of x is in γ, where (Ax)n = ∑ k ankxk for each n ∈ N. (7) By (μ : γ), we denote the class of all matrices A such that A : μ → γ. Thus, A ∈ (μ : γ) if and only if the series on the right side of (7) converges for each n ∈ N and every x ∈ μ, and we have Ax = {(Ax)n}n∈N ∈ γ for all x ∈ μ. Proposition 1 (see [15, Proposition 1.3, p. 28]). Spectra and subspectra of an operatorT ∈ B(X) and its adjointT ∈ B(X) are related by the following relations: (a) σ(T, X) = σ(T,X), (b) σc(T ∗ , X ∗ ) ⊆ σap(T,X), (c) σap(T ∗ , X ∗ ) = σδ(T,X), (d) σδ(T ∗ , X ∗ ) = σap(T,X), (e) σp(T ∗ , X ∗ ) = σco(T,X), (f) σco(T ∗ , X ∗ ) ⊇ σp(T,X), (g) 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 onHilbert spaces aremost similar tomatrices in finite dimensional spaces (see [15]). Lemma 2 (see [16, p. 60]). The adjoint operator T of T is onto if and only if T has a bounded inverse. Lemma 3 (see [16, p. 59]). T has a dense range if and only if T ∗ is one to one. Our main focus in this paper is on the triple-band matrix A(r, s, t), where A (r, s, t) = [ [ [ [ [ [ [ r s t 0 . . . 0 r s t . . . 0 0 r s . . . 0 0 0 r . . . .. .. .. .. . . . ] ] ] ] ] ]


Introduction
In functional analysis, the spectrum of an operator generalizes the notion of eigenvalues for matrices.The spectrum of an operator over a Banach space is partitioned into three parts, which are the point spectrum, the continuous spectrum, and the residual spectrum.The calculation of these three parts of the spectrum of an operator is called calculating the fine spectrum of the operator.
Over the years and different names the spectrum and fine spectra of linear operators defined by some triangle matrices over certain sequence spaces were studied.
Several authors studied the spectrum and fine spectrum of linear operators defined by some triangle matrices over some sequence spaces.We introduce knowledge in the existing literature concerning the spectrum and the fine spectrum.Cesàro operator of order one on the sequence space ℓ  was studied by Gonzàlez [1], where 1 <  < ∞.Also, weighted mean matrices of operators on ℓ  have been investigated by Cartlidge [2].The spectrum of the Cesàro operator of order one on the sequence spaces  0 and  were investigated by Okutoyi [3,4].The spectrum and fine spectrum of the Rally operators on the sequence space ℓ  were examined by Yıldırım [5].The fine spectrum of the difference operator Δ over the sequence spaces  0 and  was studied by Altay and Bas ¸ar [6].The same authors also worked out the fine spectrum of the generalized difference operator (, ) over  0 and , in [7].Recently, the fine spectra of the difference operator Δ over the sequence spaces  0 and  have been studied by Akhmedov and Bas ¸ar [8,9], where   is the space consisting of the sequences  = (  ) such that  = (  −  −1 ) ∈ ℓ  and introduced by Bas ¸ar and Altay [10] with 1 ⩽  ⩽ ∞.In the recent paper, Furkan et al. [11] have studied the fine spectrum of (, , ) over the sequence spaces ℓ  and   with 1 <  < ∞, where (, , ) is a lower triangular tripleband matrix.Later, Karakaya and Altun have determined the fine spectra of upper triangular double-band matrices over the sequence spaces  0 and , in [12].Quite recently, Karaisa [13] has determined the fine spectrum of the generalized difference operator (r, s), defined as an upper triangular double-band matrix with the convergent sequences r = (  ) and s = (  ) having certain properties, over the sequence space ℓ  , where 1 <  < ∞.
In this paper, we study the fine spectrum of the generalized difference operator (, , ) defined by a triple sequential band matrix acting on the sequence space ℓ  (0 <  < ∞), with respect to Goldberg's classification.Additionally, we give the approximate point spectrum and defect spectrum and give some applications.

Preliminaries, Background, and Notation
Let  and  be two Banach spaces and  :  →  be a bounded linear operator.By () we denote range of , that is, By () we also denote the set of all bounded linear operators on  into itself.If  ∈ () then the adjoint  * of  is a bounded linear operator on the dual  * of  defined by ( * )() = () for all  ∈  * and  ∈ .
Let  ̸ = {} be a complex normed space and  : () →  be a linear operator with domain () ⊆ .With  we associate the operator   =  − , where  is a complex number and  is the identity operator on ().If   has an inverse which is linear, we denote it by  −1  , that is, and call it the resolvent operator of .Many properties of   and  −1  depend on , and spectral theory is concerned with those properties.For instance, we shall be interested in the set of all  in the complex plane such that  −1  exists.The boundedness of  −1  is another property that will be essential.We shall also ask for what  the domain of  −1  is dense in , to name just a few aspects For our investigation of ,   , and  −1  , we need some basic concepts in spectral theory which are given as follows (see [14, pp. 370-371]).
Let  ̸ = {} be a complex normed space and  : () →  be a linear operator with domain () ⊆ .A regular value  of  is a complex number such that is defined on a set which is dense in .The resolvent set () of  is the set of all regular values  of .Its complement C \ () in the complex plane C is called the spectrum of .Furthermore, the spectrum () is partitioned into three disjoint sets as follows.The point spectrum   () is the set such that  −1  does not exist. ∈   () is called an eigenvalue of .The continuous spectrum   () is the set such that  −1  exists and satisfies (R3) but not (R2).The residual spectrum   () is the set such that  −1  exists but does not satisfy (R3).
In this section, following Appell et al. [15], we define the three more subdivisions of the spectrum called the approximate point spectrum, defect spectrum, and compression spectrum.
Given a bounded linear operator  in a Banach space , we call a sequence (  ) in  as a Weyl sequence for  if ‖  ‖ = 1 and ‖  ‖ → 0, as  → ∞.
In what follows, we call the set  ap (, ) := { ∈ C : there exists a Weyl sequence for −} the approximate point spectrum of .Moreover, the subspectrum is called defect spectrum of .
The two subspectra given by ( 3) and (4) form a (not necessarily disjoint) subdivisions of the spectrum.There is another subspectrum which is often called compression spectrum in the literature.By the definitions given above, we can illustrate the subdivisions of spectrum in Table 1.
If these possibilities are combined in all possible ways, nine different states are created.These are labelled by  1 ,  2 ,  3 ,  1 ,  2 ,  3 ,  1 ,  2 and  3 .If  is a complex number such that   ∈  1 or   ∈  1 , then  is in the resolvent set (, ) of .The further classification gives rise to the fine spectrum of .If an operator is in state  2 , for example, then () ̸ = () =  and  −1 exists but is discontinuous and we write  ∈  2 (, ).
Let  and  be two sequence spaces and let  = (  ) be an infinite matrix of real or complex numbers   , where ,  ∈ N = {0, 1, 2, . ..}.Then, we say that  defines a matrix mapping from  into  and we denote it by writing  :  →  if for every sequence  = (  ) ∈  the sequence  = {()  }, the -transform of  is in , where By ( : ), we denote the class of all matrices  such that  :  → .Thus,  ∈ ( : ) if and only if the series on the right side of (7) converges for each  ∈ N and every  ∈ , and we have  = {()  } ∈N ∈  for all  ∈ .

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 (, ) =  ap (, ) if  is a Hilbert space and  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 [15]).
Lemma 2 (see [16, p. 60]).The adjoint operator  * of  is onto if and only if  has a bounded inverse.Lemma 3 (see [16, p. 59]). has a dense range if and only if  * is one to one.
Our main focus in this paper is on the triple-band matrix (, , ), where We assume here and after that  and  are complex parameters which do not simultaneously vanish.We introduce the introduce the operator (, , ) from ℓ  to itself by where  = (  ) ∈ ℓ  . (9)

Fine Spectra of Upper Triangular
Triple-Band Matrices over the Sequence Space ℓ  (0 <  ⩽ 1) In this section, we prove that the operator (, , ) : ℓ  → ℓ  is a bounded linear operator and compute its norm.We essentially emphasize the fine spectrum of the operator (, , ) : ℓ  → ℓ  in the case 0 <  ⩽ 1.
If  : ℓ  → ℓ  is a bounded matrix operator with the matrix , then it is known that the adjoint operator  * : ℓ *  → ℓ *  is defined by the transpose of the matrix .The dual space of ℓ  is isomorphic to ℓ ∞ , where 0 <  < 1.
Before giving the main theorem of this section, we should note the following remark.In this work, here and in what follows, if  is a complex number, then by √ we always mean the square root of  with a nonnegative real part.If Re(√) = 0, then √ represents the square root of  with Im(√) > 0. The same results are obtained if √ represents the other square root.Theorem 5. Let  be a complex number such that √  2 = − and define the set  1 by Then,   ((, , ), ℓ  ) ⊆  1 .

Fine Spectra of Upper Triangular
Triple-Band Matrices over the Sequence Space ℓ  (1 <  < ∞) In the present section, we determine the fine spectrum of the operator (, , ) : ℓ  → ℓ  in case 1 ⩽  < ∞.We quote some lemmas which are needed in proving the theorems given in Section 4.
In the case 1 <  < ∞, since the proof of the theorems, in Section 4, determining the spectrum and fine spectrum of the matrix operator (, , ) on the sequence space ℓ  is similar to the case 0 <  ⩽ 1; to avoid the repetition of similar statements, we give the results by the following theorem without proof.
Theorem 19.The following statements hold:

Some Applications
In this section, we give two theorems related to Toeplitz matrix.

Table 1 :
Subdivisions of spectrum of a linear operator.
, Theorem 34.16]).The matrix  = (  ) gives rise to a bounded linear operator  ∈ (ℓ 1 ) from ℓ 1 to itself if and only if the supremum of ℓ 1 norms of the columns of  is bounded.Lemma 14 (see [17, p. 245, Theorem 34.3]).The matrix  = (  ) gives rise to a bounded linear operator  ∈ (ℓ ∞ ) from ℓ ∞ to itself if and only if the supremum of ℓ 1 norms of the rows of  is bounded.