Fine Spectra of Symmetric Toeplitz Operators

Abstract

The fine spectra of 2-banded and 3-banded infinite Toeplitz matrices were examined by several authors. The fine spectra of n-banded triangular Toeplitz matrices and tridiagonal symmetric matrices were computed in the following papers: Altun, “On the fine spectra of triangular toeplitz operators” (2011) and Altun, “Fine spectra of tridiagonal symmetric matrices” (2011). Here, we generalize those results to the (2𝑛+1)-banded symmetric Toeplitz matrix operators for arbitrary positive integer 𝑛.

1. Introduction and Preliminaries

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. Some other parts also arise by examining the surjectivity of the operator and continuity of the inverse operator. Such subparts of the spectrum are called the fine spectra of the operator.

The spectra and fine spectra of linear operators defined by some particular limitation matrices over some sequence spaces were studied by several authors. We introduce the knowledge in the existing literature concerning the spectrum and the fine spectrum. Wenger [1] examined the fine spectrum of the integer power of the Cesàro operator over 𝑐, and Rhoades [2] generalized this result to the weighted mean methods. Reade [3] worked on the spectrum of the Cesàro operator over the sequence space 𝑐₀. Gonzáles [4] studied the fine spectrum of the Cesàro operator over the sequence space ℓ_𝑝. Okutoyi [5] computed the spectrum of the Cesàro operator over the sequence space 𝑏𝑣. Recently, Rhoades and Yildirim [6] examined the fine spectrum of factorable matrices over 𝑐₀ and 𝑐. Akhmedov and Başar [7, 8] have determined the fine spectrum of the Cesàro operator over the sequence spaces 𝑐₀, ℓ_∞ and ℓ_𝑝. Altun and Karakaya [9] computed the fine spectra of Lacunary matrices over 𝑐₀ and 𝑐. Furkan et al. [10] determined the fine spectrum of 𝐵(𝑟,𝑠,𝑡) over the sequence spaces 𝑐₀ and 𝑐, where 𝐵(𝑟,𝑠,𝑡) is a lower triangular triple-band matrix. Later, Altun [11] computed the fine spectra of triangular Toeplitz matrices over 𝑐₀ and 𝑐.

The fine spectrum of the difference operator Δ over 𝑐₀ and 𝑐 was studied by Altay and Başar [12]. Recently, the fine spectra of Δ over ℓ_𝑝 and 𝑏𝑣_𝑝 are studied by Akhmedov and Başar [13, 14], where 𝑏𝑣_𝑝 is the space of 𝑝-bounded variation sequences, introduced by Başar and Altay [15] with 1≤𝑝<∞. The fine spectrum with respect to the Goldberg's classification of the operator 𝐵(𝑟,𝑠,𝑡) over ℓ_𝑝 and 𝑏𝑣_𝑝 with 1<𝑝<∞ has recently been studied by Furkan et al. [16]. Quite recently, Akhmedov and El-Shabrawy [17] 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 𝑐. In 2010, Srivastava and Kumar [18] have determined the spectra and the fine spectra of the 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 (see [19]).

In this work, our purpose is to determine the spectra of the operator, for which the corresponding matrix is a (2𝑛+1)-banded symmetric Toeplitz matrix, over the sequence spaces 𝑐₀,𝑐,ℓ₁ and ℓ_∞. We will also give the fine spectra results for the spaces 𝑐₀ and 𝑐.

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. (1.1) 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. Let 𝑋≠{𝜃} be a complex normed space and 𝑇∶𝒟(𝑇)→𝑋 be a linear operator with domain 𝒟(𝑇)⊂𝑋. With 𝑇, we associate the operator 𝑇_𝜆=𝑇−𝜆𝐼, (1.2) 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 𝑇_𝜆⁻¹, that is, 𝑇_𝜆⁻¹=(𝑇−𝜆𝐼)⁻¹, (1.3) and call it the resolvent operator of 𝑇_𝜆. If 𝜆=0, we will simply write 𝑇⁻¹. 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 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. 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 [20, pages 370-371]).

Let 𝑋≠{𝜃} be a complex normed space and 𝑇∶𝒟(𝑇)→𝑋 be a linear operator with domain 𝒟(𝑇)⊂𝑋. A regular value 𝜆 of 𝑇 is a complex number 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) of 𝑇 is the set of all regular values 𝜆 of 𝑇. Its complement 𝜎(𝑇)=ℂ⧵𝜌(𝑇) in the complex plane ℂ is called the spectrum of 𝑇. Furthermore, the spectrum 𝜎(𝑇) is partitioned into three disjoint sets as follows: the point spectrum 𝜎_𝑝(𝑇) is the set such that 𝑇_𝜆⁻¹ does not exist. A 𝜆∈𝜎_𝑝(𝑇) is called an eigenvalue of 𝑇. The continuous spectrum 𝜎_𝑐(𝑇) is the set such that 𝑇_𝜆⁻¹ exists and satisfies (R3) but not (R2). The residual spectrum 𝜎_𝑟(𝑇) is the set such that 𝑇_𝜆⁻¹ exists but does not satisfy (R3).

From Goldberg [21], if 𝑇∈𝐵(𝑋), 𝑋 a Banach space, then there are three possibilities for ℛ(𝑇), the range of 𝑇:(I)ℛ(𝑇)=𝑋,(II)ℛ(𝑇)=𝑋, but ℛ(𝑇)≠𝑋,(III)ℛ(𝑇)≠𝑋,and three possibilities for 𝑇⁻¹:(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 as I₁, I₂, I₃, II₁, II₂, II₃, III₁, III₂, and III₃. If 𝜆 is a complex number such that 𝑇_𝜆∈𝐼₁ or 𝑇_𝜆∈II₁, then 𝜆 is in the resolvent set 𝜌(𝑇,𝑋) of 𝑇, the set of all regular values of 𝑇 on 𝑋. The other classification gives rise to the fine spectrum of 𝑇. For example, we will write 𝜆∈III₁𝜎(𝑇,𝑋) if 𝑇 satisfies III and 1.

A triangle is a lower triangular matrix with all of the principal diagonal elements nonzero. We will write ℓ_∞, 𝑐 and 𝑐₀ for the spaces of all bounded, convergent, and null sequences, respectively. By ℓ_𝑝, we denote the space of all 𝑝-absolutely summable sequences, where 1≤𝑝<∞. Let 𝜇 and 𝛾 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 𝜇 into 𝛾, and we denote it by writing 𝐴∶𝜇→𝛾, if for every sequence 𝑥=(𝑥_𝑘)∈𝜇 the sequence 𝐴𝑥={(𝐴𝑥)_𝑛}, the 𝐴-transform of 𝑥, is in 𝛾, where (𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘(𝑛∈ℕ). (1.4) 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 (1.4) converges for each 𝑛∈ℕ and every 𝑥∈𝜇, and we have 𝐴𝑥={(𝐴𝑥)_𝑛}_𝑛∈ℕ∈𝛾 for all 𝑥∈𝜇.

Let an (𝑛+1)-tuple 𝑡=(𝑡₀,𝑡₁,…,𝑡_𝑛)∈ℂ^𝑛+1 be given. A symmetric infinite Toeplitz matrix is a (2𝑛+1)-band matrix of the form (𝑡)=𝑡𝑆=𝑆⁰𝑡₁⋯⋯𝑡_𝑛𝑡00⋯₁𝑡₀𝑡₁⋯⋯𝑡_𝑛0⋯⋮𝑡₁𝑡₀𝑡₁⋯⋯𝑡_𝑛⋯⋮⋮𝑡₁𝑡₀𝑡₁𝑡⋯⋯⋯_𝑛⋮⋮𝑡₁𝑡₀𝑡₁⋯⋯0𝑡_𝑛⋮⋮𝑡₁𝑡₀𝑡₁⋯00𝑡_𝑛⋮⋮𝑡₁𝑡₀⋯⋮⋮⋮⋮⋮⋮⋮⋱. (1.5) The spectral results are clear when 𝑆 is a multiple of the identity matrix, so for the sequel we will have 𝑛≥1 and 𝑡_𝑛≠0.

Let 𝑅 be the right shift operator: 𝑅=00000⋯10000⋯01000⋯⋮⋮⋮⋮⋮⋱, (1.6) and 𝐿 be the left shift operator: 𝐿=𝑅^𝑡=𝑅⁻¹. (1.7) Let 𝐹(𝑧)=𝑡_𝑛[𝑧^𝑛+𝑧^−𝑛]+𝑡_𝑛−1[𝑧^𝑛−1+𝑧^{−(𝑛−1)}]+⋯+𝑡₁[𝑧+𝑧⁻¹]+𝑡₀=𝑃(𝑧)/𝑧^𝑛, where 𝑃 is the palindromic polynomial 𝑃(𝑧)=𝑡_𝑛𝑧^2𝑛+𝑡_𝑛−1𝑧^2𝑛−1+⋯+𝑡₀𝑧^𝑛+𝑡₁𝑧^𝑛−1+𝑡₂𝑧^𝑛−2+⋯+𝑡_𝑛. Then, we can see that 𝑆=𝐹(𝑅) and we will call 𝐹 and 𝑃 as the function and polynomial associated to the operator 𝑆, respectively. We also have 𝑆=𝐿^𝑛𝑃(𝑅). (1.8) The roots of 𝑃(𝑧) are nonzero and symmetric, that is, if 𝛼 is a root, 𝛼⁻¹ is also a root. Let 𝛼₁,𝛼₂,…,𝛼_𝑛,𝛼₁⁻¹,𝛼₂⁻¹,…,𝛼_𝑛⁻¹ be the roots of 𝑃(𝑧) such that |𝛼_𝑘|≤1 for 𝑘=1,2,…,𝑛. Then 𝑆=𝑡_𝑛𝐿^𝑛𝑅−𝛼₁𝐼𝑅−𝛼₂𝐼⋯𝑅−𝛼_𝑛𝐼𝑅−𝛼₁⁻¹𝐼𝑅−𝛼₂⁻¹𝐼⋯𝑅−𝛼_𝑛⁻¹𝐼. (1.9) Now, by induction, we can see that 𝐿^𝑛𝑅−𝛼₁𝐼𝑅−𝛼₂𝐼⋯𝑅−𝛼_𝑛𝐼=𝐼−𝛼₁𝐿𝐼−𝛼₂𝐿⋯𝐼−𝛼_𝑛𝐿. (1.10) Let 𝐷 be the unit disc {𝑧∈ℂ∶|𝑧|≤1} and 𝜕𝐷 be the unit circle {𝑧∈ℂ∶|𝑧|=1}. We have the following two lemmas as a consequence of the corresponding results in [22] and [23], respectively.

Lemma 1.1. (𝐼−𝛼𝐿)∈(𝑐₀,𝑐₀) is onto if and only if 𝛼 is not on the unit circle.

Lemma 1.2. (𝑅−𝛼𝐼)∈(𝑐₀,𝑐₀) is onto if and only if 𝛼 is outside the unit disc.

Theorem 1.3. 𝑆∈(𝑐₀,𝑐₀) is onto if and only if 𝑃 has no root on the unit circle.

Proof. Suppose 𝑃 has a root on the unit circle. Let 𝛼₁,𝛼₂,…,𝛼_𝑛,𝛼₁⁻¹,𝛼₂⁻¹,…,𝛼_𝑛⁻¹ be the roots of 𝑃(𝑧) such that |𝛼_𝑘|≤1 for 𝑘=1,2,…,𝑛. We have 𝑆=𝑡_𝑛𝐼−𝛼₁𝐿𝐼−𝛼₂𝐿⋯𝐼−𝛼_𝑛𝐿𝑅−𝛼₁⁻¹𝐼𝑅−𝛼₂⁻¹𝐼⋯𝑅−𝛼_𝑛⁻¹𝐼. (1.11) Since the matrix operators (𝐼−𝛼₁𝐿),(𝐼−𝛼₂𝐿),…,(𝐼−𝛼_𝑛𝐿) commute with each other, without loss of generality, we can suppose 𝛼₁ is a root on the unit circle. Clearly, all the operators (𝐼−𝛼₁𝐿),(𝐼−𝛼₂𝐿),…,(𝐼−𝛼_𝑛𝐿),(𝑅−𝛼₁⁻¹𝐼),(𝑅−𝛼₂⁻¹𝐼),…,(𝑅−𝛼_𝑛⁻¹𝐼) are in (𝑐₀,𝑐₀). But, by Lemma 1.1 the operator (𝐼−𝛼₁𝐿) is not onto. So, 𝑆 cannot be onto.
Suppose, now, 𝑃 has no root on the unit circle. That means |𝛼_𝑘|<1 for 𝑘=1,2,…,𝑛. Then all the operators (𝐼−𝛼₁𝐿),(𝐼−𝛼₂𝐿),…,(𝐼−𝛼_𝑛𝐿),(𝑅−𝛼₁⁻¹𝐼),(𝑅−𝛼₂⁻¹𝐼),…,(𝑅−𝛼_𝑛⁻¹𝐼) are onto by Lemma 1.1 and Lemma 1.2. Hence, 𝑆=𝑡_𝑛(𝐼−𝛼₁𝐿)(𝐼−𝛼₂𝐿)⋯(𝐼−𝛼_𝑛𝐿)(𝑅−𝛼₁⁻¹𝐼)(𝑅−𝛼₂⁻¹𝐼)⋯(𝑅−𝛼_𝑛⁻¹𝐼) is onto.

Theorem 1.4 (cf. [24]). Let 𝑇 be an operator with the associated matrix 𝐴=(𝑎_𝑛𝑘).(i)𝑇∈𝐵(𝑐) if and only if ‖𝐴‖∶=sup_𝑛^∞_𝑘=1𝑎_𝑛𝑘𝑎<∞,_𝑘∶=lim_𝑛→∞𝑎_𝑛𝑘𝑒𝑥𝑖𝑠𝑡𝑠𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑘,𝑎∶=lim^∞_{𝑛→∞𝑘=1}𝑎_𝑛𝑘𝑒𝑥𝑖𝑠𝑡𝑠. (1.12)(1.13)(1.14)(ii)𝑇∈𝐵(𝑐₀) if and only if (1.12) and (1.13) with 𝑎_𝑘=0 for each 𝑘.(iii)𝑇∈𝐵(ℓ_∞) if and only if (1.12). In these cases, the operator norm of 𝑇 is ‖𝑇‖_{(ℓ∞∶ℓ∞)}=‖𝑇‖_{(𝑐∶𝑐)}=‖𝑇‖_{(𝑐0∶𝑐0)}=‖𝐴‖. (1.15)(iv)𝑇∈𝐵(ℓ₁) if and only if 𝐴^𝑡=sup_𝑘^∞_𝑛=1𝑎_𝑛𝑘<∞. (1.16)In this case, the operator norm of 𝑇 is ‖𝑇‖_{(ℓ1∶ℓ1)}=‖𝐴^𝑡‖.

Corollary 1.5. 𝑆(𝑡)∈𝐵(𝜇) for 𝜇∈{𝑐₀,𝑐,ℓ₁,ℓ_∞} and ‖𝑆(𝑡)‖_(𝜇,𝜇)=𝑡₀+2𝑡₁+𝑡₂+⋯+𝑡_𝑛. (1.17)

Theorem 1.6. Let 𝑋 be a Banach space and 𝑇∈𝐵(𝑋). Then 𝜆∈ℂ is in the spectrum 𝜎(𝑇,𝑋) if and only if 𝑇−𝜆𝐼 is not bijective.

Proof. Suppose 𝑇−𝜆𝐼 is not bijective. Then 𝑇−𝜆𝐼 is not 1-1 or not onto. If it is not 1-1, then 𝜆∈𝜎_𝑝(𝑇,𝑋)⊂𝜎(𝑇,𝑋). Suppose now 𝑇−𝜆𝐼 is 1-1. Then it is not onto and by Lemma 7.2-3 of [20], 𝜆 cannot be in 𝜌(𝑇,𝑋). Hence, 𝜆∈𝜎(𝑇,𝑋).
Now, suppose 𝑇−𝜆𝐼 is bijective. Then by the open mapping theorem (𝑇−𝜆𝐼)⁻¹ is continuous. Hence, 𝜆 is not in the spectrum 𝜎(𝑇,𝑋).

Corollary 1.7. Let 𝑋 be a Banach space and 𝑇∈𝐵(𝑋). Then 𝜆∈𝜌(𝑇,𝑋) if and only if 𝑇_𝜆 is bijective.

2. The Spectra and Fine Spectra

Lemma 2.1 (Lemma 3.4 of [11]). Let 𝑧₁,𝑧₂,…,𝑧_𝑟 be distinct complex numbers with |𝑧_𝑖|=1 for 1≤𝑖≤𝑟. Let 0≠𝑥=(𝑥_𝑘) be a sequence satisfying 𝑥_𝑘=𝛼_1,0+𝛼_1,1𝑘+⋯+𝛼_1,𝑚1−1𝑘^𝑚1−1𝑧^𝑘₁+𝛼_2,0+𝛼_2,1𝑘+⋯+𝛼_2,𝑚2−1𝑘^𝑚2−1𝑧^𝑘₂+⋯+𝛼_𝑟,0+𝛼_𝑟,1𝑘+⋯+𝛼_{𝑟,𝑚𝑟−1}𝑘^𝑚𝑟−1𝑧^𝑘_𝑟, (2.1) for 𝑘=0,1,2,…, where 𝛼_𝑖,𝑗 are constants forming the polynomials 𝑃_𝑖(𝑘)=𝛼_𝑖,0+𝛼_𝑖,1𝑘+⋯+𝛼_{𝑖,𝑚𝑖−1}𝑘^𝑚𝑖−1≠0 for 1≤𝑖≤𝑟 and 0≤𝑗≤𝑚_𝑖−1. Then 𝑥∉𝑐₀.

Lemma 2.2. Let 𝑧₁,𝑧₂,…,𝑧_𝑟 be distinct complex numbers. Let 0≠𝑥=(𝑥_𝑘)∈𝑐 be a sequence satisfying 𝑥_𝑘=𝛼_1,0+𝛼_1,1𝑘+⋯+𝛼_1,𝑚1−1𝑘^𝑚1−1𝑧^𝑘₁+𝛼_2,0+𝛼_2,1𝑘+⋯+𝛼_2,𝑚2−1𝑘^𝑚2−1𝑧^𝑘₂+⋯+𝛼_𝑟,0+𝛼_𝑟,1𝑘+⋯+𝛼_{𝑟,𝑚𝑟−1}𝑘^𝑚𝑟−1𝑧^𝑘_𝑟, (2.2) for 𝑘=0,1,2,…, where 𝛼_𝑖,𝑗 are constants forming the polynomials 𝑃_𝑖(𝑘)=𝛼_𝑖,0+𝛼_𝑖,1𝑘+⋯+𝛼_{𝑖,𝑚𝑖−1}𝑘^𝑚𝑖−1≠0 for 1≤𝑖≤𝑟 and 0≤𝑗≤𝑚_𝑖−1. Then |𝑧_𝑖|≤1 for 1≤𝑖≤𝑟, and the existence of a 𝑡≤𝑟 with |𝑧_𝑡|=1 implies 𝑧_𝑡=1 and 𝑃_𝑡 is a constant.

Proof. Let |𝑧₁|≥|𝑧₂|≥⋯≥|𝑧_𝑟|. To prove |𝑧_𝑖|≤1 for all 𝑖≤𝑟, suppose it is not true. Then let 𝑎∶=|𝑧₁|>1. Let 𝑠≤𝑟 be the largest positive integer with |𝑧₁|=|𝑧₂|=⋯=|𝑧_𝑠|. Then (𝑥_𝑘/𝑎^𝑘)∈𝑐₀. Let 𝑢_𝑘=𝛼_1,0+𝛼_1,1𝑘+⋯+𝛼_1,𝑚1−1𝑘^𝑚1−1𝑧₁𝑎^𝑘+𝛼_2,0+𝛼_2,1𝑘+⋯+𝛼_2,𝑚2−1𝑘^𝑚2−1𝑧₂𝑎^𝑘+⋯+𝛼_𝑠,0+𝛼_𝑠,1𝑘+⋯+𝛼_{𝑠,𝑚𝑠−1}𝑘^𝑚𝑠−1𝑧_𝑠𝑎^𝑘. (2.3) We have 0=(𝑥_𝑘/𝑎^𝑘−𝑢_𝑘) for 𝑠=𝑟. If 𝑠<𝑟, we have 𝑥_𝑘𝑎^𝑘−𝑢_𝑘=𝛼_𝑠+1,0+𝛼_𝑠+1,1𝑘+⋯+𝛼_{𝑠+1,𝑚𝑠+1−1}𝑘^{𝑚𝑠+1−1}𝑧_𝑠+1𝑎^𝑘+𝛼_𝑠+2,0+𝛼_𝑠+2,1𝑘+⋯+𝛼_{𝑠+2,𝑚𝑠+2−1}𝑘^{𝑚𝑠+2−1}𝑧_𝑠+2𝑎^𝑘+⋯+𝛼_𝑟,0+𝛼_𝑟,1𝑘+⋯+𝛼_{𝑟,𝑚𝑟−1}𝑘^𝑚𝑟−1𝑧_𝑟𝑎^𝑘. (2.4) Since |𝑧_𝑗/𝑎|<1 for 𝑠+1≤𝑗≤𝑟, we have (𝑥_𝑘/𝑎^𝑘−𝑢_𝑘)∈𝑐₀. Then (𝑢_𝑘)∈𝑐₀ but this contradicts with Lemma 2.1. Hence, we have |𝑧_𝑖|≤1 for 1≤𝑖≤𝑟.
Now, let us prove the second part. Suppose, there exist a positive integer 𝑞≤𝑟 such that |𝑧_𝑖|=1 for all 𝑖≤𝑞. For any 𝑖, 𝑃_𝑖 is constant means 𝑚_𝑖=1. Suppose 𝑚=max{𝑚₁,…,𝑚_𝑞}>1. Without loss of generality let 𝑚₁≥𝑚₂≥⋯≥𝑚_𝑞. Let 𝑞₀≤𝑞 be the largest integer satisfying 𝑚₁=𝑚₂=⋯=𝑚_𝑞0=𝑚. Then (𝑥_𝑘/𝑘^𝑚−1)∈𝑐₀. This means, since |𝑧_𝑖| have modulus less than 1, (𝑣_𝑘)∈𝑐₀, where 𝑣_𝑘=𝛼_1,𝑚−1𝑧^𝑘₁+𝛼_2,𝑚−1𝑧^𝑘₂+⋯+𝛼_{𝑞0,𝑚−1}𝑧^𝑘_𝑞0. (2.5) But, this again contradicts with Lemma 2.1. Hence, we have 𝑚₁=𝑚₂=⋯=𝑚_𝑞=1. Now, we have (𝑤_𝑘)∈𝑐, where 𝑤_𝑘=𝛼_1,0𝑧^𝑘₁+𝛼_2,0𝑧^𝑘₂+⋯+𝛼_𝑞,0𝑧^𝑘_𝑞. (2.6) Suppose, one of the elements in {𝑧₁,𝑧₂,…,𝑧_𝑞} is equal to 1, say 𝑧₁=1. Then, (𝑤_𝑘+1−𝑤_𝑘)∈𝑐₀, where 𝑤_𝑘+1−𝑤_𝑘=𝛼_2,0𝑧₂−1𝑧^𝑘₂+⋯+𝛼_𝑞,0𝑧_𝑞−1𝑧^𝑘_𝑞, (2.7) and this again contradicts with Lemma 2.1. Hence, 𝑞≤1 and 1 is the unique candidate for 𝑧_𝑡 with modulus 1.

Theorem 2.3. 𝜎_𝑝(𝑆,𝜇)=∅ for 𝜇∈{ℓ₁,𝑐₀,𝑐}.

Proof. Since ℓ₁⊂𝑐₀⊂𝑐, it is enough to show that 𝜎_𝑝(𝑆,𝑐)=∅. Let 𝜆 be an eigenvalue of the operator 𝑆. An eigenvector 𝑥=(𝑥₀,𝑥₁,…)∈𝑐 corresponding to this eigenvalue satisfies the linear system of equations: 𝑡₀𝑥₀+𝑡₁𝑥₁+𝑡₂𝑥₂+⋯+𝑡_𝑛𝑥_𝑛=𝜆𝑥₀𝑡₁𝑥₀+𝑡₀𝑥₁+𝑡₁𝑥₂+⋯+𝑡_𝑛𝑥_𝑛+1=𝜆𝑥₁𝑡₂𝑥₀+𝑡₁𝑥₁+𝑡₀𝑥₂+⋯+𝑡_𝑛𝑥_𝑛+2=𝜆𝑥₂⋮𝑡_𝑛𝑥₀+𝑡_𝑛−1𝑥₁+𝑡_𝑛−2𝑥₂+⋯+𝑡_𝑛𝑥_2𝑛=𝜆𝑥_𝑛𝑡_𝑛𝑥₁+𝑡_𝑛−1𝑥₂+𝑡_𝑛−2𝑥₃+⋯+𝑡_𝑛𝑥_2𝑛+1=𝜆𝑥_𝑛+1𝑡_𝑛𝑥₂+𝑡_𝑛−1𝑥₃+𝑡_𝑛−2𝑥₄+⋯+𝑡_𝑛𝑥_2𝑛+2=𝜆𝑥_𝑛+2⋮ (2.8) Since 𝑡_𝑛≠0 we can write this system of equations in the form: 𝑑₀𝑥₀+𝑑₁𝑥₁+𝑑₂𝑥₂+⋯+𝑑_𝑛𝑥_𝑛𝑑=0₁𝑥₀+𝑑₀𝑥₁+𝑑₁𝑥₂+⋯+𝑑_𝑛𝑥_𝑛+1𝑑=0₂𝑥₀+𝑑₁𝑥₁+𝑑₀𝑥₂+⋯+𝑑_𝑛𝑥_𝑛+2⋮𝑑=0_𝑛𝑥₀+𝑑_𝑛−1𝑥₁+𝑑_𝑛−2𝑥₂+⋯+𝑑_𝑛𝑥_2𝑛𝑑=0_𝑛𝑥₁+𝑑_𝑛−1𝑥₂+𝑑_𝑛−2𝑥₃+⋯+𝑑_𝑛𝑥_2𝑛+1𝑑=0_𝑛𝑥₂+𝑑_𝑛−1𝑥₃+𝑑_𝑛−2𝑥₄+⋯+𝑑_𝑛𝑥_2𝑛+2⋮=0 (2.9) where 𝑑₀=(𝑡₀−𝜆)/𝑡_𝑛, 𝑑_𝑛=1 and 𝑑_𝑗=𝑡_𝑗/𝑡_𝑛 for 1≤𝑗≤𝑛−1. This system of equations, by change of variables 𝑢_𝑛+𝑘=𝑥_𝑘 for 𝑘=0,1,2,…, is equivalent to system of equations 𝑑_𝑛𝑢_𝑘+𝑑_𝑛−1𝑢_𝑘+1+⋯+𝑑₀𝑢_𝑛+𝑘+𝑑₁𝑢_𝑛+𝑘+1+⋯+𝑑_𝑛𝑢_2𝑛+𝑘=0,𝑘=0,1,2,…, (2.10) with the initial conditions 𝑢₀=𝑢₁=⋯=𝑢_𝑛−1=0.
We see that this is a 2𝑛-th order linear homogenous difference equation with the corresponding characteristic polynomial 𝑃(𝑧)=𝑧^2𝑛+𝑑_𝑛−1𝑧^2𝑛−1+⋯+𝑑₁𝑧^𝑛+1+𝑑₀𝑧^𝑛+𝑑₁𝑧^𝑛−1+⋯+𝑑_𝑛−1𝑧+1. (2.11) Suppose 𝑃(𝑧) has 𝑟 distinct roots 𝑧₁,𝑧₂,…,𝑧_𝑟 with multiplicities 𝑚₁,𝑚₂,…,𝑚_𝑟. Then, any solution (𝑢_𝑘) of the system of equations satisfies 𝑢_𝑘=𝛼_1,0+𝛼_1,1𝑘+⋯+𝛼_1,𝑚1−1𝑘^𝑚1−1𝑧^𝑘₁+𝛼_2,0+𝛼_2,1𝑘+⋯+𝛼_2,𝑚2−1𝑘^𝑚2−1𝑧^𝑘₂+⋯+𝛼_𝑟,0+𝛼_𝑟,1𝑘+⋯+𝛼_{𝑟,𝑚𝑟−1}𝑘^𝑚𝑟−1𝑧^𝑘_𝑟. (2.12) Observe that if 𝜁 is a root of 𝑃(𝑧), then 1/𝜁 is also a root. There are two cases.
Case 1 (1 is not a root of 𝑃(𝑧)). Since (𝑢_𝑘)∈𝑐, by Lemma 2.2 we can write 𝑢_𝑘=𝛼_1,0+𝛼_1,1𝑘+⋯+𝛼_1,𝑚1−1𝑘^𝑚1−1𝑧^𝑘₁+𝛼_2,0+𝛼_2,1𝑘+⋯+𝛼_2,𝑚2−1𝑘^𝑚2−1𝑧^𝑘₂+⋯+𝛼_𝑞,0+𝛼_𝑞,1𝑘+⋯+𝛼_{𝑞,𝑚𝑞−1}𝑘^𝑚𝑞−1𝑧^𝑘_𝑞, (2.13) where |𝑧_𝑖|<1 for 1≤𝑖≤𝑞. By the symmetry of the roots we have 𝑚₁+𝑚₂+𝑚₃+⋯+𝑚_𝑞=𝑛. Now, using the initial conditions 𝑢₀=𝑢₁=⋯=𝑢_𝑛−1=0 we have 𝑉𝛼=0, (2.14) where 𝛼=[𝛼_1,0,𝛼_1,1,…,𝛼_1,𝑚1−1,𝛼_2,0,𝛼_2,1,…,𝛼_2,𝑚2−1,…,𝛼_𝑞,0,𝛼_𝑞,1,…,𝛼_{𝑞,𝑚𝑞−1}]^𝑡 and 𝑉 is the generalized Vandermonde matrix𝑧10⋯010⋯0⋯⋯10⋯0₁𝑧₁⋯𝑧₁𝑧₂𝑧₂⋯𝑧₂⋯⋯𝑧_𝑞𝑧_𝑞⋯𝑧_𝑞𝑧²₁2𝑧²₁⋯2^𝑚1−1𝑧²₁𝑧²₂2𝑧²₂⋯2^𝑚2−1𝑧²₂⋯⋯𝑧²_𝑞2𝑧²_𝑞⋯2^𝑚𝑞−1𝑧²_𝑞𝑧³₁3𝑧³₁⋯3^𝑚1−1𝑧³₁𝑧³₂3𝑧³₂⋯3^𝑚2−1𝑧³₂⋯⋯𝑧³_𝑞3𝑧³_𝑞⋯3^𝑚𝑞−1𝑧³_𝑞𝑧⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮_1𝑛−1𝔄⋯𝔅𝑧₂^𝑛−1ℭ⋯𝔇⋯⋯𝑧_𝑞^𝑛−1𝔈⋯𝔉, (2.15) where 𝔄 denotes (𝑛−1)𝑧₁^𝑛−1, 𝔅 denotes (𝑛−1)^𝑚1−1𝑧₁^𝑛−1, ℭ denotes (𝑛−1)𝑧₂^𝑛−1, 𝔇 denotes (𝑛−1)^𝑚2−1𝑧₂^𝑛−1, 𝔈 denotes (𝑛−1)𝑧_𝑞^𝑛−1, and 𝔉 denotes (𝑛−1)^𝑚𝑞−1𝑧_𝑞^𝑛−1.
The determinant of the matrix 𝑉 was explicitly given in [25, 26]: det𝑉=^𝑞_𝑖=1^𝑚𝑖−1_𝑗=0𝑗!𝑧^𝑚𝑖2_{𝑖1≤𝑖<𝑗≤𝑞}𝑧_𝑗−𝑧_𝑖^{𝑚𝑖𝑚𝑗}. (2.16) An inductive proof of this formula is given by Chen and Li [27]. Since zero is not a root of our polynomial 𝑃, we have det𝑉≠0; hence, we conclude 𝛼=0, which means the sequences (𝑢_𝑘)=0 and (𝑥_𝑘)=0. Hence, there is no eigenvalue in this case.
Case 2 (1 is a root of 𝑃(𝑧)). Since (𝑢_𝑘)∈𝑐, by Lemma 2.2 we can write 𝑢_𝑘=𝛼_1,01^𝑘+𝛼_2,0+𝛼_2,1𝑘+⋯+𝛼_2,𝑚2−1𝑘^𝑚2−1𝑧^𝑘₂+⋯+𝛼_𝑞,0+𝛼_𝑞,1𝑘+⋯+𝛼_{𝑞,𝑚𝑞−1}𝑘^𝑚𝑞−1𝑧^𝑘_𝑞, (2.17) where |𝑧_𝑖|<1 for 2≤𝑖≤𝑞. By the symmetry of the roots we have 𝑝∶=1+𝑚₂+𝑚₃+⋯+𝑚_𝑘≤𝑛. Now, using the initial conditions 𝑢₀=𝑢₁=⋯=𝑢_𝑛−1=0 we have 𝑊𝛼=0, (2.18) where 𝛼=[𝛼_1,0,𝛼_2,0,𝛼_2,1,…,𝛼_2,𝑚2−1,…,𝛼_𝑞,0,𝛼_𝑞,1,…,𝛼_{𝑞,𝑚𝑞−1}]^𝑡 and 𝑊 is an 𝑛×𝑝 submatrix of a generalized 𝑛×𝑛 Vandermonde matrix. Since the determinant of generalized Vandermonde matrix with nonzero roots is not zero, we have that the columns of 𝑊 are linearly independent. So again we can conclude that 𝛼=0, which again will mean that there is no eigenvalue.

If 𝑇∶𝜇→𝜇 (𝜇 is ℓ₁ or 𝑐₀) is a bounded linear operator represented by the matrix 𝐴, then it is known that the adjoint operator 𝑇^∗∶𝜇^∗→𝜇^∗ is defined by the transpose 𝐴^𝑡 of the matrix 𝐴. It should be noted that the dual space 𝑐^∗₀ of 𝑐₀ is isometrically isomorphic to the Banach space ℓ₁ and the dual space ℓ^∗₁ of ℓ₁ is isometrically isomorphic to the Banach space ℓ_∞.

Lemma 2.4 (see [21, page 59]). 𝑇 has a dense range if and only if 𝑇^∗ is one to one.

Corollary 2.5. If 𝑇∈(𝜇∶𝜇) then 𝜎_𝑟(𝑇,𝜇)=𝜎_𝑝(𝑇^∗,𝜇^∗)⧵𝜎_𝑝(𝑇,𝜇).

Theorem 2.6. 𝜎_𝑟(𝑆,𝑐₀)=∅.

Proof. 𝜎_𝑝(𝑆,ℓ₁)=∅ by Theorem 2.3. Now using Corollary 2.5 we have 𝜎_𝑟(𝑆,𝑐₀)=𝜎_𝑝(𝑆^∗,𝑐^∗₀)⧵𝜎_𝑝(𝑆,𝑐₀)=𝜎_𝑝(𝑆,ℓ₁)⧵𝜎_𝑝(𝑆,𝑐₀)=∅.

If 𝑇∶𝑐→𝑐 is a bounded matrix operator represented by the matrix 𝐴, then 𝑇^∗∶𝑐^∗→𝑐^∗ acting on ℂ⊕ℓ₁ has a matrix representation of the form 𝜒0𝑏𝐴^𝑡, (2.19) where 𝜒 is the limit of the sequence of row sums of 𝐴 minus the sum of the limits of the columns of 𝐴, and 𝑏 is the column vector whose 𝑘th entry is the limit of the 𝑘th column of 𝐴 for each 𝑘∈ℕ. For 𝑆∶𝑐→𝑐, the matrix 𝑆^∗ is of the form 2𝑡₁+𝑡₂+⋯+𝑡_𝑛+𝑡₀0=𝐹(1)00𝑆0𝑆. (2.20)

Theorem 2.7. 𝜎_𝑟(𝑆,𝑐)={𝑡₀+2(𝑡₁+𝑡₂+⋯+𝑡_𝑛)}={𝐹(1)}.

Proof. Let 𝑥=(𝑥₀,𝑥₁,…)∈ℂ⊕ℓ₁ be an eigenvector of 𝑆^∗ corresponding to the eigenvalue 𝜆. Then we have (2[𝑡₁+𝑡₂+⋯+𝑡_𝑛]+𝑡₀)𝑥₀=𝜆𝑥₀ and 𝑆𝑥′=𝜆𝑥′ where 𝑥′=(𝑥₁,𝑥₂,…). By Theorem 2.3  𝑥′=(0,0,…). Then 𝑥₀≠0. So 𝜆=2[𝑡₁+𝑡₂+⋯+𝑡_𝑛]+𝑡₀ is the only value that satisfies (2[𝑡₁+𝑡₂+⋯+𝑡_𝑛]+𝑡₀)𝑥₀=𝜆𝑥₀. Hence, 𝜎_𝑝(𝑆^∗,𝑐^∗)={2[𝑡₁+𝑡₂+⋯+𝑡_𝑛]+𝑡₀}. Then 𝜎_𝑟(𝑆,𝑐)=𝜎_𝑝(𝑆^∗,𝑐^∗)⧵𝜎_𝑝(𝑆,𝑐)={2[𝑡₁+𝑡₂+⋯+𝑡_𝑛]+𝑡₀}.

We will write 𝐹(1) instead of {𝐹(1)} for the sequel.

Lemma 2.8. 𝜎(𝑆,ℓ₁)=𝜎(𝑆,𝑐₀)=𝜎(𝑆,𝑐)=𝜎(𝑆,ℓ_∞).

Proof. We will use the fact that the spectrum of a bounded operator over a Banach space is equal to the spectrum of the adjoint operator. The adjoint operator is the transpose of the matrix for 𝑐₀ and ℓ₁. So 𝜎(𝑆,𝑐₀)=𝜎(𝑆^∗,𝑐^∗₀)=𝜎(𝑆,ℓ₁)=𝜎(𝑆^∗,ℓ^∗₁)=𝜎(𝑆,ℓ_∞). We know by Cartlidge [28] that if a matrix operator 𝐴 is bounded on 𝑐, then 𝜎(𝐴,𝑐)=𝜎(𝐴,ℓ_∞). Hence, we have 𝜎(𝑆,𝑐₀)=𝜎(𝑆,ℓ₁)=𝜎(𝑆,ℓ_∞)=𝜎(𝑆,𝑐).

Theorem 2.9. 𝜎(𝑆,𝜇)=𝐹(𝜕𝐷) for 𝜇∈{ℓ₁,𝑐₀,𝑐,ℓ_∞}.

Proof. Let us first consider 𝑆 as an operator on 𝑐₀. By Theorems 1.6 and 2.3  𝜆∈𝜎(𝑆,𝑐₀) if and only if 𝑆−𝜆𝐼 is not onto over 𝑐₀. By Theorem 1.3  𝑆−𝜆𝐼 is not onto over 𝑐₀ if and only if the polynomial 𝑃(𝑧)−𝜆𝑧^𝑛 has a root on the unit circle. 𝑃(𝑧)−𝜆𝑧^𝑛 has a root on the unit circle if and only if 𝜆=𝑃(𝑧)/𝑧^𝑛=𝐹(𝑧) for some 𝑧∈𝜕𝐷. We have 𝜆=𝐹(𝑧) for some 𝑧∈𝜕𝐷 if and only if 𝜆∈𝐹(𝜕𝐷). Hence, 𝜎(𝑆,𝑐₀)=𝐹(𝜕𝐷). Finally, we apply Lemma 2.8.

Corollary 2.10. 𝑆∈(𝑐,𝑐) is onto if and only if 𝑃 has no root on the unit circle.

The spectrum 𝜎 is the disjoint union of 𝜎_𝑝, 𝜎_𝑟 and 𝜎_𝑐, so we have the following theorem as a consequence of Theorems 2.3, 2.6, 2.7, and 2.9.

Theorem 2.11. 𝜎_𝑐(𝑆,𝑐₀)=𝐹(𝜕𝐷) and 𝜎_𝑐(𝑆,𝑐)=𝐹(𝜕𝐷)⧵𝐹(1).

As a result of Theorems 1.3, 2.3, 2.6, 2.7, and 2.9 and Corollary 2.10, we have the following.

Theorem 2.12. 𝐹(𝜕𝐷)=𝐼𝐼₂𝜎(𝑆,𝑐₀), 𝐹(𝜕𝐷)⧵𝐹(1)=𝐼𝐼₂𝜎(𝑆,𝑐) and 𝐹(1)=𝐼𝐼𝐼₂𝜎(𝑆,𝑐).

3. Some Applications

Now, let us give an application of Theorem 2.9. Consider the system of equations 𝑦_𝑘=𝑡₀𝑥_𝑘+^𝑛_𝑗=1𝑡_𝑗𝑥_𝑘+𝑗+𝑥_𝑘−𝑗𝑘=0,1,2,…, (3.1) where 𝑥_𝑘=0 for negative 𝑘.

Theorem 3.1. Let 𝑃(𝑧)=𝑡_𝑛𝑧^2𝑛+𝑡_𝑛−1𝑧^2𝑛−1+⋯+𝑡₀𝑧^𝑛+𝑡₁𝑧^𝑛−1+𝑡₂𝑧^𝑛−2+⋯+𝑡_𝑛, where 𝑡₀,𝑡₁,…,𝑡_𝑛 are complex numbers such that the complex sequences 𝑥=(𝑥_𝑛) and 𝑦=(𝑦_𝑛) are solutions of system (3.1). Then the following are equivalent:(i)boundedness of (𝑦_𝑛) always implies a unique bounded solution (𝑥_𝑛),(ii)convergence of (𝑦_n) always implies a unique convergent solution (𝑥_𝑛),(iii)𝑦_𝑛→0 always implies a unique solution (𝑥_𝑛) with 𝑥_𝑛→0,(iv)∑|𝑦_𝑛|<∞ always implies a unique solution (𝑥_𝑛) with ∑|𝑥_𝑛|<∞,(v)𝑃 has no root on the unit circle 𝜕𝐷.

Proof. The system of equations (3.1) holds, so we have 𝑆𝑥=𝑦. Then 𝑃 is the polynomial associated to 𝑆. Let 𝐹 be the function associated to 𝑆. Let us prove only (i)⇔(v) and omit the proofs of (ii)⇔(v), (iii)⇔(v), (iv)⇔(v) since they are similarly proved. Suppose boundedness of (𝑦_𝑛) always implies a unique bounded solution (𝑥_𝑛). Then the operator 𝑆−0𝐼=𝑆∈(ℓ_∞,ℓ_∞) is bijective. So, 𝜆=0 is not in the spectrum 𝜎(𝑆,ℓ_∞) by Theorem 1.6, which means 0∉𝐹(𝜕𝐷) and 0∉𝑃(𝜕𝐷).
For the reverse implication, suppose 𝑃(𝑧) has no root on the unit circle 𝜕𝐷. Then 𝐹(𝑧) has no zero on the unit circle. So, 𝜆=0 is in the resolvent set 𝜌(𝑆,ℓ_∞). Now, by Theorem 1.6, 𝑆=𝑆−0𝐼 is bijective on ℓ_∞, which means that the boundedness of (𝑦_𝑛) implies a bounded unique solution (𝑥_𝑛).

Example 3.2. We can see that 𝐹(𝜕𝐷)=𝑡₀+2𝑡₁cos𝜃+2𝑡₂cos2𝜃+⋯+2𝑡_𝑛cos𝑛𝜃∶𝜃∈[0,𝜋]. (3.2) When 𝑛=1, 𝑆 is a tridiagonal matrix, that is, 𝑆=𝑞𝑟0000⋯𝑟𝑞𝑟000⋯0𝑟𝑞𝑟00⋯00𝑟𝑞𝑟0⋯⋮⋮⋮⋮⋮⋮⋱, (3.3) then 𝜎(𝑆,𝜇)=𝐹(𝜕𝐷) for 𝜇∈{ℓ₁,𝑐₀,𝑐,ℓ_∞}, where 𝐹=𝑞+𝑟(𝑧+𝑧⁻¹). Therefore, 𝜎𝑆,𝜇=𝑞+2𝑟cos𝜃∶𝜃∈[0,𝜋]=𝑞−2𝑟,𝑞+2𝑟, (3.4) which is one of the main results of [29]. [𝑞−2𝑟,𝑞+2𝑟] is the closed line segment in the complex plane with endpoints 𝑞−2𝑟 and 𝑞+2𝑟.

Example 3.3. When 𝑛=2, 𝑆 is a pentadiagonal matrix, that is, 𝑆=𝑞𝑟𝑠0000⋯𝑟𝑞𝑟𝑠000⋯𝑠𝑟𝑞𝑟𝑠00⋯0𝑠𝑟𝑞𝑟𝑠0⋯00𝑠𝑟𝑞𝑟𝑠⋯⋮⋮⋮⋮⋮⋮⋮⋱, (3.5) then 𝐹=𝑞+𝑟(𝑧+𝑧⁻¹)+𝑠(𝑧²+𝑧⁻²) and 𝜎𝑆,𝜇=𝑞+2𝑟cos𝜃+2𝑠cos2𝜃∶𝜃∈[0,𝜋]. (3.6) So the spectrum is a line segment if 𝑟 is a real multiple of 𝑠. It can be proved that, the spectrum is a closed connected part of a parabola if 𝑟 is not a real multiple of 𝑠. For example, if 𝑞=𝑟=1 and 𝑠=𝑖 (the complex number 𝑖) we have 𝜎𝑆,𝜇={1+2cos𝜃+2𝑖cos2𝜃∶𝜃∈[0,𝜋]}=𝑥,𝑦∈ℝ²∶𝑦=𝑥²−2𝑥−1,𝑥∈[−1,3]. (3.7)