Fine Spectrum of the Generalized Difference Operator 𝐵(𝑟,𝑠) over the Class of Convergent Series

Abstract

We have determined the spectra of the generalized difference operator 𝐵(𝑟,𝑠) over the class of convergent series.

1. Preliminaries and Background

Spectral theory is one of the thrust areas of research in mathematical sciences. Due to its usefulness and application-oriented scope, its importance is not only confined to mathematics but also the theory finds its applications in other fields like aeronautics, electrical engineering, quantum mechanics, structural mechanics and probability theory, ecology, and some others.

Throughout 𝐵(𝑋), we denote the set of all bounded linear operators on 𝑋 into itself. If 𝑇∈𝐵(𝑋), where 𝑋 is a Banach space, then the adjoint operator 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 𝑇∶𝐷(𝑇)→𝑋 be a linear operator defined on 𝐷(𝑇)⊂𝑋, where 𝐷(𝑇) denotes the domain of 𝑇 and 𝑋 is a complex normed linear space. For 𝑇∈𝐵(𝑋) we associate the operator (𝑇−𝛼𝐼) denoted by 𝑇_𝛼 defined on the same domain 𝐷(𝑇), where 𝛼 is a complex number and 𝐼 is the identity operator on 𝐷(𝑇). The inverse operator (𝑇−𝛼𝐼)⁻¹ is denoted by 𝑇_𝛼⁻¹ and known as the resolvent operator of 𝑇. The resolvent set 𝜌(𝑇,𝑋) of 𝑇 is the set of all the regular values 𝛼 of 𝑇, such that 𝑇_𝛼⁻¹ exists, bounded and 𝑇_𝛼⁻¹ is defined on a set which is dense in 𝑋.

Its complement ℂ⧵𝜌(𝑇) in the complex plane ℂ is called the spectrum of 𝑇, denoted by 𝜎(𝑇). Thus the spectrum 𝜎(𝑇) consists of those values of 𝛼∈ℂ, for which 𝑇_𝛼 is not invertible.

The spectrum 𝜎(𝑇) is partitioned into three disjoint sets. The point spectrum 𝜎_𝑝(𝑇) is the set such that 𝑇_𝛼⁻¹ does not exist. Any 𝛼∈𝜎_𝑝(𝑇) is called the eigen value of 𝑇. The continuous spectrum 𝜎_𝑐(𝑇) is the set such that 𝑇_𝛼⁻¹ exists, unbounded and the domain of 𝑇_𝛼⁻¹ is dense in 𝑋. The residual spectrum 𝜎_𝑟(𝑇) is the set such that 𝑇_𝛼⁻¹ exists (and may be bounded or not) and the domain of 𝑇_𝛼⁻¹ is not dense in 𝑋. In finite-dimensional case, continuous spectrum coincides with the residual spectrum, is the empty set and the spectrum consists of only the point spectrum.

Throughout 𝑤, ℓ_∞, 𝑐, 𝑐₀, and ℓ_𝑝 denote the class of all, bounded, convergent, null, and p-absolutely summable sequence of fuzzy real or complex terms.

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∈𝑁₀={0,1,2,…}. We say that 𝐴 defines a matrix mapping from 𝐸 into 𝐹, denoted by 𝐴∶𝐸→𝐹, if for every sequence 𝑥=(𝑥_𝑛)∈𝐸 the sequence 𝐴𝑥={(𝐴𝑥)_𝑛} is in 𝐹, where (𝐴𝑥)_𝑛=∑^∞_𝑘=0𝑎_𝑛𝑘𝑥_𝑘 (𝑛∈𝑁 and 𝑥∈𝐸), provided the right hand side converges for every 𝑛∈𝑁 and 𝑥∈𝐸.

Spectra of some particular types of matrix operators have been investigated from different aspects by Okutoyi [1], Rhoades [2], Tripathy and Saikia [3], Tripathy and Paul [4, 5], and Dutta and Tripathy [6]. Okutoyi [1] studied the spectrum of the Cesàro operator 𝐶₁ on 𝑏𝑣₀. Akhmedov and Başar [7] worked on the spectra of the difference operator Δ over the sequence space 𝑏𝑣_𝑝 (1≤𝑝<∞). Recently Altay and Başar [8] and Furkan et al. [9] determined the spectra of 𝐵(𝑟,𝑠) over 𝑐₀, 𝑐 and ℓ₁, 𝑏𝑣, respectively.

In this paper, we have analyzed the spectra of the generalized difference operator 𝐵(𝑟,𝑠) on 𝛾 of all convergent series. By well-established convention, we define the space 𝛾 of all “convergent series” by 𝛾={(𝑥_𝑘)∈𝑤∶(∑^𝑛_𝑘=0𝑥_𝑘)∈𝑐}.

The generalized difference operator 𝐵(𝑟,𝑠) is represented by 𝐵(𝑟,𝑠)=().𝑟000⋯𝑠𝑟00⋯0𝑠𝑟0⋯⋯⋯⋯⋯⋯𝑠≠0 (1) If 𝑇∶𝛾→𝛾 is a bounded linear operator with the matrix 𝐴, then its adjoint operator 𝑇^∗∶𝛾^∗→𝛾^∗ is defined by transpose of the matrix 𝐴 and 𝛾 is isomorphic to ℓ₁ with the norm ‖𝑥‖=∑^∞_𝑘=0|𝑥_𝑘|.

The following lemma helps us to determines the norm and to verify that 𝐵(𝑟,𝑠)∈𝐵(𝛾).

Lemma 1. The matrix 𝐴=(𝑎_𝑛𝑘) 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. 𝐵(𝑟,𝑠)∶𝛾→𝛾 is a bounded linear operator with ‖𝐵(𝑟,𝑠)‖_(𝛾,𝛾)≤|𝑟|+|𝑠|.

Proof. It can be easily verified that 𝐵(𝑟,𝑠) is linear.
We have ‖𝐵(𝑟,𝑠)‖_(𝛾,𝛾)=sup_𝑛^𝑛_𝑘=0𝑠𝑥_𝑘−1+𝑟𝑥_𝑘,since𝑥_𝑘𝑐𝑠=^𝑛_𝑘=0𝑥_𝑘≤sup_𝑛|𝑠|^𝑛_𝑘=0𝑥_𝑘−1+sup_𝑛|𝑟|^𝑛_𝑘=0𝑥_𝑘≤(|𝑠|+|𝑟|)‖𝑥‖_𝛾⇒‖𝐵(𝑟,𝑠)‖_(𝛾,𝛾)≤(|𝑠|+|𝑟|)‖𝑥‖_𝛾. (2)

Lemma 3 (see [10]). A linear operator 𝑇 has a dense range if and only if its adjoint operator 𝑇^∗ is one to one.

Lemma 4 (see [10]). The adjoint operator 𝑇^∗ of  𝑇 is onto if and only if  𝑇  has a bounded inverse.

Lemma 5 (see [10]). Let 𝑇∈𝐵(𝑋), where 𝑋 is any Banach space. Then the spectrum of 𝑇^∗ is identical with the spectrum of T. Furthermore 𝑅_𝜆(𝑇)=(𝑇−𝜆𝐼)⁻¹ and 𝜌(𝑇)={𝜆∈𝐶∶(𝑇−𝜆𝐼)⁻¹𝑒𝑥𝑖𝑠𝑡}.

Now we shall determine the spectrum, point spectra, continuous spectra, and residual spectra of the operator 𝐵(𝑟,𝑠) on the space 𝛾.

2. Main Result

Theorem 6. 𝜎(𝐵(𝑟,𝑠),𝛾)={𝛼∈𝐶∶|𝛼−𝑟|≤𝑠}.

Proof. We shall prove this theorem by showing that (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹ exists and is in 𝐵(𝛾) for |𝛼−𝑟|>|𝑠| and then show that the operator (𝐵(𝑟,𝑠),𝛾) is not invertible for |𝛼−𝑟|≤|𝑠|.
Let 𝛼 be such that |𝛼−𝑟|>|𝑠|. Since (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹ is a lower triangular matrix, so (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹ exists. Solving  (𝐵(𝑟,𝑠)−𝛼𝐼)𝑥=𝑦 for 𝑥 in terms of 𝑦, we get the following system of equations: 𝑥₀=1𝑦𝑟−𝛼⁰,𝑥₁=1𝑦𝑟−𝛼¹−𝑠(𝑟−𝛼)²𝑦₀,𝑥₂=1𝑦𝑟−𝛼²−𝑠(𝑟−𝛼)²𝑦₁+𝑠²(𝑟−𝛼)³𝑦₀,⋮⋮ (3)
The 𝑛th row is given by 𝑥_𝑛=_𝑘≤𝑛(−𝑠)^𝑛−𝑘(𝑟−𝛼)^{𝑛−𝑘+1}𝑦_𝑘inthe𝑘thplace. (4)
Thus the matrix (𝑎_𝑛𝑘) determined by (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹ is defined by 𝑎_𝑛𝑘=(−𝑠)^𝑛−𝑘(𝑟−𝛼)^{𝑛−𝑘+1},for𝑛≤𝑘,0,otherwise. (5) We observe that ‖𝐵(𝑟,𝑠)−𝛼𝐼‖⁻¹_𝛾,𝛾=sup_𝑛^𝑛_𝑘=0(−𝑠)^𝑛−𝑘(𝑟−𝛼)^{𝑛−𝑘+1}=1𝑟−𝛼sup_𝑛^𝑛_𝑘=0𝑠(𝑟−𝛼)^𝑛−𝑘=1𝑟−𝛼^𝑛_𝑘=0𝑠(𝑟−𝛼)^𝑛<∞. (6) Thus (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹∈𝐵(𝛾).
Now we show that the operator (𝐵(𝑟,𝑠)−𝛼𝐼) is not invertible for |𝛼−𝑟|≤|𝑠|.
Let 𝛼 be such that |𝛼−𝑟|≤|𝑠| and 𝛼≠𝑟. Since (𝐵(𝑟,𝑠)−𝛼𝐼) is triangular, so (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹ exists. From (6) we have ‖𝐵(𝑟,𝑠)−𝛼𝐼‖⁻¹_(𝛾,𝛾)=∞, whenever |𝛼−𝑟|≤|𝑠|, that is (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹∉𝐵(𝛾). When 𝛼=𝑟, the operator (𝐵(𝑟,𝑠)−𝛼𝐼) is represented by the matrix 𝐵(𝑠) given by 𝐵(𝑠)=.000⋯𝑠00⋯0𝑠0⋯⋯⋯⋯⋯ (7) We observe that (𝐵(𝑠)−𝑠𝐼)𝑥=𝜃⇒𝑥=𝜃. Thus (𝐵(𝑠)−𝑠𝐼)∶𝛾→𝛾 is injective but its range is not a dense set. Hence (𝐵(𝑠)−𝑠𝐼) is not invertible. This shows that (𝐵(𝑟,𝑠)−𝛼𝐼) is not invertible for |𝛼−𝑟|≤|𝑠|. This completes the proof.

Theorem 7. 𝜎_𝑝(𝐵(𝑟,𝑠),𝛾)=∅.

Proof. Let 𝐵(𝑟,𝑠)𝑥=𝛼𝑥, for 𝑥≠𝜃=(0,0,0,…) in 𝛾. We have the following system of equations: 𝑟𝑥₀=𝛼𝑥₀𝑠𝑥₀+𝑟𝑥₁=𝛼𝑥₁⋮𝑠𝑥_𝑘+𝑟𝑥_𝑘+1=𝛼𝑥_𝑘+1⋮⋮ (8) If 𝑥_𝑖 is the first nonzero term of the sequence 𝑥=(𝑥_𝑛), 𝑟=𝛼, hence we have 𝑠𝑥_𝑖+𝑟𝑥_𝑖+1=𝛼𝑥_𝑖+1⇒𝑠𝑥_𝑖=0⇒𝑥_𝑖=0,since𝑠≠0. (9) This contradicts to the fact that 𝑥_𝑖≠0.
Thus (𝐵(𝑟,𝑠)−𝛼𝐼)𝑥=𝜃⇒𝑥=𝜃 and hence (𝐵(𝑟,𝑠)−𝛼𝐼)∶𝛾→𝛾 is injective.
That is, 𝜎_𝑝(𝐵(𝑟,𝑠),𝛾)=∅.

Theorem 8. 𝜎_𝑝(𝐵(𝑟,𝑠)^∗,𝛾^∗)={𝛼∈𝐶∶|𝛼−𝑟|<𝑠}.

Proof. Suppose that 𝐵(𝑟,𝑠)^∗𝑥=𝛼𝑥, for 𝑥(≠𝜃)∈𝛾^∗≅ℓ₁. Now we have the following system of equations: 𝑟𝑥₀+𝑠𝑥₁=𝛼𝑥₀𝑟𝑥₁+𝑠𝑥₂=𝛼𝑥₁⋮𝑟𝑥_𝑘+𝑠𝑥_𝑘+1=𝛼𝑥_𝑘⋮⋮ (10)
Solving the above systems of linear equations, we get 𝑥_𝑛=𝛼−𝑟𝑠^𝑛𝑥₀,for𝑛∈𝑁. (11) We observe that 𝑥=(𝑥_𝑛)∈𝛾^∗ if and only if |𝛼−𝑟|<𝑠.
This completes the proof.

Theorem 9. 𝜎_𝑟(𝐵(𝑟,𝑠),𝛾)={𝛼∈𝐶∶|𝛼−𝑟|<𝑠}.

Proof. We prove that the range of (𝐵(𝑟,𝑠)−𝛼𝐼) is not dense in 𝛾 but (𝐵(𝑟,𝑠)−𝛼𝐼) has an inverse, bounded, or unbounded.
Theorem 8 implies that (𝐵(𝑟,𝑠)^∗−𝛼𝐼) is not one-to-one and therefore by Lemma 3, (𝐵(𝑟,𝑠)−𝛼𝐼) is not dense in 𝛾. We have also seen that (𝐵(𝑟,𝑠)−𝛼𝐼)⁻¹ exist and (𝐵(𝑟,𝑠)−𝛼𝐼)≠𝛾 for those 𝛼 satisfying  |𝛼−𝑟|<𝑠. Again for 𝛼≠𝑠, (𝐵(𝑟,𝑠)−𝛼𝐼) is triangular and therefore has an inverse.
This completes the proof.

Theorem 10. 𝜎_𝑐(𝐵(𝑟,𝑠),𝛾)={𝛼∈𝐶∶|𝛼−𝑟|=𝑠}.

Proof. It is well known that 𝜎(𝐵(𝑟,𝑠),𝛾) is the disjoint union of 𝜎_𝑝(𝐵(𝑟,𝑠),𝛾), 𝜎_𝑟(𝐵(𝑟,𝑠),𝛾), and 𝜎_𝑐(𝐵(𝑟,𝑠),𝛾) since 𝜎_𝑝(𝐵(𝑟,𝑠),𝛾)=∅ and 𝜎_𝑟(𝐵(𝑟,𝑠),𝛾) = {𝛼∈𝐶∶|𝛼−𝑟|<𝑠}.
Thus 𝜎_𝑐(𝐵(𝑟,𝑠),𝛾)={𝛼∈𝐶∶|𝛼−𝑟|=𝑠}.
This completes the proof.

3. Conclusion

In this paper, the spectra of matrix operator on convergent series is established. In a similar way the spectra of other matrix operators can also be determined over the convergent series.

Acknowledgments

The authors thank the reviewer for his comments on the paper; those improved the presentation of the paper. The work of the authors is financially supported by the Council of Scientific and Industrial Research, India vide Grant no. 25(0182)/10/EMR-II.