# Fine Spectrum of the Generalized Difference OperatorB(r,s) over the Class of Convergent Series

Authors: Amar Jyoti Dutta; Binod Chandra Tripathy
Journal: International Journal of Analysis
 (2013)

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## Abstract

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


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## Body

## 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.ThroughoutB(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.LetT: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 Tα 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 Tα-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 Tα-1 exists,  bounded and Tα-1 is defined on a set which is dense in X.Its complementℂ∖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 Tα is not invertible.The spectrumσ(T) is partitioned into three disjoint sets. The point spectrum σp(T) is the set such  that Tα-1 does not exist. Any α∈σp(T) is called the eigen value of T. The continuous spectrum σc(T) is the set such that Tα-1 exists, unbounded and the domain of Tα-1 is dense in X. The residual spectrum σr(T) is the set such that Tα-1 exists (and may be bounded or not) and the domain of Tα-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.Throughoutw, ℓ∞, c, c0, and ℓp denote the class of all, bounded, convergent, null, and p-absolutely summable sequence of fuzzy real or complex terms.LetE and F be two sequence spaces and A=(ank) be an infinite matrix of real or complex numbers ank, 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=(xn)∈E the sequence Ax={(Ax)n} is in F, where (Ax)n=∑k=0∞ankxk (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 bvp (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 operatorB(r,s) on γ of all convergent series. By well-established convention, we define the space γ of all “convergent series” by γ={(xk)∈w:(∑k=0nxk)∈c}.The generalized difference operatorB(r,s) is represented by
(1)B(r,s)=(r000⋯sr00⋯0sr0⋯⋯⋯⋯⋯⋯)(s≠0).
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∞|xk|.The following lemma helps us to determines the norm and to verify thatB(r,s)∈B(γ).Lemma 1.
The matrixA=(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.
B
(
r
,
s
)
:
γ
→
γ is a bounded linear operator with ∥B(r,s)∥(γ,γ)≤|r|+|s|.Proof.
It can be easily verified thatB(r,s) is linear.
We have(2)∥B(r,s)∥(γ,γ)=supn|∑k=0nsxk-1+rxk|,since∥(xk)∥cs=|∑k=0nxk|≤supn|s||∑k=0nxk-1|+supn|r||∑k=0nxk|≤(|s|+|r|)∥x∥γ⇒∥B(r,s)∥(γ,γ)≤(|s|+|r|)∥x∥γ.Lemma 3 (see [10]).
A linear operatorT has a dense range if and only if its adjoint operator T* is one to one.Lemma 4 (see [10]).
The adjoint operatorT* of  T is onto if and only if  T  has a bounded inverse.Lemma 5 (see [10]).
LetT∈B(X), where X is any Banach space. Then the spectrum of T* is identical with the spectrum of T. Furthermore Rλ(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 operatorB(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|.
Letα 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:
(3)x0=1r-αy0,x1=1r-αy1-s(r-α)2y0,x2=1r-αy2-s(r-α)2y1+s2(r-α)3y0,⋮⋮
Thenth row is given by
(4)xn=∑k≤n(-s)n-k(r-α)n-k+1ykinthekthplace.
Thus the matrix(ank) determined by (B(r,s)-αI)-1 is defined by
(5)ank={(-s)n-k(r-α)n-k+1,forn≤k,0,otherwise.
We observe that
(6)∥B(r,s)-αI∥(γ,γ)-1=supn∑k=0n|(-s)n-k(r-α)n-k+1|=|1r-α|supn∑k=0n|s(r-α)|n-k=|1r-α|∑k=0n|s(r-α)|n<∞.
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|.
Letα 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∥(γ,γ)-1=∞, 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
(7)B(s)=[000⋯s00⋯0s0⋯⋯⋯⋯⋯].
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.
LetB(r,s)x=αx, for x≠θ=(0,0,0,…) in γ. We have the following system of equations:
(8)rx0=αx0sx0+rx1=αx1⋮sxk+rxk+1=αxk+1⋮⋮
If xi is the first nonzero term of the sequence x=(xn),  r=α, hence we have
(9)sxi+rxi+1=αxi+1⇒sxi=0⇒xi=0,sinces≠0.
This contradicts to the fact that xi≠0.
Thus(B(r,s)-αI)x=θ⇒x=θ and hence (B(r,s)-αI):γ→γ is injective.
That is,σp(B(r,s),γ)=∅.Theorem 8.
σ
p
(
B
(
r
,
s
)
*
,
γ
*
)
=
{
α
∈
C
:
|
α
-
r
|
<
s
}.Proof.
Suppose thatB(r,s)*x=αx, for x(≠θ)∈γ*≅ℓ1. Now we have the following system of equations:
(10)rx0+sx1=αx0rx1+sx2=αx1⋮rxk+sxk+1=αxk⋮⋮
Solving the above systems of linear equations, we get(11)xn=(α-rs)nx0,forn∈N.
We observe that x=(xn)∈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.
Theorem8 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 thatσ(B(r,s),γ) is the disjoint union of σp(B(r,s),γ), σr(B(r,s),γ), and σc(B(r,s),γ) since σp(B(r,s),γ)=∅ and σr(B(r,s),γ) = {α∈C:|α-r|<s}.
Thusσc(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.



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*Source: 630436-2013-01-17.xml*