The Fine Spectra of the Difference Operator Δ Over the Sequence

Acta Mathematica Sinica, English Series Oct., 2007, Vol. 23, No. 10, pp. 1757–1768 Published online: Jul. 4, 2007 DOI: 10.1007/s10114-005-0777-0 Http://www.ActaMath.com

The Fine Spectra of the Difference Operator Δ Over the Sequence Space bvp , (1 ≤ p < ∞)∗ Ali M. AKHMEDOV Baku State University, Department of Mech. & Math., Z. Khalilov Str., 23, P. O. Box 370145, Baku, Azerbaijan E-mail: ali [email protected]

Feyzi BAS ¸ AR ˙ on¨ ¨ In¨ u Universitesi, E˘gitim Fak¨ ultesi, Matematik E˘gitimi B¨ ol¨ um¨ u, Malatya-44280, T¨ urkiye E-mail: [email protected] Abstract The sequence space bvp consisting of all sequences (xk ) such that (xk −xk−1 ) belongs to the space p has recently been introduced by Ba¸sar and Altay [Ukrainian Math. J., 55(1), 136–147(2003)]; where 1 ≤ p ≤ ∞. In the present paper, some results concerning with the continuous dual and f -dual, and the AD-property of the sequence space bvp have been given and the norm of the difference operator Δ acting on the sequence space bvp has been found. The fine spectrum with respect to the Goldberg’s classification of the difference operator Δ over the sequence space bvp has been determined, where 1 ≤ p < ∞. Keywords spectrum of an operator, difference operator, the continuous dual and f -dual of a sequence space, the AD-property and the sequence space bvp MR(2000) Subject Classification 47A10, 47B37

1

Preliminaries, Background and Notation

Let X and Y be the Banach spaces and T : X → Y also be a bounded linear operator. By R(T ), we denote the range of T , i.e., R(T ) = {y ∈ Y : y = T x, 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 (T x), for all f ∈ X ∗ and x ∈ X with T  = T ∗ . Also by Ker(T ), we denote the kernel of a bounded linear operator T . Let X = {θ} be a non-trivial complex normed space and T : D(T ) → X a linear operator defined on a subspace D(T ) ⊆ X. We do not assume that D(T ) is dense in X, or that T has a closed graph {(x, T x) : x ∈ D(T )} ⊆ X × X. We mean by the expression “T is invertible” that there exists a bounded linear operator S : R(T ) → X for which ST = I on D(T ) and R(T ) = X, such that S = T −1 is necessarily uniquely determined, and linear; the boundedness of S means that T must be bounded below, in the sense that there is k > 0 for Received January 5, 2005, Revised September 6, 2005, Accepted September 19, 2005 ¨ ITAK–BAYG ˙ The authors wish to express their thanks to TUB for supplying the financial support by PC-B Programme during the preparation of the present work in winter 2004 and for their common project This work has been presented in brief in the International Workshop on Analysis and Its Applications, September 07-11, 2004, Mersin, T¨ urkiye

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Akhmedov A. M. and Ba¸sar F.

which T x ≥ kx for all x ∈ D(T ). Associated with each complex number α is the perturbed operator Tα = T − αI, defined on the same domain D(T ) as T . The spectrum σ(T, X) consists of those α ∈ C for which Tα is not invertible, and the resolvent is the mapping from the complement σ(T, X) of the spectrum into the algebra of bounded linear operators on X defined by α → Tα−1 . The name resolvent is appropriate, since Tα−1 helps to solve the equation Tα x = y. Thus, x = Tα−1 y provided Tα−1 exists. More importantly, the investigation of properties of Tα−1 will be basic for an understanding of the operator T itself. Naturally, many properties of Tα and Tα−1 depend on α, and the spectral theory is concerned with those properties. For instance, we shall be interested in the set of all α’s in the complex plane such that Tα−1 exists. Boundedness of Tα−1 is another property that will be essential. We shall also ask for what α’s the domain of Tα−1 is dense in X, to name just a few aspects. A regular value α of T is a complex number such that Tα−1 exists and is bounded, and whose domain is dense in X. For our investigation of T , Tα and Tα−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 α of T . Its complement σ(T, X) = C\ρ(T, X) in the complex plane C is called the spectrum of T . Furthermore, the spectrum σ(T, X) is partitioned into the following three disjoint sets: The point (discrete) spectrum σp (T, X) is the set such that Tα−1 does not exist. A α ∈ σp (T, X) is called an eigenvalue of T . The continuous spectrum σc (T, X) is the set such that Tα−1 exists and unbounded, and the domain of Tα−1 is dense in X. The residual spectrum σr (T, X) is the set such that Tα−1 exists (and may be bounded or not) but the domain of Tα−1 is not dense in 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 shall 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. From Goldberg [2, pp. 58–71], if X is a Banach space and T ∈ B(X), then there are three possibilities for R(T ) and T −1 : (I) R(T ) = X; (II) R(T ) = R(T ) = X; (III) R(T ) = X; and: (1) T −1 exists and is continuous; (2) T −1 exists but is discontinuous; (3) T −1 does not exist. Applying Golberg’s classification to Tα , we have three possibilities for Tα and Tα−1 : (I) Tα is surjective; (II) R(Tα ) = R(Tα ) = X;

Fine Spectra of the Difference Operator Δ

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(III) R(Tα ) = X; and: (1) Tα is injective and Tα−1 is continuous; (2) Tα is injective and Tα−1 is discontinuous; (3) Tα is not injective. If these possibilities are combined in all possible ways, nine different states are created. These are labelled by: I1 , I2 , I3 , II1 , II2 , II3 , III1 , III2 and III3 . If α is a complex number such that Tα ∈ I1 , then α is in the resolvent set ρ(T, X) of T . The further classification gives rise to the fine spectrum of T . If an operator is in state II2 for example, then R(T ) = R(T ) = X and T −1 exists but is discontinuous and we write α ∈ II2 σ(T, X). By a sequence space, we understand a linear subspace of the space w = CN of all complex sequences which contains φ, the set of all finitely non-zero sequences, where N = {0, 1, 2, . . .}. We write ∞ , c and c0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Also by 1 and p , we denote the spaces of all absolutely summable and absolutely p-summable sequences, respectively. bv is the space consisting of all sequences (xk ) such that (xk − xk+1 ) in 1 and bv0 is the intersection of the spaces bv and c0 . A sequence space λ with a linear topology is called a K-space provided each of the maps pi : λ → C defined by pi (x) = xi is continuous for all i ∈ N. A K-space λ is called an FK-space provided λ is a complete linear metric space. An FK-space, whose topology is normable, is called a BK-space. If a normed sequence space λ contains a sequence (bn ) with the property that for every x ∈ λ there is a unique sequence of scalars (αn ) such that   lim x − (α0 b0 + α1 b1 + · · · + αn bn ) = 0, n→∞  then (bn ) is called a Schauder basis (or briefly basis) for λ. The series αk bk which has the  sum x is then called the expansion of x with respect to (bn ), and written as x = αk bk . x

[n]

Given a BK-space λ ⊃ φ, we denote the nth section of a sequence x = (xk ) ∈ λ by n = k=0 xk e(k) , and we say that x has the property AK if

lim x − x[n] λ = 0 (abschnittskonvergent),

n→∞

AD if x ∈ φ (closure of φ ⊂ λ) (abschnittsdicht), (k)

is a sequence whose only non-zero term is a 1 in the kth place for each k ∈ N. If where e one of these properties holds for every x ∈ λ then we say that the space λ has that property. It is trivial that AK implies AD. For example, c0 and p are AK-spaces, and c and ∞ are not AD-spaces, where 1 ≤ p < ∞. Let n, k ∈ N and A = (ank ) be an infinite matrix of complex numbers ank , and write  (Ax)n = ank xk , (n ∈ N, x ∈ D00 (A)), (1.1) k

where D00 (A) denotes the subspace of w consisting of x ∈ w for which the sum on the right side of (1.1) exists as a finite sum. For simplicity of notation, here and in what follows, the summation without limits runs from 0 to ∞. More generally, if μ is a normed sequence space, we can write Dμ (A) for the x ∈ w for which the sum in (1.1) converges in the norm of μ. We

Akhmedov A. M. and Ba¸sar F.

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shall write (λ : μ) = {A : λ ⊆ Dμ (A)} for the space of those matrices which send the whole of the sequence space λ into the sequence space μ in this sense. A sequence x is said to be A-summable to α if Ax converges to α, which is called the A-limit of x. The matrix domain λA of an infinite matrix A in a sequence space λ is defined by   λA = x = (xk ) ∈ w : Ax ∈ λ , which is a sequence space. If A is a triangle, that is ank = 0 if k > n and ann = 0 for all n ∈ N and λ is a sequence space, then f ∈ λ∗A if and only if f = g ◦ A, g ∈ λ∗ .

We shall assume throughout unless stated otherwise that p, q > 1 with p−1 + q −1 = 1 and use the convention that any term with negative subscript is equal to naught.

We summarize the knowledge in the existing literature concerning the spectrum and the fine spectrum of the linear operators defined by some particular limitation matrices over some sequence spaces. Wenger [3] examined the fine spectrum of the integer power of the Ces`aro operator in c. Reade [4] worked the spectrum of the Ces`aro operator in the sequence space c0 . Gonz` alez [5] studied the fine spectrum of the Ces`aro operator in the sequence space p . Okutoyi [6] computed the spectrum of the Ces`aro operator on the sequence space bv. Recently, Yıldırım [7] worked the fine spectrum of the Rhally operators on the sequence spaces c0 and c. Next, Co¸skun [8] studied the spectrum and fine spectrum for the p-Ces`aro operator acting on the space c0 . Akhmedov and Ba¸sar [9, 10] have recently determined, independently of that of Gonz` alez [5], the fine spectrum of the Ces`aro operator in the sequence spaces c0 , ∞ and p , in a different way; respectively, where 1 < p < ∞. Quite recently, de Malafosse [11], and Altay and Ba¸sar [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 |xk | xsr = sup k , (r > 0). k∈N r Also, Akhmedov and Ba¸sar [13], and Altay and Ba¸sar [14] have determined the fine spectrum with respect to the Goldberg’s classification of the difference operator Δ and the generalized difference operator B(r, s) over the sequence spaces p and c0 , c, respectively. In this work, our purpose is to give some results concerning the continuous dual, f -dual and the AD-property of the space bvp and to find the norm of the difference operator Δ ∈ B(bvp ) and to investigate the fine spectrum of the difference operator Δ on the sequence space bvp which is the natural continuation of Altay and Ba¸sar [12], where 1 ≤ p < ∞. 2

The Space bvp of Sequences of p-bounded Variation

In the present section, we wish to give some required knowledge about the sequence space bvp . Prior to giving some new results, we record the main consequences related to the sequence space bvp , given in [15]. The sequence space bvp is defined, in [15], by p     bvp = x = (xk ) ∈ w : xk − xk−1  < ∞ = p (Δ). k

Fine Spectra of the Difference Operator Δ

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It is proved that bvp is a BK-space which is linearly isomorphic to the space p and the inclusion bvp ⊃ p strictly holds. The α-, β- and γ-duals of the space bvp are determined together with the fact that bv2 is the only Hilbert space among the spaces bvp . The basis of the space bvp is also constructed and given by the following lemma:  (k)  Lemma 2.1 [15, Theorem 3.1] Define the sequence b(k) = bn n∈N of the elements of the space bvp , for every fixed k ∈ N, by

0, (n < k) (k) bn = 1, (n ≥ k). Then the sequence {b(k) }k∈N is a basis for the space bvp and any x ∈ bvp has a unique representation of the form  x= λk b(k) , (2.1) k

where λk = xk − xk−1 for all k ∈ N. Now, we may give the corresponding theorem to Theorem 2.5 of [15] with the space c instead of the space ∞ . Theorem 2.2 Neither of the spaces bvp and c includes the other one while the inclusion bv1 ⊂ c strictly holds. Proof Let 1 < p < ∞. Since p ⊂ c is well known and the inclusion bvp ⊃ p strictly holds (see [15, Theorem 2.4]), we have p ⊂ bvp ∩ c. Let us consider the sequences u = (uk ) and x = (xk ) defined by k  1 1 uk = and x2k = √ , p j + 1 k +1 j=0 1 1 x2k+1 = √ + √ , p p k+1 k+2 for all k ∈ N, respectively. Then, we have u = {1/(k + 1)} ∈ p , which gives that u is in bvp but not in c. Nevertheless, x is in c but not in bvp . Hence, the sequence spaces bvp and c overlap but neither contains the other. Since the second part of theorem is trivial, we omit the details. This completes the proof. Theorem 2.3

Define the spaces d1 and dq consisting of all sequences a = (ak ) normed by    n   (2.2) ad1 = sup  aj  < ∞ k,n∈N j=k

and adq =

   ∞ q 1/q   a < ∞ , (1 < q < ∞). j  k

(2.3)

j=k

Then, bv1∗ and bvp∗ are isometrically isomorphic to d1 and dq , respectively. Proof Since it is a routine verification to show that d1 and dq are the Banach spaces with the norms defined by (2.2) and (2.3), we omit the details. To prove the theorem, we should show the existence of a linear and norm preserving bijection from the spaces bv1∗ and bvp∗ to the spaces d1 and dq , respectively.

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Let f ∈ bv1∗ and x = (xk ) ∈ bv1 . Then, we derive, by applying f to (2.1), that ∞    (Δx)k f (b(k) ) = (Δx)k fj , f (x) = k

k

j=k

where fj = f (e(j) ) for each j ∈ N. Therefore, we observe that    n   f  ≤ sup  fj , k,n∈N

(2.4)

j=k

since |f (x)| ≤ xbv1 On the other hand,

   n   · sup  fj . k,n∈N

j=k

   ∞   fj  = |f (b(k) )| ≤ f b(k) bv1 = f   j=k

holds for any k ∈ N, which leads to the consequence that    n   fj . f  ≥ sup  k,n∈N

(2.5)

j=k

Thus, (2.5) shows that f˜ = (f0 , f1 , f2 , . . .) ∈ d1 . Combining (2.4) and (2.5), we obtain that    n  f  = sup  fj , k,n∈N j=k

which is the norm on d1 . Let us take any g˜ = (g0 , g1 , g2 , . . .) in d1 and define f on the space bv1 by  ∞  (Δx)k gj . f (x) = k

j=k

Then, the linearity of f is clear and since       n   ∞  |(Δx)k | gj  ≤ xbv1 · sup  gj , |f (x)| ≤ f is continuous. Hence,

bv1∗

k

k,n∈N

j=k

j=k

is congruent to the space d1 .

Af = a = Let 1 < p < ∞ and define the transformation A : bvp∗ → dq by f → (a0 , a1 , a2 , . . .). The linearity of A is obvious. Additionally, since Af = θ = (0, 0, 0, . . .) trivially implies f = θ = (0, 0, 0, . . .), A is injective. Let a = (ak ) ∈ dq . Consider the functional f defined on the space bvp such that ∞   f (x) = (Δx)k aj . k

Then, f is linear and since |f (x)| ≤

 ∞     (Δx)k  a j  k

≤

j=k



j=k

|(Δx)k |p

k

= xbvp

 1/p      ∞ q 1/q   a j  k

j=k

   ∞ q 1/q   · a , j  k

j=k

Fine Spectra of the Difference Operator Δ

which yields that

1763

   ∞ q 1/q  f  ≤ aj  = adq ,  k

(2.6)

j=k

f is bounded, i.e., f ∈ bvp∗ . Therefore, A−1 a = f ∈ bvp∗ whenever a ∈ dq , which means that A is surjective. Let us take any x = (xk ) ∈ bvp and f ∈ bvp∗ . Then, we have, by applying f to (2.1), that ∞      f (x) = (Δx)k f b(k) = (Δx)k fj , k (j)

where fj = f (e

k

j=k

) for each j ∈ N. Thus, we observe that    ∞ q 1/q  f  ≤ fj  ,  k

since |f (x)| ≤ xbvp

(2.7)

j=k

   ∞ q 1/q  · fj  .  k

j=k

(n) {xk } q−1

(n)

∈ bvp by Let us define the sequence x = ⎧   n n   ⎪  ⎨  fj  sgn fj , (0 ≤ k ≤ n)  (n) (Δx)k = j=k j=k ⎪ ⎩ 0, (k > n). Then, it is clear that q n    n      f |f x(n) | = j  k=0 j=k

≤ f x(n) bvp q 1/p  n   n    = f  · f , j  k=0 j=k

which leads to the consequence that f  ≥

   ∞ q 1/q   f . j  k

(2.8)

j=k

Therefore, we have by combining the results (2.7) and (2.8) that    ∞ q 1/q   f , f  = j  k

j=k

which is the norm on dq , i.e., A is norm preserving. Hence, bvp∗ is isometrically isomorphic to dq and this step completes the proof. Theorem 2.4 Although the sequence space bvp has not the AD-property for p = 1, it has the AD-property for 1 < p < ∞. Proof Note that if f ∈ bvp∗ , then f (x) = g(Δx) for some g ∈ ∗p . Since p has the AK-property  and ∗p ∼ = q , f (x) = k tk (Δx)k for some t = (tk ) ∈ q . For any f ∈ bvp∗ and e(k) ∈ φ, we have  f (e(k) ) = tj (Δe(k) )j = tk − tk+1 ; (k ∈ N). j

Akhmedov A. M. and Ba¸sar F.

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Hence, an application of the Hahn–Banach theorem yields the fact that φ ⊂ bvp is dense in bvp if and only if (tk − tk+1 ) = θ for t ∈ q implies t = θ. For p = 1, consider the sequence e = (1, 1, 1, . . .) ∈ β1 = ∞ . Since Δt e = θ, bv1 does not have the AD-property, where Δt denotes the transpose of the matrix Δ. Since the null space of the operator Δt on q is {θ}, bvp has the AD-property. Therefore, the proof of the theorem is completed. Subsequent to defining the concept of f -duality of a sequence space, we give the f -dual of the space bvp . By λf , we denote the f -dual of a sequence space λ which is defined by   λf = {f (e(k) )} : f ∈ λ∗ . Lemma 2.5 [16, p. 108]

Let λ be an FK-space which contains φ. Then, λ has AD iff λf = λ∗ .

Thus, as an easy consequence of combining Theorem 2.3 or Theorem 2.4 with Lemma 2.5, we have Corollary 2.6

Let 1 < p < ∞. Then q   ∞     f ∗   bvp = bvp = a = (ak ) ∈ w : aj  < ∞ .  k

3

j=k

The Fine Spectrum of the Difference Operator Δ On the Sequence Space bvp

In this section, the fine spectrum with respect to the Goldberg’s classification of the difference operator Δ on the sequence space bvp has been examined and the norm of the difference operator Δ has been found. The difference operator Δ is represented by the matrix ⎤ ⎡ 1 0 0 ... ⎥ ⎢ ⎢ −1 1 0 . . . ⎥ ⎥ ⎢ Δ=⎢ ⎥. ⎢ 0 −1 1 . . . ⎥ ⎣ . .. .. . . ⎦ .. . . . Theorem 3.1

σ(Δ, bvp ) = {α ∈ C : |α − 1| ≤ 1}.

Proof Solving the equation (Δ − αI)x = y, for y ∈ bv1 , we can formally derive that ⎡ 1/(1 − α) ⎢ ⎢ 1/(1 − α)2 ⎢ (Δ − αI)−1 = ⎢ ⎢ 1/(1 − α)3 ⎣ .. .

(3.1) ⎤

0

0

1/(1 − α)

0

1/(1 − α)2 .. .

1/(1 − α) .. .

...

⎥ ... ⎥ ⎥ ⎥. ... ⎥ ⎦ .. ..

Therefore, one can obtain by (3.1) that x = (Δ − αI)−1 y. Thus, we observe that  1 < ∞, (Δ − αI)−1 (bv1 :bv1 ) = |(1 − α)k+1 | k

if and only if |α − 1| > 1. At this step, it is easy to show that k  yj − yj−1 , (k ∈ N), xk − xk−1 = (1 − α)k−j+1 j=0

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and (Δ − αI)−1 is bounded if and only if |α − 1| > 1. It follows from here that σ(Δ, bv1 ) = {α ∈ C : |α − 1| ≤ 1} . Let y = (yk ) ∈ bvp and p > 1. Solve the equation (Δ − αI)x = y. One can calculate that xk =

k  j=0

yj ; (k ∈ N). (1 − α)k+1−j

Therefore, we obtain, for all k ∈ N, that k  |yj − yj−1 |p . |1 − α|p(k+1−j) j=0  Thus, the d’Alambert criterion yields the fact that the series k (k + 1)p−1 /|1 − α|kp converges if |1 − α| > 1. Hence, the equation (Δ − αI)x = y has a unique solution x ∈ bvp if |1 − α| > 1 and (Δ − αI)−1 is a bounded linear operator for such α’s, which is what we wished to prove.

|xk − xk−1 |p ≤ (k + 1)p−1

Prior to giving our theorem related to the norm of the operator Δ acting on the sequence space bvp , we wish to define the spectral radius of a bounded linear operator, which is needed in the proof. The spectral radius rσ (T ) of an operator T ∈ B(X) on a complex Banach space X is the radius of the smallest closed disk centered at the origin of the complex α-plane, i.e., rσ (T ) =

sup

|α|

α∈σ(T, X)

and containing σ(T, X), (see [1, p. 378]). It is obvious that the inequality rσ (T ) ≤ T 

(3.2)

holds for the spectral radius of a bounded linear operator T on a complex Banach space. Theorem 3.2

Δ ∈ B(bvp ) with the norm Δ(bvp :bvp ) = 2.

Proof The linearity of the operator Δ : bvp → bvp is trivial and so is omitted. Let us take any x = (xk ) ∈ bvp . Then, since  1/p |xk − xk−1 − (xk−1 − xk−2 )|p Δxbvp = k

≤



p

|xk − xk−1 |

1/p +

 ∞

k

p

1/p

|xk−1 − xk−2 |

k=1

= 2xbvp , we derive from here that Δ(bvp :bvp ) ≤ 2.

(3.3)

Furthermore, because of rσ (Δ) = 2 by Theorem 3.1 we have, with (3.2), that 2 ≤ Δ(bvp :bvp ) .

(3.4)

Therefore, we obtain, by combining the inequalities (3.3) and (3.4), that Δ(bvp :bvp ) = 2, as desired. This step concludes the proof. We should remark for the reader that from now on the index p has different meanings in the notation of the spaces bvp , bvp∗ and in the point spectrums σp (Δ, bvp ), σp (Δ∗ , bvp∗ ), which occur in the following two theorems.

Akhmedov A. M. and Ba¸sar F.

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Theorem 3.3

σp (Δ, bvp ) = ∅.

Proof Suppose that Δx = αx for x = θ = (0, 0, 0, . . .) in bvp . Then, by solving the system of linear equations ⎫ x0 = αx0 ⎪ ⎪ ⎪ ⎪ x1 − x0 = αx1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ x2 − x1 = αx2 ⎪ , .. ⎪ . ⎪ ⎪ ⎪ ⎪ xk − xk−1 = αxk ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎭ . we find that, if xn0 is the first non-zero entry of the sequence x = (xn ), then α = 1 and xn0 +1 − xn0 = xn0 +1 , which implies that xn0 = 0. This contradicts the fact that xn0 = 0, which means that σp (Δ, bvp ) = ∅ and this completes the proof. Theorem 3.4

σp (Δ∗ , bvp∗ ) = {α ∈ C : |α − 1| < 1}.

Proof Suppose Δ∗ f = αf for f = θ in bvp∗ , which is isometrically isomorphic to dq by Theorem 2.3. Then, by solving the system of linear equations f0 − f1

=

f1 − f2

=

f2 − f3

= .. .

fk − fk+1

= .. .

⎫ αf0 ⎪ ⎪ ⎪ ⎪ αf1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ αf2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ αfk ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

,

(3.5)

we obtain that fk = (1 − α)k f0 , (k ∈ N). This shows that    q  q   ∞    ∞ q  f0 (1 − α)k q q  q kq  j j  ,    fj  = |f0 |  (1 − α)  = |f0 | |1 − α|  (1 − α)  ≤    α j=k

j

j=k

if |α − 1| < 1. Therefore, one can see that q  q  ∞      f0   fj  ≤   |α − 1|kq < ∞,  α k

j=k

k

if |α − 1| < 1. This means that f ∈ bvp∗ if f0 = 0 and |α − 1| < 1, as asserted. Since the case p = 1 may be proved by analogy to the case p > 1, we omit it. Now, we may give the following lemmas which are needed in the proof of the next theorems: Lemma 3.5 [2, p. 59] of T is one to one.

A linear operator T has a dense range if and only if the adjoint T ∗

Lemma 3.6 [2, p. 60] inverse.

The adjoint operator T ∗ of T is onto if and only if T has a bounded

Theorem 3.7

σr (Δ, bvp ) = {α ∈ C : |α − 1| < 1}.

Proof For |α − 1| < 1, the operator Δ − αI is one to one, hence it has an inverse. However,

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Δ∗ −αI is not one to one by Theorem 3.4. Now, Lemma 3.5 yields the fact that R(Δ − αI) = bvp and this step concludes the proof. Theorem 3.8

1 ∈ III1 σ(Δ, bvp ).

Proof By Theorem 3.7, Δ − I ∈ III. Additionally, α = 1 is not in σp (Δ, bvp ) by Theorem 3.3. Hence, Δ − I has an inverse. Thus, Δ − I ∈ 1 ∪ 2. For establishing the fact Δ − I ∈ 1, it is enough to show by Lemma 3.6 that Δ∗ − I is onto. For a given g = (gk ) ∈ bvp∗ we must find that f = (fk ) ∈ bvp∗ such that (Δ∗ − I)f = g. A direct calculation yields that fn = −gn−1 for all n ∈ N. This means that Δ∗ − I is onto, as desired. Theorem 3.9

If α = 1 and α ∈ σr (Δ, bvp ), then α ∈ III2 σ(Δ, bvp ).

Proof Since α = 1, the operator Δ − αI is a triangle, hence it has an inverse. Take e(0) = (1, 0, 0, . . .) ∈ bvp . Then, the (Δ − αI)−1 -transform of the sequence e(0) is the sequence y = ((1 − α)−1 , (1 − α)−2 , (1 − α)−3 , . . .) which is not in bvp . This shows that (Δ − αI)−1 is discontinuous. Therefore, Δ − αI ∈ 2. Furthermore, Δ∗ − αI is not one to one by Theorem 3.4 and Δ − αI does not have a dense range by Lemma 3.5. Hence, Δ − αI ∈ III. This completes the proof of the Theorem. Theorem 3.10

σc (Δ, bvp ) = {α ∈ C : |α − 1| = 1}.

Proof Since α = 1, Δ − αI is a triangle and has an inverse. Therefore, Δ∗ − αI is one to one from Lemma 3.5 which is what we wished to prove. Theorem 3.11

If α ∈ σc (Δ, bvp ) then α ∈ II2 .

Proof Let us take α ∈ σc (Δ, bvp ). In order to prove that α ∈ II2 we must show that Δ − αI is not onto, R(Δ − αI) = bvp and (Δ − αI)−1 is discontinuous. In the case α = 0, since the proof is similar to the proof of Theorem 3.9, we omit the details. Also in the case α = 0, the (Δ − αI)−1 -transform of the sequence e = (1, 1, 1, . . .) is the sequence y = (k + 1) which is not in the space bvp . These mean that Δ − αI is in 2 but not in I. Besides, we have by Theorem 3.10 that R(Δ − αI) = bvp . This step completes the proof. Combining Theorems 3.1–3.3 and Theorems 3.7 and 3.10, we have the following main theorem: Theorem 3.12

(a) Δ ∈ B(bvp ) with the norm Δ(bvp :bvp ) = 2;

(b) σ(Δ, bvp ) = {α ∈ C : |α − 1| ≤ 1}; (c) σp (Δ, bvp ) = ∅; (d) σr (Δ, bvp ) = {α ∈ C : |α − 1| < 1}; (e) σc (Δ, bvp ) = {α ∈ C : |α − 1| = 1}. Acknowledgements We have benefited a lot from discussions with Dr. Bilˆ al Altay, Matem˙ ¨ atik E˘ gitimi B¨ ol¨ um¨ u, In¨ on¨ u Universitesi, Malatya-44280/T¨ urkiye, about this work, especially in proving Theorem 2.4 and obtaining the inequality (3.4). We would like to express our gratitude for this valuable help. Finally, we thank the reviewers for their careful reading and making some useful comments and suggesting the revision of the manuscript with respect to the Goldberg’s classification, which have improved the presentation of the paper.

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References [1] Kreyszig, E.: Introductory Functional Analysis with Applications, John Wiley & Sons Inc. New YorkChichester-Brisbane-Toronto, 1978 [2] Goldberg, S.: Unbounded Lineer Operators, Dover Publications Inc. New York, 1985 [3] Wenger, R. B.: The fine spectra of H¨ older summability operators. Indian J. Pure Appl. Math., 6, 695–712 (1975) [4] Reade, J. B.: On the spectrum of the Cesaro operator. Bull. Lond. Math. Soc., 17, 263–267 (1985) [5] Gonz` alez, M.: The fine spectrum of the Ces` aro operator in p (1 < p < ∞). Arch. Math., 44, 355–358 (1985) [6] Okutoyi, J. T.: On the spectrum of C1 as an operator on bv. Commun. Fac. Sci. Univ. Ank., Ser. A1 , 41, 197–207 (1992) [7] Yıldırım, M.: On the spectrum and fine spectrum of the compact Rhally operators. Indian J. Pure Appl. Math., 27(8), 779–784 (1996) [8] Co¸skun, C.: The spectra and fine spectra for p-Ces` aro operators. Turkish J. Math., 21, 207–212 (1997) [9] Akhmedov, A. M., Ba¸sar, F.: On spectrum of the Cesaro operator. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 19, 3–8 (2003) [10] Akhmedov, A. M., Ba¸sar, F.: On the fine spectrum of the Ces` aro operator in c0 . Math. J. Ibaraki Univ., 36, 25–32 (2004) [11] de Malafosse, B.: Properties of some sets of sequences and application to the spaces of bounded difference sequences of order μ. Hokkaido Math. J., 31, 283–299 (2002) [12] Altay, B., Ba¸sar, F.: On the fine spectrum of the difference operator on c0 and c. Inform. Sci., 168, 217–224 (2004) [13] Akhmedov, A. M., Ba¸sar, F.: On the fine spectra of the difference operator Δ over the sequence space p , (1 ≤ p < ∞). Demonstratio Math., 39 (2006), to appear [14] Altay, B., Ba¸sar, F.: On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c0 and c. Internat. J. Math. & Math. Sci., 18, 3005–3013 (2005) [15] Ba¸sar, F., Altay, B.: On the space of sequences of p-bounded variation and related matrix mappings. Ukrainian Math. J., 55(1), 136–147 (2003) [16] Wilansky, A.: Summability through Functional Analysis, North-Holland Mathematics Studies, 85, Amsterdam-New York-Oxford, 1984