Math. Sci. Lett. 3, No. 3, 215-221 (2014)
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Mathematical Sciences Letters An International Journal http://dx.doi.org/10.12785/msl/030313
Spectrum and Fine Spectrum Generalized Difference Operator Over The Sequence Space ℓ1 Ali Karaisa∗ Department of Mathematics-Computer Science, Faculty of Sciences, Necmettin Erbakan University, Meram Campus , 42090 Meram, Konya, Turkey Received: 28 Mar. 2014, Revised: 8 May 2014, Accepted: 10 May 2014 Published online: 1 Sep. 2014
Abstract: In this paper, we examined the fine spectrum of upper triangular double-band matrices over the sequence spaces ℓ1 . Also, we determined the point spectrum, the residual spectrum and the continuous spectrum of the operator A(e r, se) on ℓ1 . Further, we derived the approximate point spectrum, defect spectrum and compression spectrum of the matrix operator A(e r, se) over the space ℓ1 . Keywords: Spectrum of an operator, double sequential band matrix, spectral mapping theorem, the sequence space ℓ1 , Goldberg’s classification.
1 Introduction, notations and known results 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. 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. Ces`aro operator of order one on the sequence space ℓ p studied by Gonz`alez [16], where 1 < p < ∞. Also, weighted mean matrices of operators on ℓ p have been investigated by Cartlidge [12]. The spectrum of the Ces`aro operator of order one on the sequence spaces bv0 and bv investigated by Okutoyi [21, 22]. The spectrum and fine spectrum of the Rhally operators on the sequence spaces ℓ p , examined by Yıldırım [24]. The fine spectrum of the difference operator ∆ over the sequence spaces c0 and c studied by Altay and Bas¸ar [4]. The same authors also worked the fine spectrum of the generalized difference operator B(r, s) over c0 and c, in [5]. Recently, the fine spectra of the difference operator ∆ over the sequence spaces ℓ p and ∗ Corresponding
bv p studied by Akhmedov and Bas¸ar [1,2], where bv p is the space consisting of the sequences x = (xk ) such that x = (xk − xk−1 ) ∈ ℓ p and introduced by Bas¸ar and Altay [9] with 1 ≤ p ≤ ∞. In the recent paper, Furkan [13] has studied fine spectrum of B(r, s,t) over the sequence spaces ℓ p and bv p with 1 < p < ∞, where B(r, s,t) is a lower triangular triple-band matrix. Later, Karakaya and Altun have determined the fine spectra of upper triangular double-band matrices over the sequence spaces c0 and c, in [19]. Quite recently, Karaisa [6] have determined the fine spectrum of the generalized difference operator A(e r, se), defined as a upper triangular double-band matrix with the convergent sequences e r = (rk ) and se = (sk ) having certain properties, over the sequence space ℓ p , where 1 < p < ∞. Finally, Karaisa and Bas¸ar [17,18] have determined the fine spectrum of the upper triangular triple-band matrix A(r, s,t) over the sequence space ℓ p , where 0 < p < ∞. Further informations on the spectrum and fine spectra of different operators over some sequence spaces can be found in the list of references [3,7,10,11, 14,23] In this paper, we study the spectrum and fine spectrum of the generalized difference operator A(e r, se) defined by a double sequential band matrix acting on the sequence space ℓ1 with respect to the Goldberg’s classification. Additionally, we give the approximate point spectrum, defect spectrum.
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By ω , we denote the space of all complex valued sequences. Any vector subspace of ω is called a sequence space. We write ℓ∞ , c0 , c and bv for the spaces of all bounded, convergent, null and bounded variation sequences, respectively, which are the Banach spaces with the sup-norm kxk∞ = sup|xk | and kxkbv
∞
=∑
k∈N
|xk − xk+1 |,
respectively,
Table 1: Subdivisions of spectrum of a linear operator. 2 Tα−1 exists and is unbounded –
3 Tα−1 does not exist α ∈ σ p (T, X) α ∈ σap (T, X)
α ∈ σc (T, X) α ∈ σap (T, X) α ∈ σδ (T, X) α ∈ σr (T, X) α ∈ σap (T, X) α ∈ σδ (T, X) α ∈ σco (T, X)
α ∈ σ p (T, X) α ∈ σap (T, X) α ∈ σδ (T, X) α ∈ σ p (T, X) α ∈ σap (T, X) α ∈ σδ (T, X) α ∈ σco (T, X)
A
R(α I − T ) = X
α ∈ ρ (T, X)
B
R(α I − T ) = X
α ∈ ρ (T, X)
C
R(α I − T ) 6= X
α ∈ σr (T, X) α ∈ σδ (T, X)
where
k=0
N = {0, 1, 2, . . .}. Also by ℓ1 and ℓ p , we denote the spaces of all absolutely summable and p-absolutely summable sequences, which are the Banach spaces with the norm 1/p ∞ p kxk p = ∑ |xk | , respectively, where 1 ≤ p < ∞.
1 Tα−1 exists and is bounded
α ∈ σco (T, X)
k=0
Let X and Y is a Banach space and T : X → Y be a bounded linear operator. By R(T ), we denote 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 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. Let X 6= {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 Tα = T − α I, where α is a complex number and I is the identity operator on D(T ). If Tα has an inverse that is linear, we denote it by Tα−1 , that is Tα−1
= (T − α I)
−1
and call it the resolvent operator of T . Many properties of Tα and Tα−1 depend on α , and spectral theory is concerned with those properties. For instance, we shall be interested in the set of all α in the complex plane such that Tα−1 exists. The boundedness of Tα−1 is another property that will be essential. We shall also ask for what α the domain of Tα−1 is dense in X, to name just a few aspects For our investigation of T , Tα and Tα−1 , we need some basic concepts in spectral theory which are given as follows (see [20, pp. 370-371]): Let X 6= {θ } be a complex normed space and T : D(T ) → X be a linear operator with domain D(T ) ⊆ X. A regular value α of T is a complex number such that (R1)Tα−1 (R2)Tα−1 (R3)Tα−1
exists, is bounded, is defined on a set which is dense in X.
The resolvent set ρ (T ) of T is the set of all regular values α of T . Its complement 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 Tα−1 does not exist. α ∈ σ p (T ) is called an eigenvalue of T. The continuous spectrum σc (T ) is the set such that Tα−1 exists and satisfies (R3) but not (R2). The residual spectrum σr (T ) is the set such that Tα−1 exists but not satisfy (R3).
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In this section, following Appell et al. [8], we define the three more subdivisions of the spectrum called as the approximate point spectrum, defect spectrum and compression spectrum. Given a bounded linear operator T in a Banach space X, we call a sequence (xk ) in X as a Weyl sequence for T if kxk k = 1 and kT xk k → 0, as k → ∞. In what follows, we call the set
σap (T, X) := {α ∈ C : there exists a Weyl sequence for α I − T } (1) the approximate point spectrum of T . Moreover, the subspectrum
σδ (T, X) := {α ∈ C : α I − T is not surjective}
(2)
is called defect spectrum of T . The two subspectra given by (1) and (2) form a (not necessarily disjoint) subdivisions
σ (T, X) = σap (T, X) ∪ σδ (T, X) of the spectrum. There is another subspectrum,
σco (T, X) = {α ∈ C : R(α I − T ) 6= X} which is often called compression spectrum in the literature. By the definitions given above, we can illustrate the subdivisions spectrum in the following table: From Goldberg [15] if T ∈ B(X), X a Banach space, then there are three possibilities for R(T ) the range of T : (A) R(T ) = X. (B) R(T ) 6= R(T ) = X. (C) R(T ) = 6 X. and 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 by: A1 , A2 , A3 , B1 , B2 , B3 , C1 , C2 , C3 . If α is a complex number such that Tα ∈ A1 or Tα ∈ B1 then α is in the resolvent set ρ (X, T ) of T . The further classification gives
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rise to the fine spectrum of T . If an operator is in state B2 for example, then R(T ) 6= R(T ) = X and T −1 exists but is discontinuous and we write α ∈ B2 σ (X, T ). Let µ and γ be two sequence spaces and A = (ank ) be an infinite matrix of real or complex numbers ank , where n, k ∈ N = {0, 1, 2, . . .}. 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 = (xk ) ∈ µ the sequence Ax = {(Ax)n }, the A−transform of x is in γ ; where (Ax)n = ∑ ank xk for each n ∈ N.
(3)
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 (3) converges for each n ∈ N and every x ∈ µ , and we have Ax = {(Ax)n }n∈N ∈ γ for all x ∈ µ . Proposition 1.1. [8, Proposition 1.3, p. 28] Spectra and subspectra of an operator T ∈ B(X) and its adjoint T ∗ ∈ 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)σ p (T ∗ , X ∗ ) = σco (T, X). (f)σco (T ∗ , X ∗ ) ⊇ σ p (T, X). (g)σ (T, X) = σap (T, X) σ p (T, X) ∪ σap(T ∗ , X ∗ ).
Therefore, we introduce the operator A(e r, se) from ℓ1 to itself by A(e r, se)x = (rk xk + sk xk+1 )∞ k=0 where x = (xk ) ∈ ℓ1 .
2 The fine spectrum of the operator A(e r, se) over the sequence space ℓ1
Theorem 2.1. The operator A(e r, se) : ℓ1 → ℓ1 is a bounded linear operator and kA(e r, sekℓ1 = sup |rk | + sup |sk |. k∈N
k
∪
σ p (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 on Hilbert spaces are most similar to matrices in finite dimensional spaces (see [8]). Lemma 1.1.[15, p. 60] The adjoint operator T ∗ of T is onto if and only if T is a bounded operator. Let e r = (rk ) and se = (sk ) be sequences whose entries either constants or distinct none-zero real numbers satisfying the following conditions: lim rk = r,
k→∞
lim sk = s 6= 0,
k→∞
|rk − r| 6= |s|. Then, we define the sequential generalized difference matrix A(e r , se) by r0 s0 0 0 . . . 0 r1 s1 0 . . . 0 0 r s ... 2 2 A(e r, se) = 0 0 0 r ... . 3 .. .. .. .. . . . . . . .
217
k∈N
Proof. The proof is simple. So we omit detail. Throughout the paper, by C and S D we denote the set of constant sequences and the set of sequences of distinct none-zero real numbers, respectively. Theorem 2.2. (i) If e r, se ∈ C , r , se), ℓ1 ) = {α ∈ C : |r − α | < |s|} . σ p (A(e (ii) If e r , se ∈ S D, o n n < 1 ⊆ σ p (A(e α ∈ C : supn∈N α −r r, se), ℓ1 ). sn (iii) If e r, se ∈ S D,n o n σ p (A(e r , se), ℓ1 ) ⊆ α ∈ C : infn∈N α −r < 1 . sn iv) If e r, se ∈ S D, r, se), ℓ1 ). {rk : k ∈ N} ⊆ σ p (A(e (v)If e r, se ∈ S D, r, se), ℓ1 ). {α ∈ C : |r − α | < |s|} ⊆ σ p (A(e Proof. Let A(e r, se)x = α x for θ 6= x ∈ ℓ1 . Then, by solving linear equation r0 x0 + s0 x1 = α x0 r1 x1 + s1 x2 = α x1 r2 x2 + s2 x3 = α x2 .. . rk−1 xk−1 + sk−1 xk = α xk .. . α −rk−1 xk = sk−1 xk−1 for all k ≥ 1 and (α − rk−1 )(α − rk−2 ) · · · (α − r1 )(α − r0 ) x0 . xk = sk−1 sk−2 . . . s1 s0 (i) Assume that e r , se ∈ C . Let rk = r and sk = s for all k k ∈ N. We observe that xk = α −r x0 . This shows that s x ∈ ℓ1 if and only if |α − r| < |s|, as asserted. n (ii) Let e r , se ∈ S D and for α ∈ C, supn∈N α −r < 1. So sn we have ∞
∑ |xk |
k=0
(rk−1 − α )(rk−2 − α ) · · · (r0 − α ) |x0 | = |x0 | + ∑ sk−1 sk−2 . . . s0 k=1 ∞ α − rn k |x0 |. ≤ |x0 | + ∑ sup sn k=1 n∈N ∞
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Hence, x = (xk ) ∈ ℓ1 . (iii) Let e r , se ∈ S D and x = (xk ) ∈ ℓ1 . Thus, ∞
∑ |xk |
k=0
(rk−1 − α )(rk−2 − α ) · · · (r0 − α ) |x0 | = |x0 | + ∑ sk−1 sk−2 . . . s0 k=1 ∞ α − rn k |x0 |. ≥ |x0 | + ∑ inf n∈N sn k=1 ∞
(4)
If we use of (4) and we consider x = (xk ) ∈ ℓ1 , inequality n infn∈N α −r < 1. sn (iv) Let e r, se ∈ S D. It is clear that, for all k ∈ N, the vector x = (x0 , x1 , . . . , xk , 0, 0, . . .) is an eigenvector of the operator A(e r, se) corresponding to the eigenvalue α = rk , where x0 6= 0 and xn =
α −rn sn−1
xn−1 , for 1 ≤ n ≤ k. Thus
{rk : k ∈ N} ⊆ σ p (A(e r, se), ℓ1 ). (v) Let e r, se∈ S D and |α − r| < |s|. Since limk→∞ x xk = k−1 r −α limk→∞ k−1 = r−α < 1, x ∈ ℓ1 . This completes the sk−1
s
proof.
Theorem 2.3. σ p (A(e r , se)∗ , ℓ∗1 ) =
0/ , se,e r ∈ C, where, B , se,e r∈SD
B = {rk : k ∈ N, |r − rk | > |s|}. Proof. We prove the theorem by dividing into two parts. Part 1. Assume that se,e r ∈ C . Consider A(e r, se)∗ f = α f ∗ for f 6= θ = (0, 0, 0, . . .) in ℓ1 = ℓ∞ . Then, by solving the system of linear equations r0 f0 = α f0 s0 f0 + r1 f1 = α f1 s1 f1 + r2 f2 = α f2 .. .
for all k ≥ 1. Since ℓ1 ⊆ ℓ∞ , we can applying ratio test and we have sk−1 s fk ≤ 1. = = lim lim k→∞ rk − r0 k→∞ f k−1 r − r0 s But our assumption r−r 6= 1. Hence, 0 α = r0 ∈ {rk : k ∈ N, |rk − r| > |s|} = B. Similarly we can prove that α = rk ∈ {rk : k ∈ N, |rk − r| > |s|} = B, for α = rk 6= r for all k ∈ N1 . Conversely, let α ∈ B. Then, exists k ∈ N, α = rk 6= r and fn s sn = lim = lim r − r < 1. n→∞ f n−1 n→∞ rn+1 − r k
xk =
syk−1 yk sk−1 y0 + ···− + . (r − α )k (r − α )2 r − α
We get, k
k−i
sk−1 fk−1 + rk fk = α fk .. .
1 xk = r−α
we find that f0 = 0 if α 6= r = rk and f1 = f2 = · · · = 0 if f0 = 0 which contradicts f 6= θ . If fn0 is the first non zero entry of the sequence f = ( fn ) and α = r, then we get s fn0 + r fn0 +1 = α fn0 +1 which implies fn0 = 0 which contradicts the assumption fn0 6= 0. Hence, the equation A(e r, se)∗ f = α f has no solution f 6= θ . Part 2. Assume that e r, se ∈ S D. Then, by solving the equation A(e r, se)∗ f = α f for f 6= θ = (0, 0, 0, . . .) in ℓ∞ we obtain (r0 − α ) f0 = 0 and (rk+1 − α ) fk+1 + sk fk = 0 for / {rk : k ∈ N}, we have fk = 0 all k ∈ N. Hence, for all α ∈ / for all k ∈ N, which contradicts our assumption. So, α ∈ r , se)∗ , ℓ∞ ). This shows that σ p (A(e r , se)∗ , ℓ∞ ) ⊆ {rk : σ p (A(e k ∈ N}\{r}. Now, we prove that
for all k ∈ N. Hence, ∞ 1 s i |xk | ≤ ∑ r − α kyk∞. |r − α | i=0
r, se)∗ , ℓ∞ ) if and only if α ∈ B. α ∈ σ p (A(e r, se)∗ , ℓ∞ ). Then, by solving the equation Let α ∈ σ p (A(e ∗ A(e r, se) f = α f for f 6= θ = (0, 0, 0, . . .) in ℓ1 with α = r0 fk =
s0 s1 s2 . . . sk−1 f0 (r0 − rk )(r0 − rk−1 )(r0 − rk−2 ) · · · (r0 − r1 )
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k
That is f ∈ ℓ1 . Since ℓ1 ⊆ ℓ∞ , f ∈ ℓ∞ . So we have r , se)∗ , ℓ∞ ). This completes the proof. B ⊆ σ p (A(e Theorem 2.4 σr (A(e r , se), ℓ1 ) = σ p (A(e r, se)∗ , ℓ∗1 )\σ p (A(e r, se), ℓ1 ). Proof. The proof is obvious so is omitted. Theorem 2.5. Let (rk ), (sk ) in S D and C . r , se), ℓ1 ) = 0. / σr (A(e Proof. r, se), ℓ1 ) = 0. / By Theorem 2.2-2.4, we get σr (A(e Theorem 2.6. σ (A(e r , se), ℓ1 ) = A ∪ B, where A = {α ∈ C : |r − α | ≤ |s|}. Proof. We prove the theorem by dividing into two parts. Part 1. Assume that e r , se ∈ C and y = (yk ) ∈ ℓ∞ . Then, by solving the equation A((e r, se) − α I)∗ x = y for x = (xk ) in terms of y, we obtain
∑
i=0
s r−α
yi
For |s| < |r − α |, we can observe that kxk∞ ≤
1 kyk∞ . |r − α | − |s|
r, se)∗ − α I is onto and by Lemma Thus for |s| < |r − α |, A(e 1.1, A(e r, se) − α I bounded inverse. This means that r , se), ℓ1 ) ⊆ {α ∈ C : |r − α | ≤ |s|}. σc (A(e
Combining this with Theorem 2.2 and Theorem 2.5, we get r , se), ℓ1 ). {α ∈ C : |r − α | < |s|} ⊆ σ (A(e
Since the spectrum of any bounded operator is closed, we have
σ (A(e r, se), ℓ1 ) = {α ∈ C : |r − α | ≤ |s|}.
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Part 2. Assume that e r, se ∈ S D and y = (yk ) ∈ ℓ∞ . Then, by solving the equation A((e r , se) − α I)∗ x = y terms of y, we obtain (−1)k s0 s1 s2 · · · sk−1 y0 xk = + ··· (r0 − α )(r1 − α )(r2 − α ) · · · (rk − α ) yk sk−1 yk−1 + − . (rk − α )(rk−1 − α ) rk − α Then, |xk | ≤ Sk kyk∞ , where 1 sk−1 Sk = + rk − α (rk−1 − α )(rk − α ) sk−1 sk−2 + (rk−2 − α )(rk−1 − α )(rk − α ) s0 s1 . . . sk−1 . +··· + (r0 − α )(r1 − α ) · · · (rk − α )
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Since the spectrum of any bounded operator is closed, we have
σ (A(e r, se), ℓ1 ) = A ∪ B.
This completes the proof. Theorem 2.7. Let (rk ), (sk ) ∈ S D, r , se), ℓ1 ) = {α ∈ C : |r − α | < |s|} ∪ B ∪ H . Where; σ p (A(e ) ( ∞ k α − r i−1 max{k0 , k1 }. Hence, supk∈N Sk < ∞. This shows that (α − r1 )(α − r2 ) · · · (α − rk−1 )y0 kxk∞ ≤ k(Sk )k∞ kyk∞ < ∞, since (yk ) ∈ ℓ∞ . Thus for |s| < + ··· xk = ∗ s0 s1 · · · sk−1 |r − α |, A(e r , se) − α I is onto and by Lemma 1.1 A(e r, se) − α I bounded inverse. This means that (rk−2 − α )yk−2 yk−1 + . + σc (A(e sk−1 sk−2 sk−1 r , se), ℓ1 ) ⊆ {α ∈ C : |r − α | ≤ |s|}. Combining this with Theorem 2.2 and Theorem 2.5, we get B ∪ {α ∈ C : |r − α | < |s|} ⊆ σ (A(e r , se), ℓ1 ).
Then, ∑k |xk | ≤ supk (Tk ) ∑k |yk |, where 1 (rk+1 − α ) (rk+1 − α )(rk+2 − α ) + ··· Tk = + + s ss ss s k
k k+1
k k+1 k+2
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for all k ∈ N. Since |(rk+1 − α )/sk+1 | −→ |s/(r − α )| < 1, as k −→ ∞, then there exists k0 ∈ N and a real number z0 such that |sk+1 /(rk+1 − α )| < z0 for all k ≥ k0 . Then, for all k ≥ k0 + 1, 1 Tk ≤ 1 + z0 + z20 + · · · . |sk | But, there exists k1 ∈ N and a real number z1 such that |1/sk | < z1 for all k ≥ k1 . Then, T k ≤ z1 /(1 − z0 ), for all k > max{k0 , k1 }. Thus, supk∈N T k < ∞. Therefore, (Tk ) ∑ |yk | < ∞. ∑ |xk | ≤ sup k∈N k
k
This shows that x = (xk ) ∈ ℓ1 . Thus A(e r , se) − α I) is r, se), ℓ1 )A3 . onto. So we have α ∈ σ (A(e Theorem 2.10. Let (rk ), (sk ) ∈ C with rk = r, sk = s for all k ∈ N. Then, the following statements hold: r, se), ℓ1 ), (i)σap (A(e r, se), ℓ1 ) = σ (A(e (ii)σδ (A(e r, se), ℓ1 ) = {α ∈ C : |r − α | = |s|}, r, se), ℓ1 ) = 0. / (iii)σco (A(e Proof. (i) From Table 1, we obtain
r , se), ℓ1 ) = σ (A(e σap (A(e r, se), ℓ1 ) \σ (A(e r, se), ℓ1 )C1 . We have by Theorem 2.5
σ (A(e r , se), ℓ1 )C1 = σ (A(e r, se), ℓ1 )C2 = 0. /
Hence;
r , se), ℓ1 ) = A . σap (A(e
(ii) Since the following equality
σδ (A(e r, se), ℓ1 ) = σ (A(e r , se), ℓ1 )\σ (A(e r, se), ℓ1 ) A3
holds from Table 1, we derive by Theorem 2.6 and Theorem 2.9 that σδ (A(e r , se), ℓ1 ) = {α ∈ C : |r − α | = |s|}. (iii) From Table 1, we have
r , se), ℓ1 ) σco (A(e = σ (A(e r, se), ℓ1 )C1 ∪ σ (A(e r , se), ℓ1 )C2 ∪ σ (A(e r, se), ℓ1 )C3 r, se), ℓ1 ) = 0. / by Theorem 2.3 it is immediate that σco (A(e Theorem 2.11. Let e r, se ∈ S D. Then σap (A(e r , se), ℓ1 ) = A ∪ B, σδ (A(e r , se), ℓ1 ) = {α ∈ C : |r − α | = |s|} ∪ B, σco (A(e r , se), ℓ1 ) = B.
Proof. We have by Theorem 2.3 and Part (e) of Proposition 1.1 that
σ p (A(e r , se)∗ , ℓ∗1 ) = σco (A(e r, se), ℓ1 ) = B.
By Theorem 2.5 and Theorem 2.3, we must have
σ (A(e r , se), ℓ1 )C1 = σ (A(e r, se), ℓ p )C2 = 0. /
Hence, σ (A(e r , se), ℓ1 )C3 = {rk }. Therefore, we derive from Table 1, Theorem 2.6 and Theorem 2.9 that r , se), ℓ1 ) = σ (A(e σap (A(e r, se), ℓ1 ) \σ (A(e r, se), ℓ1 )C1 r, se), ℓ1 ) , = σ (A(e σδ (A(e r , se), ℓ1 ) = σ (A(e r, se), ℓ1 ) \σ (A(e r, se), ℓ1 ) A3 = {α ∈ C : |r − α | = |s|} ∪ B. c 2014 NSP
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Acknowledgement We thank the referees for their careful reading of the original manuscript and for the valuable comments.
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Ali Karaisa has obtained his B.Sc. from Erciyes University, Faculty of Science, Kayseri Turkey, in 1998 − 2003, M.Sc. Fatih University,the institute of sciences and engineering, Istanbul, Turkey in 2008 − 2010. He is working toward the Ph.D. at Fatih University, the institute of sciences and engineering, Istanbul, Turkey. His research interests include Functional Analysis and Topology.
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