7_178_dutta.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 4, 2009, (554-563) ISSN 1307-5543 – www.ejpam.com On Köthe-Toeplitz and Null Duals of Some Differ- ence Sequence Spaces Defined by Orlicz Functions Hemen Dutta Department of Mathematics, A.D.P. College, Nagaon-782002, Assam, India Abstract. The main aim of this paper is to compute Köthe-Toeplitz and Null duals of some difference sequence spaces, defined by means of a fixed sequence of multiplier and by an Orlicz function. Further the coincidence for three pairs of analogous spaces is established. 2000 Mathematics Subject Classifications: 40A05, 40C05, 46A45. Key Words and Phrases: Difference sequence spaces, Orlicz function, Köthe-Toeplitz dual, Null dual. 1. Introduction and Preliminaries Throughout this section w, ℓ∞, ℓ1, c and c0 denote the spaces of all, bounded, absolutel y summable, conver gent and null sequences x = (xk)with complex terms respectively. Email address: hemen_dutta08�rediffmail. om http://www.ejpam.com 554 c© 2009 EJPAM All rights reserved. H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 555 An Orlicz function is a function M : [0,∞) −→ [0,∞), which is continuous, non- decreasing and convex with M(0) = 0, M(x) > 0, for x > 0 and M(x) → ∞, as x →∞. An Orlicz function M is said to satisfy the ∆2-condition for all values of u, if there exists a constant K > 0, such that M(2u)≤ KM(u) (u≥ 0). The above ∆2-condition implies M(lu) ≤ Kl l og2K M(u), for all u> 0, l > 1. For details on integral representation of Orlicz function as well as on complemen- tary Orlicz functions one may refer to [7, 12]. For an Orlicz function M , we have the following inequality: M(λx)< λM(x), for all x ≥ 0 and λ with 0< λ < 1. Lindenstrauss and Tzafriri [9] used the Orlicz function and introduced the se- quence space ℓM as follows: ℓM = {(xk) ∈ w : ∞ ∑ k=1 M( |xk| ρ )<∞, for some ρ > 0}. They proved that ℓM is a Banach space normed by ‖(xk)‖= inf{ρ > 0 : ∞ ∑ k=1 M( |xk| ρ ) ≤ 1}. Let Λ = (λk) be a sequence of non-zero scalars. Then for E a sequence space, the multiplier sequence space E(Λ), associated with the multiplier sequence Λ is defined as E(Λ) = {(xk) ∈ w : (λk xk) ∈ E}. The scope for the studies on sequence spaces was extended by using the notion of associated multiplier sequences. Goes and Goes [4] defined the differentiated H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 556 sequence space dE and integrated sequence space ∫ E for a given sequence space E, using the multiplier sequences (k−1) and (k) respectively. A multiplier sequence can be used to accelerate the convergence of the sequences in some spaces. In some sense, it can be viewed as a catalyst, which is used to accelerate the process of chemical reaction. The notion of difference sequence space was introduced by Kizmaz [6], who stud- ied the difference sequence spaces Z(∆), for Z = ℓ∞, c, c0 and defined as follows: Z(∆) = {x = (xk) ∈ w : (∆xk) ∈ Z}, where ∆x = (∆xk) = (xk − xk+1), for all k ∈ N . In this paper our aim is to investigate some important structures of some spaces which are defined using an Orlicz function and a multiplier sequence. These spaces generalize the spaces Z(∆), for Z = ℓ∞, c, c0 introduced and studied by Kizmaz [6]. Let Λ = (λk) be a non-zero sequence of scalars. Then we define the following sequence spaces for an Orlicz function M : c0(M ,Λ,∆) = {x = (xk) : lim k M( |∆λk xk| ρ ) = 0, for some ρ > 0}, c(M ,Λ,∆) = {x = (xk) : lim k M( |∆λk xk − L| ρ ) = 0, for some L and ρ > 0}, ℓ∞(M ,Λ,∆) = {x = (xk) : sup k M( |∆λk xk| ρ )<∞, for some ρ > 0}, where ∆λk xk = λk xk−λk+1xk+1, for all k ∈ N . It is obvious that c0(M ,Λ,∆)⊂ c(M ,Λ,∆)⊂ ℓ∞(M ,Λ,∆). Throughout the paper X will denote one of the sequence spaces c0, c and ℓ∞. The sequence spaces X (M ,Λ,∆) are Banach spaces normed by ‖x‖∆ = |λ1x1|+ inf{ρ > 0 : sup k M( |∆λk xk| ρ )≤ 1}. Now we shall write∆−1xk = xk− xk−1, for all k ∈ N . It is trivial that (∆λk xk) ∈ X (M) H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 557 if and only if (∆−1λk xk) ∈ X (M). Now for x ∈ X (M ,Λ,∆−1), we define ‖x‖∆−1 = inf{ρ > 0 : sup k M( |∆−1λk xk| ρ ) ≤ 1}. It can be shown that X (M ,Λ,∆) is a BK-space under the norms ‖.‖∆ and ‖.‖∆−1 respectively and it is obvious that the norms ‖.‖∆ and ‖.‖∆−1 are equivalent. Obviously ∆−1 : X (M ,Λ,∆−1) −→ X (M), defined by ∆−1x = y = (∆−1λk xk), is isometric isomorphism. Hence c0(M ,Λ,∆−1), c(M ,Λ,∆−1) and ℓ∞(M ,Λ,∆−1) are isometrically isomor- phic to c0(M), c(M) and ℓ∞(M) respectively. From abstract point of view X (M ,Λ,∆−1) is identical with X (M), for X = c0, c and ℓ∞. The results obtained in the next section also hold for the spaces c0(M ,Λ,∆−1), c(M ,Λ,∆−1) and ℓ∞(M ,Λ,∆−1) as well as for the spaces associated with these three spaces. Now we define the spaces c̃0(M ,Λ,∆), c̃(M ,Λ,∆) and ℓ̃∞(M ,Λ,∆) as follows: c̃0(M ,Λ,∆) is a subspace of c0(M ,Λ,∆) consisting of those x ∈ c0(M ,Λ,∆) such that lim k M( |∆λk xk| d ) = 0 f or each d > 0. Similarly we can define c̃(M ,Λ,∆) and ℓ̃∞(M ,Λ,∆) as subspace of c(M ,Λ,∆) and ℓ∞(M ,Λ,∆) respectively. It is obvious that c̃(M ,Λ,∆) ⊂ c̃(M ,Λ,∆) ⊂ ℓ̃∞(M ,Λ,∆). Also as above we can show that c̃0(M ,Λ,∆), c̃(M ,Λ,∆) and ℓ̃∞(M ,Λ,∆) are isometrically isomorphic to c̃0(M), c̃(M) and ℓ̃∞(M) respectively. Moreover X (M ,Λ) ⊂ X (M ,Λ,∆) and X̃ (M ,Λ) ⊂ X̃ (M ,Λ,∆) which can be shown by using the following inequality: M( |∆λk xk| 2ρ ) ≤ 1 2 M( |λk xk| ρ ) + 1 2 M( |λk+1xk+1| ρ ). H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 558 2. Köthe-Toeplitz and Null Dual Spaces In this section we compute Köthe-Toeplitz or α-dual and Null or N - dual of some difference sequence spaces as described in the preceding section. Let E and F be two sequence spaces. Then the F dual of E is defined as EF = {(xk) ∈ w : (xk yk) ∈ F for all (yk) ∈ E}. For F = ℓ1 and c0, the duals are termed as α-(or Köthe-Toeplitz) dual and N -(or Null) dual of E and denoted by Eα and EN respectively. If X ⊂ Y , then Y z ⊂ X z for z = α, N . Lemma 1. x ∈ ℓ∞(M ,Λ,∆) implies sup k M( |k−1λk xk | ρ )<∞, for some ρ > 0. Proof. Let x ∈ ℓ∞(M ,Λ,∆), then sup k M( |λk xk −λk+1xk+1| ρ )<∞, for some ρ > 0. Then there exists a U > 0 such that M( |λk xk−λk+1xk+1| ρ ) < U , for all k ∈ N . Taking η = kρ, for an arbitrary fixed positive integer k, by the subadditivity of mod- ulus, the monotonicity and convexity of M : M( |λ1x1−λk+1xk+1| η )< 1 k k ∑ l=1 M( |λl x l −λl+1x l+1| ρ )< U . Then the above inequality, the inequality |λk+1xk+1| (k+ 1)ρ ≤ 1 k+ 1 ( |λ1x1| ρ + k |λ1 x1−λk+1xk+1| kρ ) and the convexity of M imply M( |λk+1xk+1| (k+ 1)ρ ) ≤ 1 k+ 1 (M( |λ1 x1| ρ ) + kM( |λ1 x1−λk+1xk+1| kρ )) H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 559 ≤ max{M( |λ1 x1| ρ ), U}<∞ Hence we have the desired result. Lemma 2. x ∈ ℓ∞(M ,Λ,∆) implies sup k k−1|λk xk| <∞. Proof. Proof is obvious by using Lemma 1. Remark 1. Similar results as in Lemma 1 and Lemma 2 hold for ℓ̃∞(M ,Λ,∆) also, where the statement ’for some ρ > 0’ should be replaced by ’for every ρ > 0’. For the next theorem, let D1 = {a = (ak) : ∞ ∑ k=1 k|λ−1 k ak| < ∞}, D2 = {b = (bk) : sup k k−1|λk bk| <∞}. Theorem 1. Let M be an Orlicz function. Then (i) [c(M ,Λ,∆)]α = [ℓ∞(M ,Λ,∆)]α = D1, (ii) [c̃(M ,Λ,∆)]α = [ℓ̃∞(M ,Λ,∆)]α = D1, (iii) Dα 1 = D2. Proof. (i) Let a ∈ D1, then ∞ ∑ k=1 |kλ−1 k ak| < ∞. Now for any x ∈ ℓ∞(M ,Λ,∆) we have sup k |k−1λk xk|<∞. Then we have ∞ ∑ k=1 |ak xk| ≤ sup k |k−1λk xk| ∞ ∑ k=1 |kλ−1 k ak|<∞. Hence a ∈ [ℓ∞(M ,Λ,∆)]α. Thus D1 ⊆ [ℓ∞(M ,Λ,∆)]α (1) Again we know [ℓ∞(M ,Λ,∆)]α ⊆ [c(M ,Λ,∆)]α ⊆ [c0(M ,Λ,∆)]α (2) H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 560 Conversely suppose that a ∈ [c(M ,Λ,∆)]α. Then ∞ ∑ k=1 |ak xk| < ∞, for each x ∈ c(M ,Λ,∆). So we take xk = λ −1 k k, k ≥ 1 then ∞ ∑ k=1 |kλ−1 k ak| = ∞ ∑ k=1 |ak xk|<∞. This implies that a ∈ D1. Thus [c(M ,Λ,∆)]α ⊆ D1. (3) Combining (3) with (1), (2) it follows [c(M ,Λ,∆)]α = [ℓ∞(M ,Λ,∆)]α = D1 This completes the proof of part(i). (ii) Proof is similar to that of part (i). (iii) The proof of the inclusion Dα 1 ⊇ D2 is similar to that of D1 ⊆ [ℓ∞(M ,Λ,∆)]α. For the converse part suppose a ∈ Dα 1 and a /∈ D2. Then we have sup k |k−1λkak|=∞ Hence we can find a strictly increasing sequence (k j) of positive integers k j such that |k−1 j λk j ak j |> j2 for all j ≥ 1 We define the sequence x by xk =    |a−1 k j |, if k = k j 0, otherwise H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 561 Then x ∈ D1, because ∞ ∑ k=1 |kλ−1 k xk|= ∞ ∑ j=1 |k jλ −1 k j a−1 k j | ≤ ∞ ∑ j=1 j−2 <∞ Thus x ∈ D1 but ∞ ∑ k=1 |ak xk|= ∞ ∑ j=1 |ak j xk j |=∞. This is a contradiction to a ∈ Dα 1 . Hence a ∈ D2. This completes the proof. If we take λk = 1, for all k ∈ N in Theorem 1, then we obtain the following corollary. Corollary 1. For X = c and ℓ∞, (i) [X (M ,∆)]α = [X̃ (M ,∆)]α = H1, (ii) Hα 1 = H2, where H1 = {a = (ak) : ∞ ∑ k=1 |kak|<∞} and H2 = {b = (bk) : sup k |k−1 bk|<∞}. For the next theorem, let G1 = {a = (ak) : lim k kλ−1 k ak = 0}. Theorem 2. Let M be an Orlicz function. Then (i) [c(M ,Λ,∆)]N = [ℓ∞(M ,Λ,∆)]N = G1, (ii) [c̃(M ,Λ,∆)]N = [ℓ̃∞(M ,Λ,∆)]N = G1. Proof. (i) Proof is immediate using Lemma 2. (ii) Proof is similar to that of part (i). If we take λk = 1, for all k ∈ N in Theorem 2, then we obtain the following corollary. H. Dutta / Eur. J. Pure Appl. Math, 2 (2009), (554-563) 562 Corollary 2. For X = c and ℓ∞, (i) [X (M ,∆)]N = [X̃ (M ,∆)]N = L1, where L1 = {a = (ak) : lim k kak = 0}. Theorem 3. If M satisfies the ∆2-condition, then we have X (M ,Λ,∆) = X̃ (M ,Λ,∆), for every X = c0, c and ℓ∞. Proof. We give the proof for X = ℓ∞ and for other spaces it will follow on applying similar arguments. To prove the theorem, it is enough to show that ℓ∞(M ,Λ,∆) is a subspace of ℓ̃∞(M ,Λ,∆). Let x ∈ ℓ∞(M ,Λ,∆), then for some ρ > 0, sup k M( |∆λk xk| ρ ) <∞ Therefore M( |∆λk xk| ρ )<∞, for every k ∈ N . Choose an arbitrary η > 0. If ρ ≤ η then M( |∆λk xk | η ) < ∞ for every k ∈ N . Let now η < ρ and put l = ρ η > 1. Since M satisfies the ∆2-condition, there exists a constant K such that M( |∆λk xk| η )≤ K( ρ η )log2 K M( |∆λk xk| ρ ) <∞ for every k ∈ N . Now let us denote S = sup k M( |∆λk xk| ρ )<∞, for the fixed ρ > 0. Then it follows that for every η > 0, we have sup k M( |∆λk xk| η )≤ K( ρ η )log2 K .S <∞. 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