ON SOME NEW SEQUENCE SPACES OF NON-ABSOLUTE TYPE RELATED TO THE SPACES l p AND l I. M. Mursaleen and Abdullah K. Noman

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1 Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: htt:// Filomat 25:2 20, 33 5 DOI: /FIL02033M ON SOME NEW SEQUENCE SPACES OF NON-ABSOLUTE TYPE RELATED TO THE SPACES l AND l I M. Mursalee ad Abdullah K. Noma Abstract I the reset aer, we itroduce the sequece sace l λ of o-absolute tye ad rove that the saces l λ ad l are liearly isomorhic for 0 <. Further, we show that l λ is a -ormed sace ad a BK-sace i the cases of 0 < < ad, resectively. Furthermore, we derive some iclusio relatios cocerig the sace l λ. Fially, we costruct the basis for the sace l λ, where <. Itroductio By w, we deote the sace of all real or comlex valued sequeces. Ay vector subsace of w is called a sequece sace. A sequece sace X with a liear toology is called a K-sace rovided each of the mas : X C defied by x = x is cotiuous for all N, where C deotes the comlex field ad N = {0,, 2,...}. A K-sace X is called a F K- sace rovided X is a comlete liear metric sace. A F K-sace whose toology is ormable is called a BK-sace [9, ]. We shall write l, c ad c 0 for the sequece saces of all bouded, coverget ad ull sequeces, resectively, which are BK-saces with the same su-orm give by x l = su x k, k where, here ad i the sequel, the suremum su k is take over all k N. Also by l 0 < <, we deote the sequece sace of all sequeces associated with -absolutely coverget series. It is kow that l is a comlete -ormed sace ad a BK-sace i the cases of 0 < < ad < with resect to the usual -orm ad l -orm defied by x l = k x k ; 0 < < 200 Mathematics Subject Classificatios. 40C05, 40H05, 46A45. Key words ad Phrases. Sequece sace; BK-sace; Schauder basis; Matrix maig. Received: December 2, 200 Commuicated by Vladimir Rakocevic

2 34 M. Mursalee ad A. K. Noma ad x l = x k / ; <, k resectively see [,.27-28]. For simlicity i otatio, here ad i what follows, the summatio without limits rus from 0 to. Let X ad Y be sequece saces ad A = a k be a ifiite matrix of real or comlex umbers a k, where, k N. The, we say that A defies a matrix maig from X ito Y if for every sequece x = x k X the sequece Ax = {A x}, the A-trasform of x, exists ad is i Y, where A x = k a k x k ; N. By X : Y, we deote the class of all ifiite matrices that ma X ito Y. Thus A X : Y if ad oly if the series o the right side of coverges for each N ad every x X, ad Ax Y for all x X. For a sequece sace X, the matrix domai of a ifiite matrix A i X is defied by X A = { x w : Ax X } 2 which is a sequece sace. We shall write e k for the sequece whose oly o-zero term is a i the k th lace for each k N. The aroach of costructig a ew sequece sace by meas of the matrix domai of a articular limitatio method has recetly bee emloyed by several authors, e.g., Wag [9], Ng ad Lee [8], Malkowsky [2], Başar ad Altay [7], Malkowsky ad Savaş [3], Aydı ad Başar [3, 4, 5, 6], Altay ad Başar [], Altay, Başar ad Mursalee [2, 4] ad Mursalee ad Noma [5, 6], resectively. They itroduced the sequece saces l Nq ad c Nq i [9], l C = X ad l C = X i [8], l R t = r t, c R t = r t c ad c 0 R t = r t 0 i [2], l = bv i [7], µ G = Zu, v; µ i [3], c 0 A r = a r 0 ad c A r = a r c i [3], [c 0 u, ] A r = a r 0u, ad [cu, ] A r = a r cu, i [4], a r 0 = a r 0 ad a r c = a r c i [5], l A r = a r ad l A r = a r i [6], c 0 E r = e r 0 ad c E r = e r c i [], l E r = e r ad l E r = e r i [2, 4], c 0 Λ = c λ 0 ad c Λ = c λ i [5] ad c λ 0 = c λ 0 ad c λ = c λ i [6], where N q, C, R t ad E r deote the Nörlud, Cesàro, Riesz ad Euler meas, resectively, deotes the bad matrix defiig the differece oerator, G ad A r are defied i [3] ad [3], resectively, Λ is defied i Sectio 2, below, µ {c 0, c, l } ad <. Also c 0 u, ad cu, deote the sequece saces geerated from the Maddox s saces c 0 ad c by Başarır [8]. The mai urose of the reset aer, followig [, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 8] ad [9], is to itroduce the sequece saces l λ ad l λ of o-absolute tye ad is to derive some results related to them. Further, we establish some iclusio relatios cocerig the saces l λ ad l λ, where 0 < <. Moreover, we costruct the basis for the sace l λ, where <.

3 O some ew sequece saces of o-absolute tye 35 2 λ-boudedess ad -absolute covergece of tye λ Throughout this aer, let λ = λ k be a strictly icreasig sequece of ositive reals tedig to, that is 0 < λ 0 < λ < ad λ k as k. 3 We say that a sequece x = x k w is λ-bouded if su Λ x <, where Λ x = λ k λ k x k ; N. 4 Also, we say that the associated series k x k is -absolutely coverget of tye λ if Λ x <, where 0 < <. Here ad i the sequel, we shall use the covetio that ay term with a egative subscrit is equal to zero, e.g., λ = 0 ad x = 0. Now, let x = x k be a bouded sequece i the ordiary sese of boudedess, i.e., x l. The, there is a costat M > 0 such that x k M for all k N. Thus, we have for every N that Λ x M λ k λ k x k λ k λ k = M which shows that x is λ-bouded. Therefore, we deduce that the ordiary boudedess imlies the λ-boudedess. This leads us to the followig basic result: Lemma 2.. Every bouded sequece is λ-bouded. We shall later show that the coverse imlicatio eed ot be true. Further, we shall show that for every 0 < < there is a sequece λ = λ k satisfyig 3 such that the covergece of the series x k does ot imly the covergece of the series Λ x, ad coversely. Before that, we defie the ifiite matrix Λ = k, by λ k λ k ; 0 k, k = 5 0 ; k > for all, k N. The, for ay sequece x = x k w, the Λ-trasform of x is the sequece Λx = {Λ x}, where Λ x is give by 4 for all N. Therefore, the sequece x is λ-bouded if ad oly if Λx l. Also, the otio of -absolute covergece of tye λ of the sequece x is equivalet to say that Λx l, where 0 < <. Further, it is obvious by 5 that the matrix Λ = k is a triagle, i.e., 0 ad k = 0 for all k > N.

4 36 M. Mursalee ad A. K. Noma Recetly, the sequece saces c λ 0 ad c λ have bee defied i [5] as the matrix domais of the triagle Λ i the saces c 0 ad c, resectively, that is { } c λ 0 = x = x k w : λ k λ k x k = 0 ad c λ = { x = x k w : lim lim } λ k λ k x k exists. Also, it has bee show that the iclusios c 0 c λ 0 ad c c λ hold ad the iclusio c λ 0 c λ strictly holds. Fially, we defie the sequece yλ = {y k λ}, which will be frequetly used, as the Λ-trasform of a sequece x = x k, i.e., yλ = Λx ad so we have y k λ = k j=0 λj λ j x j ; k N. 6 λ k 3 The sequece saces l λ ad lλ of o-absolute tye I the reset sectio, as a atural cotiuatio of Mursalee ad Noma [5], we itroduce the sequece saces l λ ad l λ, as the sets of all sequeces whose Λ-trasforms are i the saces l ad l, resectively, where 0 < <, that is { l λ } = x = x k w : λ k λ k x k < ; 0 < < ad l λ = { =0 x = x k w : su } λ k λ k x k <. With the otatio of 2, we ca redefie the saces l λ ad l λ as follows: l λ = l Λ 0 < < ad l λ = l Λ. 7 The, it is obvious by 7 that l λ ad l λ 0 < < are sequece saces cosistig of all sequeces which are λ-bouded ad -absolutely coverget of tye λ, resectively. Further, we have the followig result which is essetial i the text. Theorem 3.. We have the followig: a If 0 < <, the l λ is a comlete -ormed sace with the -orm x l λ = Λx l, that is x l λ = Λ x ; 0 < <. 8

5 O some ew sequece saces of o-absolute tye 37 b If, the l λ is a BK-sace with the orm x l λ = Λx l, that is x l λ = Λ x / ; < 9 ad x l λ = su Λ x. 0 Proof. Sice the matrix Λ is a triagle, this result is immediate by 7 ad Theorem of Wilasky [20,.63]. Remark 3.2. Oe ca easily check that the absolute roerty does ot hold o the sace l λ, that is x l λ x l λ for at least oe sequece i the sace l λ, ad this tells us that l λ is a sequece sace of o-absolute tye, where x = x k ad 0 <. Theorem 3.3. The sequece sace l λ of o-absolute tye is isometrically isomorhic to the sace l, that is l λ = l for 0 <. Proof. To rove this, we should show the existece of a isometric isomorhism betwee the saces l λ ad l, where 0 <. For, let 0 < ad cosider the trasformatio T defied, with the otatio of 6, from l λ to l by x yλ = T x. The, we have T x = yλ = Λx l for every x l λ. Also, the liearity of T is trivial. Further, it is easy to see that x = 0 wheever T x = 0 ad hece T is ijective. Furthermore, let y = y k l be give ad defie the sequece x = {x k λ} by x k λ = k j=k λ j k j y j ; k N. λ k λ k The, by usig 4 ad, we have for every N that Λ x = = = = y. λ k λ k x k λ k j=k k j λ j y j λ k y k λ k y k This shows that Λx = y ad sice y l, we obtai that Λx l. Thus, we deduce that x l λ ad T x = y. Hece T is surjective.

6 38 M. Mursalee ad A. K. Noma Moreover, for ay x l λ, we have by 8, 9 ad 0 of Theorem 3. that T x l = yλ l = Λx l = x l λ which shows that T is -orm ad orm reservig i the cases of 0 < < ad, resectively. Hece T is isometry. Cosequetly, the saces l λ ad l are isometrically isomorhic for 0 <. This cocludes the roof. Now, oe may exect the similar result for the sace l λ as was observed for the sace l, ad ask the atural questio: Is ot the sace l λ a Hilbert sace with 2? The aswer is ositive ad is give by the followig theorem: Theorem 3.4. Excet the case = 2, the sace l λ is ot a ier roduct sace, hece ot a Hilbert sace for <. Proof. We have to rove that the sace l λ 2 is the oly Hilbert sace amog the l λ saces for <. Sice the sace l λ 2 is a BK-sace with the orm x l λ 2 = Λx l2 by Theorem 3. ad its orm ca be obtaied from a ier roduct, i.e., the equality x l λ = x, x /2 = Λx, Λx /2 2 2 holds for every x l λ 2, the sace l λ 2 is a Hilbert sace, where, 2 deotes the ier roduct o l 2. Let us ow cosider the sequeces ad The, we have u = {u k λ} = v = {v k λ} = λ,,, 0, 0,... λ 2 λ, λ + λ 0 λ,, 0, 0,.... λ λ 0 λ 2 λ Λu =,, 0, 0,... ad Λv =,, 0, 0,.... Thus, it ca easily be see that u + v 2 l + u λ v 2 l = 8 λ 422/ = 2 u 2 l + λ v 2 l ; 2, λ that is, the orm of the sace l λ with 2 does ot satisfy the arallelogram equality which meas that this orm caot be obtaied from a ier roduct. Hece, the sace l λ with 2 is a Baach sace which is ot a Hilbert sace, where <. This comletes the roof. Remark 3.5. It is obvious that l λ is also a Baach sace which is ot a Hilbert sace.

7 O some ew sequece saces of o-absolute tye 39 4 Some iclusio relatios I the reset sectio, we establish some iclusio relatios cocerig the saces l λ ad l λ, where 0 < <. We essetially rove that the iclusio l l λ holds ad characterize the case i which the iclusio l l λ holds for <. We may begi with quotig the followig two lemmas see [5] which are eeded i the roofs of our mai results. Lemma 4.. For ay sequece x = x k w, the equalities ad S x = S x = x Λ x; N 2 [ Λ x Λ x ] ; N 3 hold, where Sx = {S x} is the sequece defied by S 0 x = 0 ad S x = λ k x k x k ;. Lemma 4.2. For ay sequece λ = λ k satisfyig 3, we have λ k λ k+ a / l if ad oly if lim if =. λ k λ k k λ k λ k λ k+ b l if ad oly if lim if >. λ k λ k k λ k k= It is obvious that Lemma 4.2 still holds if the sequece {λ k /λ k λ k } is relaced by {λ k /λ k+ λ k }. Now, we rove the followig: Theorem 4.3. If 0 < < q <, the the iclusio l λ l λ q strictly holds. Proof. Let 0 < < q <. The, it follows by the iclusio l l q that the iclusio l λ l λ q holds. Further, sice the iclusio l l q is strict, there is a sequece x = x k i l q but ot i l, i.e., x l q \ l. Let us ow defie the sequece y = y k i terms of the sequece x as follows: The, we have for every N that y k = λ kx k λ k x k λ k λ k ; k N. Λ y = λ k x k λ k x k = x which shows that Λy = x ad hece Λy l q \ l. Thus, the sequece y is i l λ q but ot i l λ. Hece, the iclusio l λ l λ q is strict. This cocludes the roof.

8 40 M. Mursalee ad A. K. Noma Theorem 4.4. The iclusios l λ c λ 0 c λ l λ strictly hold, where 0 < <. Proof. Sice the iclusio c λ 0 c λ strictly holds [5, Theorem 4.], it is eough to show that the iclusios l λ c λ 0 ad c λ l λ are strict, where 0 < <. Firstly, it is trivial that the iclusio l λ c λ 0 holds for 0 < <, sice x l λ imlies Λx l ad hece Λx c 0 which meas that x c λ 0. Further, to show that this iclusio is strict, let 0 < < ad cosider the sequece x = x k defied by x k = ; k N. 4 / k + The x c 0 ad hece x c λ 0, sice the iclusio c 0 c λ 0 holds. O the other had, we have for every N that Λ x = = λ k λ k k + / + / + / λ k λ k which shows that Λx / l ad hece x / l λ. Thus, the sequece x is i c λ 0 but ot i l λ. Therefore, the iclusio l λ c λ 0 is strict for 0 < <. Similarly, it is also clear that the iclusio c λ l λ holds. To show that this iclusio is strict, we defie the sequece y = y k by y k = k λk + λ k ; k N. λ k λ k The, we have for every N that Λ y = k λ k + λ k = which shows that Λy l \ c. Thus, the sequece y is i l λ but ot i c λ ad hece c λ l λ is a strict iclusio. This comletes the roof. Lemma 4.5. The iclusio l λ l holds if ad oly if Sx l for every sequece x l λ, where 0 <. Proof. Suose that the iclusio l λ l holds, where 0 <, ad take ay x = x k l λ. The x l by the hyothesis. Thus, we obtai from 2 that which yields that Sx l. Sx l x l + Λx l = x l + x l λ <

9 O some ew sequece saces of o-absolute tye 4 Coversely, let x l λ be give, where 0 <. The, we have by the hyothesis that Sx l. Agai, it follows by 2 that x l Sx l + Λx l = Sx l + x l λ < which shows that x l. Hece, the iclusio l λ l holds ad this cocludes the roof. Theorem 4.6. The iclusio l l λ holds. Further, the equality holds if ad oly if Sx l for every sequece x l λ. Proof. The first art of the theorem is immediately obtaied from Lemma 2., ad so we tur to the secod art. For, suose firstly that the equality l λ = l holds. The, the iclusio l λ l holds which leads us with Lemma 4.5 to the cosequece that Sx l for every x l λ. Coversely, suose that Sx l for every x l λ. The, we deduce by Lemma 4.5 that the iclusio l λ l holds. Combiig this with the iclusio l l λ, we get the equality l λ = l. This comletes the roof. Now, the followig theorem gives the ecessary ad sufficiet coditio for the matrix Λ to be stroger tha boudedess, i.e., for the iclusio l l λ to be strict. Theorem 4.7. The iclusio l l λ strictly holds if ad oly if lim if + / =. Proof. Suose that the iclusio l l λ is strict. The, Theorem 4.6 imlies the existece of a sequece x l λ such that Sx = {S x} l. Sice x l λ, we have Λx = {Λ x} l ad hece {Λ x Λ x} l. Combiig this with the fact that {S x} / l, we obtai by 3 that { / } l ad hece { / } l. This leads us with Lemma 4.2 a to the cosequece that lim if + / = which shows the ecessity of the coditio. Coversely, suose that lim if + / =. The, we have by Lemma 4.2 a that { / } l. Let us ow cosider the sequece x = x k defied by x k = k λ k /λ k λ k for all k N. The, it is obvious that x / l. O the other had, we have for every N that Λ x = k λ k λ k λ k = which shows that Λx l ad hece x l λ. Thus, the sequece x is i l λ but ot i l. Therefore, by combiig this with the iclusio l l λ, we deduce that this iclusio is strict. This cocludes the roof. Now, as a cosequece of Theorem 4.7, the followig corollary resets the ecessary ad sufficiet coditio for the matrix Λ to be equivalet to boudedess.

10 42 M. Mursalee ad A. K. Noma Corollary 4.8. The equality l λ = l holds if ad oly if lim if + / >. Proof. The ecessity follows immediately from Theorem 4.7. For, if the equality l λ = l holds, the lim if + / ad hece lim if + / >. Coversely, suose that lim if + / >. The, Lemma 4.2 b gives us the bouded sequece { / } ad so { / } l. Now, let x l λ. The Λx = {Λ x} l ad hece {Λ x Λ x} l. Thus, we obtai by 3 that {S x} l. This shows that Sx l for every x l λ, which leads us with Theorem 4.6 to the equality l λ = l. Although the iclusios c 0 c λ 0, c c λ ad l l λ always hold, the iclusio l l λ eed ot be held, where 0 < <. I fact, we are goig to show, i the followig lemma, that if /λ / l, the the iclusio l l λ fails, where /λ = /λ k ad 0 < <. Lemma 4.9. The saces l ad l λ overla. Further, if /λ / l the either of them icludes the other oe, where 0 < <. Proof. Obviously, the saces l ad l λ overla, sice λ λ 0, λ 0, 0, 0,... l l λ for 0 < <. Now, suose that /λ / l, where 0 < <, ad cosider the sequece x = e 0 =, 0, 0,... l. The, we have for every N that Λ x = λ k λ k e 0 k = λ 0 which shows that Λx / l ad hece x / l λ. Thus, the sequece x is i l but ot i l λ. Hece, the iclusio l l λ does ot hold whe /λ / l 0 < <. O the other had, let < ad defie the sequece y = y k by ; k is eve, λ k y k = λk λ k 2 ; k is odd λ k λ k λ k for all k N. Sice /λ / l, we have y / l. Besides, we have for every N that λ λ ; is eve, Λ y = 0 ; is odd

11 O some ew sequece saces of o-absolute tye 43 ad hece Λ y = = λ 0 λ 0 = λ 0 = 2 λ 0 Λ 2 y λ λ2 λ 2 λ 2 λ = 2 2 λ = 2 2 = <. λ 2 2 λ2 λ 2 2 λ 2 λ 2 λ 2 2 λ 2 λ 2 This shows that Λy l ad so y l λ. Thus, the sequece y is i l λ but ot i l, where <. Similarly, oe ca costruct a sequece belogig to the set l λ \l for 0 < <. Therefore, the iclusio l λ l also fails whe /λ / l 0 < <. Hece, if /λ / l the either of the saces l ad l λ icludes the other oe, where 0 < <. This comletes the roof. Lemma 4.0. If the iclusio l l λ holds, the /λ l for 0 < <. Proof. Suose that the iclusio l l λ holds, where 0 < <, ad cosider the sequece x = e 0 =, 0, 0,... l. The x l λ ad hece Λx l. Thus, we obtai that λ 0 = Λ x < which shows that /λ l ad this cocludes the roof. We shall later show that the coditio /λ l is ot oly ecessary but also sufficiet for the iclusio l l λ to be held, where <. Before that, by takig ito accout the defiitio of the sequece λ = λ k give by 3, we fid that 0 < λ k λ k < ; 0 k for all, k N with + k > 0. Furthermore, if /λ l the we have the followig lemma which is easy to rove. Lemma 4.. If /λ l, the su k λ k λ k =k <.

12 44 M. Mursalee ad A. K. Noma Theorem 4.2. The iclusio l l λ holds if ad oly if /λ l. Proof. The ecessity is immediate by Lemma 4.0. [ Coversely, suose /λ l. The M = su k λk λ k =k /] < by Lemma 4.. Also, let x = x k l be give. The, we have x l λ = Λ x = λ =0 M λ k λ k x k x k λ k λ k x k = M x l <. =k This shows that x l λ. Hece, the iclusio l l λ holds. Corollary 4.3. If /λ l, the the iclusio l l λ holds for <. Proof. The iclusio trivially holds for =, which is obtaied by Theorem 4.2, above. Thus, let < < ad take ay x = x k l. The, for every N, we obtai by alyig the Hölder s iequality that [ ] Λ x λk λ k x k [ = Therefore, we derive that ] [ λk λ k x k λ k λ k x k. λ k λ k ] Λ x = λ =0 λ k λ k x k x k λ k λ k =k ad hece x l λ M x k = M x l <,

13 O some ew sequece saces of o-absolute tye 45 [ where M = su k λk λ k =k /] < by Lemma 4.. This shows that x l λ. Hece, we deduce that the iclusio l l λ also holds for < <. This comletes the roof. Corollary 4.4. The iclusio l l λ holds if ad oly if /λ l, where <. Proof. The ecessity is immediate by Lemma 4.0. Coversely, suose that /λ l, where <. The /λ = /λ k l. Thus, it follows by Lemma 4. that λ k λ k λ k λ k <. su k λ =k su k λ =k Further, we have for every fixed k N that λ k λ k Λ e k ; k, = 0 ; < k. N Thus, we obtai that e k l λ = λ k λ k λ =k < ; k N which yields that e k l λ for every k N, i.e., every basis elemet of the sace l is i l λ. This shows that the sace l λ cotais the Schauder basis of the sace l such that su k e k l λ <. Hece, we deduce that the iclusio l l λ holds ad this cocludes the roof. Now, i the followig examle, we give a imortat secial case of the sace l λ, where <. Examle 4.5. Cosider the secial case of the sequece λ = λ k give by λ k = k + for all k N. The /λ / l while /λ l for < <. Hece, the iclusio l l λ does ot hold by Lemma 4.9. O the other had, by alyig the well-kow iequality see [0,.239] x k < x ; < <, + =0 =0 we immediately deduce that the iequality x l λ < x l

14 46 M. Mursalee ad A. K. Noma holds for every x l, where < <. This shows that the iclusio l l λ holds for < <. Further, this iclusio is strict. For examle, the sequece y = { k } is ot i l but i l λ, sice Λ y = + k = 2 + < ; < <. Remark 4.6. I the secial case of the sequece λ = λ k give i Examle 4.5, i.e., λ k = k+ for all k N, we may ote that the saces l λ ad l λ are resectively reduced to the Cesàro sequece saces X ad X of o-absolute tye, which are defied as the saces of all sequeces whose C -trasforms are i the saces l ad l, resectively, where < see [7, 8]. Now, let x = x k be a ull sequece of ositive reals, that is x k > 0 for all k N ad x k 0 as k. The, as is easy to see, for every ositive iteger m there is a subsequece x kr r=0 of the sequece x such that x kr = O r + m+ ad hece r + x kr = O r + m Further, this subsequece ca be chose such that k r+ k r 2 for all r N. I geeral, if x = x k is a sequece of ositive reals such that lim if k x k = 0, the there is a subsequece x = x k r r=0 of the sequece x such that lim r x k r = 0. Thus x is a ull sequece of ositive reals. Hece, as we have see above, for every ositive iteger m there is a subsequece x kr r=0 of the sequece x, ad hece of the sequece x, such that k r+ k r 2 for all r N ad r + x kr = O r + m where k r = k θr ad θ : N N is a suitable icreasig fuctio. Now, let 0 < <. The, we ca choose a ositive iteger m such that m >. I this situatio, the sequece {r + x kr } r=0 is i the sace l. Obviously, we observe that the subsequece x kr r=0 deeds o the ositive iteger m which is, i tur, deedig o. Thus, our subsequece deeds o. Hece, from the above discussio, we coclude the followig result: Lemma 4.7. Let x = x k be a ositive real sequece such that lim if k x k = 0. The, for every ositive umber 0 < < there is a subsequece x = x kr r=0 of x, deedig o, such that k r+ k r 2 for all r N ad r r+x k r <..,

15 O some ew sequece saces of o-absolute tye 47 Now, the followig theorem gives the ecessary ad sufficiet coditios for the matrix Λ to be stroger tha -absolute covergece, i.e., for the iclusio l l λ to be strict, where <. Theorem 4.8. The iclusio l l λ strictly holds if ad oly if /λ l ad lim if + / =, where <. Proof. Suose that the iclusio l l λ is strict, where <. The, the ecessity of the first coditio is immediate by Lemma 4.0. Further, sice the iclusio l λ l caot be held, Lemma 4.5 imlies the existece of a sequece x l λ such that Sx = {S x} / l. Sice x l λ, we have Λ x <. Thus, it follows by alyig the Mikowski s iequality that Λ x Λ x <. This meas that {Λ x Λ x} l ad sice {S x} / l, it follows by the relatio 3 that { / } / l ad hece { / } / l. This leads us with Lemma 4.2 a to the ecessity of the secod coditio. Coversely, sice /λ l, we have by Corollary 4.4 that the iclusio l l λ holds. Further, sice lim if k λ k+ /λ k =, we obtai by Lemma 4.2 a that λk λ k lim if = 0. k λ k Thus, it follows by Lemma 4.7 that there is a subsequece λ = λ kr r=0 of the sequece λ = λ k, deedig o, such that k r+ k r 2 for all r N ad λkr λ kr r + <. 5 λ kr r Let us ow defie the sequece y = y k for every k N by r + ; k = k r, λk λ k 2 y k = r + ; k = k r +, r N λ k λ k 0 ; otherwise. The, it is clear that y / l. O the other had, we have for every N that λ λ r + ; = k r, Λ y = r N 0 ; k r. This ad 5 imly that Λy l ad hece y l λ. Thus, the sequece y is i l λ but ot i l. Therefore, we deduce by combiig this with the iclusio l l λ that this iclusio is strict, where <. This comletes the roof. Now, as a immediate cosequece of Theorem 4.8, the followig corollary resets the ecessary ad sufficiet coditio for the matrix Λ to be equivalet to -absolute covergece, where <. 6

16 48 M. Mursalee ad A. K. Noma Corollary 4.9. The equality l λ = l holds if ad oly if lim if + / >, where <. Proof. The ecessity follows from Theorem 4.8. For, if the equality holds, the the iclusio l l λ holds ad hece /λ l by Lemma 4.0. Further, sice the iclusio l l λ caot be strict, we have by Theorem 4.8 that lim if + / ad hece lim if + / >. Coversely, suose that lim if + / >. The, there exists a costat a > such that + / a ad hece λ 0 a for all N. This shows that /λ l which leads us with Corollary 4.3 to the cosequece that the iclusio l l λ holds for <. O the other had, we have by Lemma 4.2 b that { / } l ad hece { / } l. Now, let x l λ. The Λx = {Λ x} l ad hece {Λ x Λ x} l. Thus, we obtai by the relatio 3 that {S x} l, i.e., Sx l for every x l λ. Therefore, we deduce by Lemma 4.5 that the iclusio l λ l also holds. Hece, by combiig the iclusios l l λ ad l λ l, we get the equality l λ = l, where <. This cocludes the roof. Remark It ca easily be show that Corollary 4.9 still holds for 0 < <. Fially, we ed this sectio with the followig corollary: Corollary 4.2. Although the saces l λ, c 0, c ad l overla, the sace l λ does ot iclude ay of the other saces. Furthermore, if lim if + / =, the oe of the saces c 0, c ad l icludes the sace l λ, where 0 < <. Proof. Let 0 < <. The, it is obvious that the saces l λ, c 0, c ad l overla, sice the sequece λ λ 0, λ 0, 0, 0,... belogs to all these saces. Further, the sace l λ does ot iclude the sace c 0, sice the sequece x defied by 4, i the roof of Theorem 4.4, is i c 0 but ot i l λ. Hece, the sace l λ does ot iclude ay of the saces c 0, c ad l. Furthermore, if lim if + / = the the sace l does ot iclude the sace l λ. To see this, let 0 < <. The, Lemma 4.7 imlies that the sequece y defied by 6, i the roof of Theorem 4.8, is i l λ but ot i l. Therefore, oe of the saces c 0, c ad l icludes the sace l λ whe lim if + / =, where 0 < <. This comletes the roof. 5 The basis for the sace l λ I this fial sectio, we give a sequece of the oits of the sace l λ which forms a basis for this sace, where <. If a ormed sace X cotais a sequece b with the roerty that for every x X there is a uique sequece α of scalars such that lim x α 0b 0 + α b + + α b = 0,

17 O some ew sequece saces of o-absolute tye 49 the b is called a Schauder basis or briefly basis for X. The series k α kb k which has the sum x is the called the exasio of x with resect to b, ad writte as x = k α kb k. Now, because of the trasformatio T defied from l λ to l, i the roof of Theorem 3.3, is a isomorhism, the iverse image of the basis e k of the sace l is the basis for the ew sace l λ, where <. Therefore, we have the followig: Theorem 5.. Let < ad defie the sequece e k λ l λ for every fixed k N by k e λ = k λ k ; k k +, 0 ; otherwise. N The, the sequece e k λ is a basis for the sace lλ ad every x l λ has a uique reresetatio of the form 7 x = k Λ k x e k λ. 8 Proof. Let <. The, it is obvious by 7 that Λe k λ = ek l k N ad hece e k λ lλ for all k N. Further, let x l λ be give. For every o-egative iteger m, we ut The, we have that Λx m = x m = m m Λ k x Λe k λ Λ k x e k λ. m = Λ k x e k ad hece 0 ; 0 m, Λ x x m = Λ x ; > m., m N Now, for ay give ɛ > 0 there is a o-egative iteger m 0 such that ɛ. Λ x 2 =m 0+

18 50 M. Mursalee ad A. K. Noma Therefore, we have for every m m 0 that / x x m l λ = Λ x =m+ =m 0+ Λ x / ɛ 2 < ɛ which shows that lim m x x m l λ = 0 ad hece x is rereseted as i 8. Fially, let us show the uiqueess of the reresetatio 8 of x l λ. For this, suose that x = k α kx e k λ. Sice the liear trasformatio T defied from lλ to l, i the roof of Theorem 3.3, is cotiuous, we have Λ x = k α k x Λ e k λ = k α k x δ k = α x; N. Hece, the reresetatio 8 of x l λ is uique. This comletes the roof. Now, it is kow by Theorem 3. b that l λ < is a Baach sace with its atural orm. This leads us together with Theorem 5. to the followig corollary: Corollary 5.2. The sequece sace l λ of o-absolute tye is searable for <. Fially, we coclude our work by exressig from ow o that the aim of the ext aer is to determie the α-, β- ad γ-duals of the sace l λ ad is to characterize some related matrix classes, where. Refereces [] B. Altay ad F. Başar, Some Euler sequece saces of o-absolute tye, Ukraiia Math. J [2] B. Altay, F. Başar ad M. Mursalee, O the Euler sequece saces which iclude the saces l ad l I, Iform. Sci [3] C. Aydı ad F. Başar, O the ew sequece saces which iclude the saces c 0 ad c, Hokkaido Math. J [4] C. Aydı ad F. Başar, Some ew araormed sequece saces, Iform. Sci [5] C. Aydı ad F. Başar, Some ew differece sequece saces, Al. Math. Comut [6] C. Aydı ad F. Başar, Some ew sequece saces which iclude the saces l ad l, Demostratio Math

19 O some ew sequece saces of o-absolute tye 5 [7] F. Başar ad B. Altay, O the sace of sequeces of -bouded variatio ad related matrix maigs, Ukraiia Math. J [8] M. Başarır, O some ew sequece saces ad related matrix trasformatios, Idia J. Pure Al. Math [9] B. Choudhary ad S. Nada, Fuctioal Aalysis with Alicatios, Joh Wiley & Sos Ic., New Delhi, 989. [0] G. H. Hardy, J. E. Littlewood ad G. Polya, Iequalities, Cambridge Uiversity Press, 952. [] I. J. Maddox, Elemets of Fuctioal Aalysis, Cambridge Uiversity Press 2d editio, 988. [2] E. Malkowsky, Recet results i the theory of matrix trasformatios i sequece saces, Mat. Vesik [3] E. Malkowsky ad E. Savaş, Matrix trasformatios betwee sequece saces of geeralized weighted meas, Al. Math. Comut [4] M. Mursalee, F. Başar ad B. Altay, O the Euler sequece saces which iclude the saces l ad l II, Noliear Aalysis: TMA [5] M. Mursalee ad A. K. Noma, O the saces of λ-coverget ad bouded sequeces, Thai J. Math [6] M. Mursalee ad A. K. Noma, O some ew differece sequece saces of oabsolute tye, Math. Com. Mod [7] P.-N. Ng, O modular sace of a oabsolute tye, Nata Math [8] P.-N. Ng ad P.-Y. Lee, Cesàro sequece saces of o-absolute tye, Commet. Math. Prace Mat [9] C.-S. Wag, O Nörlud sequece saces, Tamkag J. Math [20] A. Wilasky, Summability through Fuctioal Aalysis, North-Hollad Mathematics Studies 85, Elsevier Sciece Publishers, Amsterdam; New York; Oxford, 984. M. Mursalee ad Abdullah K. Noma Deartmet of Mathematics, Aligarh Muslim Uiversity, Aligarh , Idia mursaleem@gmail.com; akaoma@gmail.com