On the Domain of Riesz Mean in the Space L s *

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1 Filomat 3:4 207, DOI /FIL704925Y Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: On the Domain of Riesz Mean in the Space L s * Medine Yeşilkayagil a, Feyzi Başar b a Department of Mathematics, Uşak University, Eylül Campus, Uşak, Turkey b Kısıklı Mah. Alim Sok. Alim Apt. No:7/6 Üsküdar, İstanbul, Turkey Abstract. Let 0 < s <. In this study, we introduce the double sequence space R qt L s as the domain of four dimensional Riesz mean R qt in the space L s of absolutely s-summable double sequences. Furthermore, we show that R qt L s is a Banach space and a barrelled space for s < and is not a barrelled space for 0 < s <. We determine the α- and βϑ-duals of the space L s for 0 < s and βbp-dual of the space R qt L s for < s <, where ϑ {p, bp, r}. Finally, we characterize the classes L s : M u, L s : C bp, R qt L s : M u and R qt L s : C bp of four dimensional matrices in the cases both 0 < s < and s < together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.. Introduction We denote the set of all real or complex valued double sequences by Ω which is a vector space with coordinatewise addition and scalar multiplication. Any vector subspace of Ω is called as a double sequence space. A double sequence x = x mn of complex numbers is said to be bounded if x = N x mn <, where N = {0,, 2,...}. Consider the sequence x = x mn Ω. If for every ε > 0 there exists n 0 = n 0 ε N and l C such that x mn l < ε for all m, n > n 0, then we call that the double sequence x is convergent in the Pringsheim s sense to the limit l and write p lim x mn = l; where C denotes the complex field. We give the set definitions of the spaces M u, C p and L s of bounded, convergent in the Pringsheim s sense and absolutely s-summable double sequences, respectively, as follows: M u := x = x kl Ω : x = x kl <, C p := L s := N { } x = x mn Ω : l C such that p lim x mn = l, x = x kl Ω : x kl s <, 0 < s <. 200 Mathematics Subject Classification. 46A45; 40C05 Keywords. Summability theory; double sequences, double series, double sequence spaces, alpha-, beta- and gamma-duals, 4- dimensional matrices and matrix transformations Received: 2 December 205; Revised: 24 June 206; 9 September 206; Accepted: 9 September 206 Communicated by Eberhard Malkowsky addresses: medine.yesilkayagil@usak.edu.tr Medine Yeşilkayagil, feyzibasar@gmail.com Feyzi Başar

2 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, M u is a Banach space with the norm. One can easily see that there are such sequences in the space C p but not in the space M u. Indeed, if we define the sequence x = x kl by k, k N, l = 0, x kl := l, l N, k = 0, 0, k, l N \ {0} for all k, l N, then it is trivial that x C p \ M u, since p lim x kl = 0 but x =. So, we can consider the space C bp of the double sequences which are both convergent in the Pringsheim s sense and bounded, i.e., C bp = C p M u. A sequence in the space C p is said to be regularly convergent if it is a single convergent sequence with respect to each index and denote the space of all such sequences by C r. Let us consider a double sequence x = x mn and define the sequence s = s mn via x by s mn = =0 x kl for all m, n N. Then, the pair x, s and the sequence s = s mn are called as a double series and the sequence of partial sums of the double series, respectively. Here and after, unless stated otherwise we assume that ϑ denotes any of the symbols p, bp or r. If the double sequence s mn is convergent in the ϑ-sense, then the double series x kl is said to be convergent in the ϑ-sense and it is showed that ϑ x kl = ϑ lim s mn. Also, we find some criteria about the convergence of a double series in Limaye and Zeltser []. Throughout the text we use the notation x kl instead of =0 x kl. By L s, we denote the space of absolutely s-summable double sequences defined by Başar and Sever [2]. Throughout the text, we assume that 0 < s < and s denotes the conjugate of s, that is, s = s/s for < s <, s = for s = or s = for s =. Also, by L u, we mean the space of absolutely convergent double series. The α-dual λ α, βϑ-dual λ βϑ with respect to the ϑ-convergence and the γ-dual λ γ of a double sequence space λ are respectively defined by λ α := λ βϑ := λ γ := a kl Ω : a kl x kl < for all x kl λ, a kl Ω : ϑ a kl x kl exists for all x kl λ, a kl Ω : N =0 a kl x kl < for all x kl λ. It is easy to see for any two spaces λ, µ of double sequences that µ α λ α whenever λ µ and λ α λ γ. Additionally, it is known that the inclusion λ α λ βϑ holds while the inclusion λ βϑ λ γ does not hold, since the ϑ-convergence of a sequence of partial sums of a double series does not imply its boundedness. Let λ and µ be two double sequence spaces, and A = a mnkl be any four-dimensional real or complex infinite matrix. Then, we say that A defines a matrix mapping from λ into µ and we write A : λ µ, if for every sequence x = x kl λ the A-transform Ax = {Ax mn } N of x exists and is in µ; where Ax mn = ϑ a mnkl x kl for each m, n N. We define the ϑ-summability domain λ ϑ of A in a space λ of double sequences by A λ ϑ A := x = x kl Ω : Ax = ϑ a mnkl x kl exists and is in λ. N We say with the notation that A maps the space λ into the space µ if λ µ ϑ and we denote the set of all A four dimensional matrices, transforming the space λ into the space µ, by λ : µ. Thus, A = a mnkl λ : µ if and only if the double series on the right side of converges in the sense of ϑ for each m, n N, i.e,

3 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, A mn λ βϑ for all m, n N and every x λ, and we have Ax µ for all x λ; where A mn = a mnkl N for all m, n N. We say that a four-dimensional matrix A is C ϑ -conservative if C ϑ C ϑ A, and is C ϑ -regular if it is C ϑ -conservative and ϑ lim A x = ϑ lim Ax mn = ϑ lim x mn, where x = x mn C ϑ. In this paper, we only consider bp summability domain. Using the notation of Zeltser [3], we define the double sequences e kl = e kl mn, e l, e k and e by e kl mn = if k, l = m, n and e kl mn = 0 otherwise, and e l := k e kl, e k := l e kl and e := e kl coordinatewise sum for all k, l, m, n N and we denote Φ by Φ = span{e kl : k, l N}. For all m, n, k, l N, we say that A = a mnkl is a triangular matrix if a mnkl = 0 for k > m or l > n or both, [4]. Following Adams [4], we also say that a triangular matrix A = a mnkl is called a triangle if a mnmn 0 for all m, n N. Referring to Cooke [5, Remark a, p. 22], one can conclude that every triangle matrix has an unique inverse which is also a triangle. Zeltser [3] essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences in her PhD thesis. Altay and Başar [6] have defined the spaces BS and CS ϑ of double series whose sequence of partial sums are in the spaces M u, C ϑ, respectively. Mursaleen and Başar [7] have introduced the spaces M u, C ϑ and L s of double sequences whose Cesàro transforms are in M u, C ϑ and L s, respectively. The reader can refer to Başar [8] and Mursaleen and Mohiuddine [9] for relevant terminology and required details on the double sequences and related topics. Following Mursaleen and Başar [7] and Alotaibi and Çakan [0], Yeşilkayagil and Başar [] have defined the double sequence spaces R qt M u, R qt C p, R qt C bp and R qt C r as the domain of four dimensional Riesz mean R qt in the spaces M u, C p, C bp and C r, respectively. Also, they have characterized the matrix class M u : M u in [2] and have introduced the some topological property of the double spaces C f0 and C f of almost null and almost convergent double sequences, respectively, in [3]. In [4] Tuǧ and Başar have introduced some new double sequence spaces BM u, BC ϑ, and BL s as the domain of four-dimensional generalized difference matrix Br, s, t, u in the spaces M u, C ϑ and L s, respectively. Let q = q k, t = t l be two sequences of non-negative numbers which are not all zero and Q m = m q k, q 0 > 0, T n = n by the matrix R qt = r qt mnkl r qt mnkl = { t l, t 0 > 0. Then, the Riesz mean with respect to the sequences q = q k and t = t l is defined as follows qktl, 0 k m, 0 l n, 0, otherwise for all m, n, k, l N. It is known by Theorem 2.8 of Yeşilkayagil and Başar [] that the four dimensional Riesz mean R qt is RH-regular if and only if lim Q m = and lim T n =. The Riesz transform R qt of a m n double sequence x = x kl is given by y mn = R qt x mn = x kl 2 =0 for all m, n N. Throughout the paper, we pose that the terms of the double sequences x = x kl and y = y mn are connected with the relation 2 and the term with negative index is zero. If p limr qt x mn = s, s C, then the sequence x = x kl is said to be Riesz convergent to s see [0]. Note that in the case q k = for all k and t l = for all l, the Riesz mean R qt is reduced to the four { dimensional Cesàro mean C of order one., m, n = k, l, Let I = δ mnkl is four dimensional unit matrix, that is, δ mnkl =. Using the equality 0, otherwise δ mnkl = i,j r mnij d ijkl = i,j=0 q it j d ijkl, one can obtain by a straightforward calculation that the inverse R qt = d mnkl of the triangle matrix R qt is given, as follows: { m+n k+l Q k T l d mnkl = q m t n, m k m, n l n, 0, otherwise

4 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, for all m, n, k, l N. In the present paper, referring Başar and Sever [2] we introduce the new space R qt L s defined by R qt L s := { x = x kl Ω : {R qt x mn } L s }, 0 < s <. 2. The Space R qt L s of Double Sequences In this section, we give some results on the space R qt L s. Theorem 2.. The set R qt L s is the linear space with the coordinatewise addition and scalar multiplication, and the following statements hold: i If 0 < s <, then R qt L s is a complete s-normed space with s x s = x kl =0 which is s-norm isomorphic to the space L s. ii If s <, then R qt L s is a Banach space with /s x s = x kl s =0 3 which is norm isomorphic to the space L s. Proof. Since, Part i can be proved in the similar way, we give the proof only for Part ii. The first part is a routine verification and so we omit it. To prove the fact R qt L s is norm isomorphic to the space L s, we should show the existence of a linear bijection between the spaces R qt L s and L s. Consider the transformation U defined from R qt L s to L s by x Ux = {R qt x mn }. It is trivial that U is linear. We get from the equation Ux = t x 0 x 00 +t x 0 t 0 x 00 +t x 0 +t 2 x T q 0 x 00 +q x 0 Q q 0 x 00 +q x 0 +q 2 x 20 Q 2. m q k x k0 Q m. 2 m q k t 0 x k0 +t x k Q T q k t 0 x k0 +t x k Q 2 T. q k t 0 x k0 +t x k Q m T. 2 m T 2 q k t 0 x k0 +t x k +t 2 x k2 Q T 2 q k t 0 x k0 +t x k +t 2 x k2 Q 2 T 2. q k t 0 x k0 +t x k +t 2 x k2 Q m T 2. = θ that x = θ whenever Ux = θ, where θ denotes the zero vector. This shows that U is injective. Let y = y kl L s and define the sequence x = x kl via y by x kl = Q k T l y kl Q k T l y k,l Q k T l y + Q k T l y k,l 4

5 for all k, l N. Then, we have R qt x mn = M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Q k T l y kl Q k T l y k,l Q k T l y + Q k T k y k,l =0 n = Q 0 T l y 0l T l y 0,l and so R qt x mn = ymn n Q T l y l Q 0 T l y 0l Q T l y,l Q 0 T l y 0,l n Q 2 T l y 2l Q T l y l Q 2 T l y 2,l Q T l y,l n Q m T l y m,l Q m 2 T l y m 2,l Q m T l y m,l Q m 2 T l y m 2,l n Q m T l y ml Q m T l y m,l Q m T l y m,l Q m T l y m,l n = Q m T l y ml T l y m,l = y mn which yields that R qt s x mn = y mn s. 5 Since y L s, we have x R qt L s, that is, U is surjective. Also, we see from 5 that U is a norm preserving transformation. Now, we can show that R qt L s is a Banach space with the norm s defined by 3. To prove this, we use Part b of Corollary in [5] which says that Let X, p and Y, q be semi-normed spaces and U : X, p Y, q be an isometric isomorphism. Then, X, p is complete if and only if Y, q is complete. In particular, X, p is a Banach space if and only if Y, q is a Banach space. Since the transformation U defined above from R qt L s to L s is an isometric isomorphism and the space L s is a Banach space from Theorem 2. in [2], we conclude that the space R qt L s is a Banach space. This step completes the proof. A non-empty subset S of a locally convex space X is called fundamental if the closure of the linear span of S equals X, [5]. Using this definition, we define the set S L s as S := {e kl : k, l N}. Then, we have Φ = spans. We shall show that Φ is dense in L s, that is, clφ = L s. Let the relation clφ = L s does not hold. Hence, there exists a ball in L s, no matter how small, does not contain any points of Φ, i.e, there does not exist a y Φ such that x y ε s 6 for a point x L s.then, by 6 we have that x y = x ij e kl ij s = x kl s ε s, i,j that is, x kl s ε s. Choose ε = /2. Then, we have either x kl /2 or 3/2 x kl for all k, l N. For both statement, we can find x L s, a contradiction. Since x L s is arbitrary, every ball in L s contains a point of

6 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Φ, i.e, Φ is dense in L s. Therefore, S is fundamental set of L s. Using this fact, we define the double sequence b kl = b kl mn by b kl mn := Q k T l, m = k, n = l, Q kt l +, m = k, n = l + Q kt l q k+ t l, m = k +, n = l, Q k T l q k+ t l+, m = k +, n = l +, 0, otherwise for all k, l, m, n N. Then, {b kl ; k, l N} is the fundamental set of the space R qt L s ; since R qt b kl = e kl with 0 < s <. Theorem 2.2. If Ls, then L s R qt L s holds. Proof. Let Ls. We take the sequence e 00. Obviously, e 00 L s. For all m, n N, we have that 7 R qt e 00 mn = q 0 t 0. Since Ls, R qt e 00 L s. So, e 00 R qt L s, as desired. Theorem 2.3. Let < s < r <. Then, the inclusion R qt L s R qt L r strictly holds. Proof. Let < s < r < and x = x kl R qt L s. Then, the following inequality i,j /r i,j /s x kl r < x kl s =0 =0 =0 =0 8 holds by Jensen s inequality. Therefore, one can see by applying p-limit to 8, as i, j that x r < x s < which means that x R qt L r, as desired. Now, consider the sequence x = x kl defined by x kl = { Q k T l [k + 2l + 2] /s for all k, l N. Using 9, we have R qt x mn = [m + 2n + 2] /s and so { R qt x mn s = Q k T l [k + l + 2] /s [m + 2n + 2] /s } s = Q k T l [k + 2l + ] /s + m + 2n + 2 =, that is, x R qt L s. Since < s < r <, < r/s. So, we have { } r R qt x mn r = = <, [m + 2n + 2] /s [m + 2n + 2] r/s that is, x R qt L r. This step completes the proof. } Q k T l [k + l + ] /s Let λ be a locally convex space. Then, a subset is called barrel if it is absolutely convex, absorbing and closed in λ. Moreover, λ is called a barrelled space if each barrel is a neighborhood of zero; [5, p. 336]. Lemma 2.4. [7] Every Banach space and every Fréchet space is a barrelled space. 9

7 Theorem 2.5. The following statements hold: i Let s <. Then, R qt L s is a barrelled space. ii Let 0 < s <. Then, R qt L s is not a barrelled space. M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Proof. i By Lemma 2.4 and Part ii of Theorem 2., we say that R qt L s is a barrelled space for s <. ii We show that the space L s is not a locally convex space for 0 < s <. Let U := {x : x s }. We shall show that U includes no convex neighborhood of 0. Let V be a convex neighborhood of 0. For some ε > 0, V {x : x s ε}. In particular, ε /s e kl V for each k, l N. Choose integers m, n > and define ε /2 s the sequence x = x kl by x kl := { ε /s m+n+, 0 k m and 0 l n, 0, otherwise. Then, by choosing of ε we see that x V and x s = m n ε /s s ε = m + n + [m + n + ] s m n ε = m + n + ] = ε[m + n + ] s [m + n + ] s > ε =. ε /2 ε/2 So, V U. Since the space L s is not a locally convex space for 0 < s <, the space R qt L s is not, too. Therefore, the space R qt L s is not a barrelled space. A double sequence space λ is said to be solid if and only if λ := {ukl Ω : x kl λ such that u kl x kl for all k, l N} λ, [2, p. 53]. A double sequence space λ is said to be monotone if xu = x kl u kl λ for every x = x kl λ and u = u kl {0, } N N, where {0, } N N denotes the set of all double sequences of zeros and ones. If λ is monotone, then λ α = λ βϑ ; [3, p. 36] and λ is monotone whenever λ is solid. Theorem 2.6. Let 0 < s <. Then, the space L s is monotone. Proof. Let 0 < s <, x = x kl L s and u = u kl {0, } N N. Then, we have x kl u kl s = x kl s u kl s x kl s for each k, l N. So, we have that x kl u kl s x kl s, that is, xu L s. Theorem 2.7. Let 0 < s <. If Ls, then the space R qt L s is not monotone. Proof. Let 0 < s < and Ls. Choose the sequence x = x kl R qt L s such that x 00 0 and take the sequence u = u kl = e 00 {0, } N N. Hence, for the sequence z = ux = e 00 x we derive that R qt z mn = q 0 t 0 x 00. Since Ls, R qt z L s. So, z R qt L s, as desired.

8 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Dual Spaces In this section, we determine the α- and βϑ duals of the space L s in the case 0 < s and βbp dual of the space R qt L s for < s <. Theorem 3.. Let 0 < s. Then, the α-dual of the space L s is the space M u. Proof. Since L u M u, L α u = M u and L s L u for 0 < s, we have that M u L α s. Conversely, pose that z = z kl L α s \M u. Then, z kl x kl < for all x = x kl L s and N z kl =. Hence, there exist sequences k m and l m such that at least one is strictly increasing for m N. So, we can take z km l m > m + 2/s. If we define x = x kl by x kl := { m + 2/s, k = k m and l = l m, 0, k k m or l l m for all k, l, m N, then we have x L s. But z kl x kl = m z km l m x km l m > m =, that is, z L α s, a contradiction. Therefore, the inclusion L α s M u holds. By combining the inclusions M u L α s and L α s M u, we get L α s = M u, as desired. Corollary 3.2. Let 0 < s. Then, the βϑ-dual of the space L s is the space M u. Theorem 3.3. Let 0 < s. Then, the inclusion {R qt L s } α M u holds. Proof. Suppose that z = z kl {R qt L s } α \ M u. Then, zx L u for all x R qt L s. We take the sequence b kl as in 7. So, we have R qt b kl mn s = e kl mn s = for all k, l N. Hence, b kl R qt L s and so zx = z ij b kl L ij u. With some calculation, we have following five cases; Case. z ij b kl Q = z k T l ij kl for i, j = k, l. Case 2. z ij b kl Q = z k T l ij + + for i, j = k, l +. Case 3. z ij b kl ij Q = z k T l k+,l q k+ t l for i, j = k +, l. Case 4. z ij b kl Q = z k T l ij k+,l+ q k+ t l+ for i, j = k +, l +. Case 5. z ij b kl = 0 for otherwise. ij For example, in case, we write that Q z k T l kl Lu so, is in M u. But, we know that Q k or T l is a positive increasing sequence, that is, it is not bounded. Therefore, z kl M u, a contradiction. Hence, the inclusion {R qt L s } α M u holds, as desired. Theorem 3.4. Let < s < and define the sets d, d 2 and d 3, as follows: d = a = a s kl Ω : Q akl kt l <, d 2 = a = a s kl Ω : n N Q akn kt n 0 < q k t n, k d 3 = a = a s kl Ω : m N Q aml mt l 0 < and q m t l Then, { R qt L s } βbp = d d 2 d 3. l a mn q m t n s M u.

9 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Proof. Let x = x mn R qt L s. Then, there exists a double sequence y = y mn L s by Part ii of Theorem 2.. Also, we have s = s mn from 4 such that s mn = =0 x kl = =0 Qk T l y kl Q k T l y k,l Q k T l y + Q k T l y k,l for all m, n N. Now, by the generalized Abel transformation for double sequences we obtain that z mn = a kl x kl = =0 m,n =0 m s kl a kl + n s kn 0 a kn + s ml 0 a ml + s mn a mn 0 for all m, n N. With some straightforward calculation, we can rewrite the relation 0 as follows z mn = + a kl x kl = =0 m,n =0 n aml Q m T l 0 q m t l akl Q k T l m y kl + y ml + a mn q m t n y mn = By mn akn Q k T n 0 y kn q k t n for all m, n N, where the four-dimensional matrix B = b mnkl is defined by b mnkl = Q k T l akl, 0 k m and 0 l n, Q k T n akn 0 q k t n, 0 k m and l = n, Q m T l aml 0 q m t l, k = m and 0 l n, a Q m T mn n q m t n, k = m and l = n, 0, otherwise for all m, n, k, l N. Thus, we see that ax = a mn x mn CS bp whenever x = x mn R qt L s if and only if z = z mn C bp whenever y = y mn L s. This leads us to the fact that B L s : C bp. Hence, from Part ii of Theorem 4.3, the following statement N b mnkl s = N m,n =0 Q kt l akl s m + holds. Therefore, we derive that s Q akl kt l <, s n N Q akn kt n 0 <, q k t n k s m N Q aml mt l 0 <, q m t l l Q a mn s mt n M u. q m t n Hence, { R qt L s } βbp = d d 2 d 3. Q kt n 0 akn q k t n s n + Q mt l 0 aml q m t l s + Q a mn s mt n <. q m t n

10 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Charactarization of Some Classes of Matrix Mappings In this section, we characterize the classes L s : M u, L s : C bp, R qt L s : M u and R qt L s : C bp of four dimensional matrices, in the cases both 0 < s and < s <. We also characterize the class L s : L s of four dimensional matrices in the cases 0 < s and s <. Theorem 4.. Let A = a mnkl be any four dimensional matrix. Then, the following statements are satisfied: i Let 0 < s. Then, A L s : M u if and only if N = a mnkl <., N ii Let < s <. Then, A L s : M u if and only if M = N a mnkl s <. 3 Proof. i Let 0 < s and A = a mnkl L s : M u. Then, Ax exists and belongs to M u for all x L s, and A mn M u by Corollary 3.2 for each m, n N. Therefore, we obtain for e kl L s that Ae kl = a mnkl < N for each fixed k, l N. That is to say that the condition 2 is necessary. Conversely, pose that 2 holds and take any x = x kl L s. Then, A mn M u by Corollary 3.2 for each m, n N which implies the existence of Ax. Let m, n N be fixed. Then, since s s a mnkl x kl a mnkl x kl s s a mnkl x kl N a mnkl N s x kl s one can obtain by taking remum over m, n N that Ax = a mnkl x kl N x s /s. N This shows the sufficiency of the condition 2. ii Let < s < and A = a mnkl L s : M u. Then, Ax exists and is in M u for all x L s. We assume that M =. Then, we may choose the sequences m i, k i, n j and l j in N with k i < k i+ and l j < l j+ for all i, j N such that a mi n j k i l j s > ij s. Let us define the double sequence x = x kl L s by { s n ami n x kl := j kl, k = k i and l = l j, 0, otherwise for all k, l N. Since s >, using the inequality 4 we see that Ax mi n j = a mi n j klx kl = ami ami n j k i l j x ki l j = n j k i l j > ij 2 4

11 and so, Ax mi n j >, i,j N M. Yeşilkayagil, F. Başar / Filomat 3:4 207, a contradiction. Therefore, the condition 3 is necessary. Conversely, pose that 3 holds and take any x = x kl L s. Then Ax exists, since A mn L s for each m, n N by Theorem 2.7 in [2]. Therefore, we obtain by Hölder s inequality that Ax = a mnkl x kl N a mnkl /s /s s x kl s N < M x s, as desired. This completes the proof. Theorem 4.2. Let 0 < s and s <. Then, A = a mnkl L s : L s if and only if a mnkl s <. N 5 Proof. Let 0 < s, s < and A L s : L s. Then, Ax exists and belongs to L s for all x L s, and A mn M u by Corollary 3.2 for each m, n N. Therefore, we obtain for e kl L s that Ae kl s = a mnkl s /s < for each fixed k, l N. That is to say that the condition 5 is necessary. Conversely, pose that the condition 5 is satisfied and take any x = x kl L s. Then, A mn M u by Corollary 3.2 for each m, n N which implies the existence of Ax. Then, i,j =0 Ax mn s /s = = i,j =0 s a mnkl x kl /s i,j =0 x kl N i,j =0 a mnkl x kl s i,j =0 /s a mnkl s a mnkl s /s /s x kl <. Since i, j N s are arbitrary, we obtain that Ax s <, as desired. Theorem 4.3. Let A = a mnkl be any four dimensional matrix. Then, the following statements hold: i Let 0 < s. Then, A L s : C bp if and only if 2 holds and there exists α kl Ω such that bp lim a mnkl = α kl. 6

12 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, ii Let < s <. Then, A L s : C bp if and only if 3 and 6 hold. Proof. i Let 0 < s and pose that A = a mnkl L s : C bp. Then, since the inclusion C bp M u holds, the necessity of the condition 2 is obtained from Part i of Theorem 4.. Besides, since Ax exists and belongs to C bp for every x L s by hypothesis, this also holds for e kl L s which gives that Ae kl = a mnkl N C bp for each fixed k, l N. Hence, the condition 6 is necessary. Conversely, pose that 2 and 6 hold, and x = x kl be any sequence in the space L s. Then, since A mn L βϑ s for each m, n N, Ax exists. Therefore, we get by 6 for each fixed k, l N with 2 that α kl = bp lim a mnkl a mnkl N which gives that α kl M u. Hence, the series α kl x kl converges for every x L s. Additionally, for every ε > 0 there exists n 0 = n 0 ε N such that a mnkl α kl < ε for all m, n > n 0 by 6. Then, we obtain that s s a mnkl x kl α kl x kl = a mnkl α kl x kl s a mnkl α kl x kl < ε s x kl < ε s x kl s. s This shows that bp lim Ax mn = α kl x kl, as desired. ii Let s >. Since the necessity of the conditions can be easily seen in the similar way used in Part i, we omit the details. It is obtained with 3 for all i, j N that i,j =0 α kl s = bp lim i,j =0 a mnkl s i,j N =0 a mnkl s <. 7 This means that α kl L s. Hence, the double series α kl x kl converges for every x L s. For any given ε > 0, let us choose fixed k 0, l 0 N such that k 0,,l 0 s x kl s + x kl s + x kl s ε <. 8 =0,l 0 + =k 0 +,0 =k 0 +,l 0 + Then, there exist an n 0 N by 6 such that k 0,l 0 a mnkl α kl x kl < ε 2 =0 2M /s for every m, n > n 0. Therefore, by applying Hlder s inequality with using relations 7-9 we have that a mnkl x kl α kl x kl = a mnkl α kl x kl < ε for all sufficiently large m, n. Hence, Ax C bp. This step completes the proof. 9

13 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Theorem 4.4. Let A = a mnkl be any four dimensional matrix. Then, the following statements hold: i Let 0 < s. Then, A R qt L s : M u if and only if Q a mnkl kt l, N <, 20 N Q kt l kl amnkl <, 2 lim Q kt l kl amnkl 0 = 0 for each l N, 22 k lim Q kt l kl amnkl 0 = 0 for each k N. 23 l ii Let < s <. Then, A R qt L s : M u if and only if the conditions hold and s Q kt l kl amnkl <. 24, N Proof. i Let 0 < s and x = x mn R qt L s. Then, there exists a sequence y = y mn L s. For the i, jth rectangular partial sum of the series a mnkl x kl, we have Ax [i,j] mn = + i,j =0 a mnkl x kl = i,j =0 j s il il 0 a mnil + s ij a mnij i s kl kl a mnkl + s kj kj 0 a mnkj for all m, n N, where s mn = i,j=0 x ij. Now, using the relation 4 we derive that Ax [i,j] mn = + i,j =0 a mnkl x kl = j Q i T l il 0 i,j =0 amnil q i t l Q k T l kl for all m, n, i, j N. Define the matrix B mn = b mn ijkl = Q k T l kl Q k T j kj Q i T l il 0 amnkl amnkj 0 q k t j amnil q i t l amnkl i y kl + Q k T j kj q k t 0 l amnkj q k t j y il + Q i T j a mnij q i t j y ij 25 b mn ijkl by, 0 k i and 0 l j, 0 k i and l = j, k = i and 0 l j a Q i T mnij j q i t j, k = i and l = j 0, otherwise. Therefore, 25 can be written as Ax [i,j] mn = B mn y [i,j]. Then, the bp-convergence of the rectangular partial sums Ax [i,j] mn for all m, n N and for all x R qt L s is equivalent to the statement that B mn L s : C bp and y kj 26

14 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, hence the conditions Q a mnkl kt l N <, 27 Q kt l kl amnkl <, 28 lim Q kt l kl amnkl 0 = 0 for each l N, 29 k lim Q kt l kl amnkl 0 = 0 for each k N 30 l must be satisfied for every fixed m, n N. If we take bp-limit in the terms of the matrix B mn = bp lim i,j b mn ijkl = Q kt l kl amnkl b mn ijkl as i, j, we have. 3 Using the relation 3 we can define a four dimensional matrix B = b mnkl by b mnkl = Q k T l kl amnkl for all m, n, k, l N. So, by the relations 25, 29, 30 and 3 we have bp lim i,j Ax [i,j] mn = bp limby mn. Thus, it is seen by combining the fact A = a mnkl R qt L s : M u if and only if B L s : M u with Part i of Theorem 4. that Q kt l kl amnkl <. 33, N Therefore, from the conditions 27-33, we see that A = a mnkl R qt L s : M u if and only if the conditions hold. ii Let < s <. With the similar way used in the proof of Part i, we have the bp-convergence of the rectangular partial sums Ax [i,j] mn for all m, n N and for all x R qt L s is equivalent to the statement that B mn L s : C bp and hence the conditions 2-23 and s Q kt l kl amnkl < 34 must be satisfied for every fixed m, n N. Also, by the definition of the matrix B mn = b mn ijkl 32 in 26 we have the relation 32. Thus, it is seen by combining the fact A = a mnkl R qt L s : M u if and only if B L s : M u with Part ii of Theorem 4. that, N Q kt l kl amnkl s <. 35 Also, the condition 35 contains the conditions 2 and 34. Therefore, we see that A = a mnkl R qt L s : M u if and only if the conditions hold. This completes the proof.

15 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, Since Theorems 4.5 and 4.6 can be proved in a similar way to that used in the proof of Theorem 4.4, we give them without proof. Theorem 4.5. Let A = a mnkl be any four dimensional matrix. Then, the following statements hold: i Let 0 < s. Then, A R qt L s : C bp if and only if the conditions hold and α kl Ω such that bp lim Q kt l kl amnkl = α kl. 36 ii Let < s <. Then, A R qt L s : C bp if and only if the conditions and 36 hold. Theorem 4.6. Let 0 < s < and < s <. Then, A = a mnkl R qt L s : L s if and only if the conditions 2-23 hold and Q a mnkl kt l N <, 37 s Q kt l kl amnkl <. 38 N Theorem 4.7. Let λ, µ be two double sequence spaces, A = a mnkl be any four dimensional matrix and B = b mnij also be a four dimensional triangle matrix such that b mnij = 0 if i > m and j > n for all m, n, i, j N. Then, A λ : µ B if and only if BA λ : µ. Proof. Suppose that λ, µ are two double sequence spaces, A = a ijkl is any four dimensional matrix and B = b mnij is also a four dimensional triangle matrix such that b mnij = 0 if i > m and j > n for all m, n, i, j N. Let x = x kl λ. Then, since the equality r,t r,t b mnij a ijkl x kl = b mnij a ijkl x kl 39 i,j=0 =0 =0 i,j=0 holds for all m, n, r, t N one can obtain by letting r, t in 39 that BAx = BAx. Therefore, it is immediate that Ax µ B whenever x λ if and only if BAx µ whenever x λ. This completes the proof. Now, we define the four dimensional matrices C = c mnkl, D = d mnkl and E = e mnkl by c mnkl = a ijkl, d mnkl = i,j=0 i,j=0 a ijkl m + n + and e mnkl = for all m, n, k, l N. One can derive several new results from Theorems Corollary 4.8. Let 0 < s. Then, the following statements hold: i,j=0 q i t j a ijkl i A = a mnkl L s : BS if and only if 2 holds with c mnkl instead of a mnkl. ii A = a mnkl L s : M u if and only if 2 holds with d mnkl instead of a mnkl. iii A = a mnkl L s : R qt M u if and only if 2 holds with e mnkl instead of a mnkl. iv A = a mnkl L s : CS bp if and only if 2 and 6 hold with c mnkl instead of a mnkl. v A = a mnkl L s : C bp if and only if 2 and 6 hold with d mnkl instead of a mnkl. vi A = a mnkl L s : R qt C bp if and only if 2 and 6 hold with e mnkl instead of a mnkl. Corollary 4.9. Let < s <. Then, the following statements hold: i A = a mnkl L s : BS if and only if 3 holds with c mnkl instead of a mnkl.

16 M. Yeşilkayagil, F. Başar / Filomat 3:4 207, ii A = a mnkl L s : M u if and only if 3 holds with d mnkl instead of a mnkl. iii A = a mnkl L s : R qt M u if and only if 3 holds with e mnkl instead of a mnkl. iv A = a mnkl L s : CS bp if and only if 3 and 6 hold with c mnkl instead of a mnkl. v A = a mnkl L s : C bp if and only if 3 and 6 hold with d mnkl instead of a mnkl. vi A = a mnkl L s : R qt C bp if and only if 3 and 6 hold with e mnkl instead of a mnkl. Corollary 4.0. Let 0 < s. Then, the following statements hold: i A = a mnkl R qt L s : BS if and only if hold with c mnkl instead of a mnkl. ii A = a mnkl R qt L s : CS bp if and only if and 36 hold with c mnkl instead of a mnkl. iii A = a mnkl R qt L s : M u if and only if hold with d mnkl instead of a mnkl. iv A = a mnkl R qt L s : C bp if and only if and 36 hold with d mnkl instead of a mnkl. v A = a mnkl R qt L s : R qt M u if and only if hold with e mnkl instead of a mnkl. vi A = a mnkl R qt L s : R qt C bp if and only if and 36 hold with e mnkl instead of a mnkl. Corollary 4.. Let < s <. Then, the following statements hold: i A = a mnkl R qt L s : BS if and only if hold with c mnkl instead of a mnkl. ii A = a mnkl R qt L s : CS bp if and only if and 36 hold with c mnkl instead of a mnkl. iii A = a mnkl R qt L s : M u if and only if hold with d mnkl instead of a mnkl. iv A = a mnkl R qt L s : C bp if and only if and 36 hold with d mnkl instead of a mnkl. v A = a mnkl R qt L s : R qt M u if and only if hold with e mnkl instead of a mnkl. vi A = a mnkl R qt L s : R qt C bp if and only if and 36 hold with e mnkl instead of a mnkl. Acknowledgements We have a lot benefited from the excellent review, so we are indebted to three anonymous referees for helpful advices and constructive criticism on the earlier version of this paper. Also, we would like to thank Hüsamettin Çapan for his careful reading and valuable suggestions on the revised version of this paper which improved the presentation and readability. References [] B.V. Limaya, M. Zeltser, On the Pringsheim convergence of double series, Proceedings of the Estonian Academy of Sciences [2] F. Başar, Y. Sever, The space L q of double sequences, Mathematical Journal of Okayama University [3] M. Zeltser, Investigation of Double sequence spaces by soft and hard analitical methods, Dissertationes Mathematicae Universtaties Tartuensis 25, Tartu University Press, University of Tartu, Faculty of Mathematics and Computer Science, Tartu, 200. [4] C.R. Adams, On non-factorable transformations of double sequences, Proceedings of the National Academy of Sciences of the United States of America [5] R.C. Cooke, Infinite Matrices and Sequence Spaces, Macmillan and Company Limited, London, 950. [6] B. Altay, F. Başar, Some new spaces of double sequences, Journal of Mathematical Analysis and Applications [7] M. Mursaleen, F. Başar, Domain of Cesàro mean of order one in some spaces of double sequences, Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences [8] F. Başar, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, stanbul, 202. [9] M. Mursaleen, S.A. Mohiuddine, Convergence Methods For Double Sequences and Applications, Springer, New Delhi Heidelberg New York Dordrecht London, 204. [0] A.M. Alotaibi, C. Çakan, The Riesz convergence and Riesz core of double sequences, Journal of Inequalities and Applications :56, 8 pp. [] M. Yeşilkayagil, F. Başar, Domain of Riesz mean in some spaces of double sequences, under communication. [2] M. Yeşilkayagil, F. Başar, Mercerian theorem for four dimensional matrices, Communications de la Faculté des Sciences de l Université d Ankara. Séries A. Mathematics and Statistics [3] M. Yeşilkayagil, F. Başar, Some topological properties of the spaces of almost null and almost convergent double sequences, Turkish Journal of Mathematics [4] O. Tuǧ, F. Başar, Four-dimensional generalized difference matrix and some double sequence spaces, AIP Conference Proceedings [5] J. Boos, Classical and Modern Methods in Summability, Oxford University Press Inc., New York, [6] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, 978. [7] H.H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Volume 3, 5th printing, 986.