Linear Algebra and its Applications

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1 Linear Algebra and its Applications Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: Compactness of matrix operators on some new difference sequence spaces M. Mursaleen, Abdullah K. Noman Department of Mathematics, Aligarh Muslim University, Aligarh , India ARTICLE INFO ABSTRACT Article history: Received 1 August 2010 Accepted 13 June 2011 Available online 13 July 2011 Submitted by V. Nikiforov AMS classification: 46B15 46B45 46B50 Keywords: BK space Matrix transformation Compact operator Hausdorff measure of noncompactness The difference sequence spaces c λ 0 and lλ have recently been introduced by Mursaleen and Noman 2010 [M. Mursaleen, A.K. Noman, On some new difference sequence spaces of nonabsolute type, Math. Comput. Modelling ]. In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the spaces c λ 0 and lλ.further, by using the Hausdorff measure of noncompactness, we characterize some classes of compact operators on these spaces Elsevier Inc. All rights reserved. 1. Background, notations and preliminaries It seems to be quite natural, in view of the fact that matrix operators between BK spaces are continuous, to find necessary and sufficient conditions for the entries of an infinite matrix to define a compact operator between BK spaces. This can be achieved in many cases by applying the Hausdorff measure of noncompactness. In this section, we give some related definitions, notations and preliminary results. Corresponding author. addresses: mursaleenm@gmail.com M. Mursaleen, akanoman@gmail.com A.K. Noman /$ - see front matter 2011 Elsevier Inc. All rights reserved. doi: /j.laa

2 42 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications Compact operators and matrix transformations Let X and Y be Banach spaces. Then, we write BX, Y for the set of all bounded continuous linear operators L : X Y, which is a Banach space with the operator norm given by L =sup x SX Lx Y for all L BX, Y, where S X denotes the unit sphere in X, that is S X ={x X : x =1}. A linear operator L : X Y is said to be compact if the domain of L is all of X and for every bounded sequence x n in X,thesequenceLx n has a subsequence which converges in Y.ByCX, Y, we denote the class of all compact operators in BX, Y.AnoperatorL BX, Y is said to be of finite rank if dim RL <, where RL is the range space of L. An operator of finite rank is clearly compact. By w, we shall denote the space of all complex sequences. If x w,thenwewritex = x k instead of x = x k.also,wewriteφ for the set of all finite sequences that terminate in zeros. Further, we shall use the conventions that e = 1, 1, 1,...and e n is the sequence whose only non-zero term is 1inthen th place for each n N, where N ={0, 1, 2,...}. Any vector subspace of w is called a sequence space. Weshallwritel, c and c 0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Further, by cs and l 1 we denote the spaces of all sequences associated with convergent and absolutely convergent series, respectively. The β-dual of a subset X of w is defined by X β = { a = a k w : ax = a k x k cs for all x = x k X }. Throughout this paper, the matrices are infinite matrices of complex numbers. If A is an infinite matrix with complex entries a nk n, k N,thenwewriteA = a nk instead of A = a nk n,.also, we write A n for the sequence in the nth row of A, that is A n = a nk for every n N. In addition, if x = x k w then we define the A-transform of x as the sequence Ax = A n x n=0, where A n x = a nk x k ; n N 1.1 provided the series on the right converges for each n N. For arbitrary subsets X and Y of w, wewritex, Y for the class of all infinite matrices that map X into Y. ThusA X, Y if and only if A n X β for all n N and Ax Y for all x X. Moreover, the matrix domain of an infinite matrix A in X is defined by X A = { x w : Ax X }. The theory of BK spaces is the most powerful tool in the characterization of matrix transformations between sequence spaces. A sequence space X is called a BK space if it is a Banach space with continuous coordinates p n : X C n N, where C denotes the complex field and p n x = x n for all x = x k X and every n N [1, p. 193]. Thesequencespacesl, c and c 0 are BK spaces with the usual sup-norm given by x l = sup k x k, where the supremum is taken over all k N. Also, the space l 1 is a BK space with the usual l 1 -norm defined by x l1 = x k [7,Example1.13]. If X φ is a BK space and a = a k w, thenwewrite a = X sup a k x k 1.2 x S X provided the expression on the right is defined and finite which is the case whenever a X β [9, p. 381]. An infinite matrix T = t nk is called a triangle if t nn = 0 and t nk = 0forallk > n n N. The study of matrix domains of triangles in sequence spaces has a special importance due to the various properties which they have. For example, if X is a BK space then X T is also a BK space with the norm given by x XT = Tx X for all x X T [3, Lemma 2.1]. The following known results are fundamental for our investigation.

3 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications Lemma 1.1 [7, Theorem 1.29]. Let X denote any of the spaces c 0,corl. Then, we have X β = l 1 and a X = a l 1 for all a l 1. Lemma 1.2 [6, Lemma 15 a]. Let X φ and Y be BK spaces. Then, we have X, Y BX, Y, that is, every matrix A X, Y defines an operator L A BX, Y by L A x = Ax for all x X. Lemma 1.3 [4, Lemma 5.2]. Let X φ be a BK space and Y be any of the spaces c 0,corl.IfA X, Y, then we have L A = A X,l = sup A n <. X n Lemma 1.4 [7, Theorem 3.8]. Let T be a triangle. Then, we have a For arbitrary subsets X and Y of w, A X, Y T if and only if B = TA X, Y. b Further, if X and Y are BK spaces and A X, Y T,then L A = L B The Hausdorff measure of noncompactness Let S and M be subsets of a metric space X, d and ε>0. Then S is called an ε-net of M in X if for every x M there exists s S such that dx, s <ε. Further, if the set S is finite, then the ε-net S of M is called a finite ε-net of M, and we say that M has a finite ε-net in X. A subset of a metric space is said to be totally bounded if it has a finite ε-net for every ε>0. By M X, we denote the collection of all bounded subsets of a metric space X, d. IfQ M X,then the Hausdorff measure of noncompactness of the set Q, denotedbyχq, isdefinedby χq = inf { ε>0 : Qhasafiniteε-net in X }. The function χ : M X [0, is called the Hausdorff measure of noncompactness [9, p. 387]. The basic properties of the Hausdorff measure of noncompactness can be found in [6, Lemma 2]. For example, if Q, Q 1 and Q 2 are bounded subsets of a metric space X, d, thenwehave χq = 0 if and only if Q is totally bounded, Q 1 Q 2 implies χq 1 χq 2. Further, if X is a normed space then the function χ has some additional properties connected with the linear structure, e.g. χq 1 + Q 2 χq 1 + χq 2, χαq = α χq for all α C. The following result shows how to compute the Hausdorff measure of noncompactness in the BK space c 0. Lemma 1.5 [5, Theorem 3.3]. Let Q M c0 and P r : c 0 c 0 r N be the operator defined by P r x = x 0, x 1,...,x r, 0, 0,...for all x = x k c 0. Then, we have χq = lim sup I P r r x l, x Q where I is the identity operator on c 0. Further, we know by [7, Theorem 1.10] that every z = z n c has a unique representation z = ze+ n=0 z n ze n, where z = lim n z n. Thus, we define the projectors P r : c c r N

4 44 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications by P r z = ze + r z n ze n ; r N 1.3 n=0 for all z = z n c with z = lim n z n. In this situation, the following result gives an estimate for the Hausdorff measure of noncompactness in the BK space c. Lemma 1.6. [6, Theorem5b]Let Q M c and P r : c c r N be the projector onto the linear span of { e, e 0, e 1,...,e r}. Then, we have 1 2 r lim sup I P r x l χq lim sup I P x Q r r x l, x Q where I is the identity operator on c. Moreover, we have the following result concerning with the Hausdorff measure of noncompactness in the matrix domains of triangles in normed sequence spaces. Lemma 1.7 [2, Theorem 2.6]. Let X be a normed sequence space, T a triangle and χ T and χ denote the Hausdorff measures of noncompactness on M XT and M X, the collections of all bounded sets in X T and X, respectively. Then χ T Q = χtq for all Q M XT. The most effective way in the characterization of compact operators between the Banach spaces is by applying the Hausdorff measure of noncompactness. This can be achieved as follows see [7, Theorem 2.25; Corollary 2.26]. Let X and Y be Banach spaces and L BX, Y. Then, the Hausdorff measure of noncompactness of L, denotedby L χ,canbegivenby L χ = χls X and we have L is compact if and only if L χ = Remark 1.8. Let X and Y be BK spaces, T a triangle, A X, Y T and B = TA. Then, we have Bx = TAx = TAx for all x X [7, p. 181]. Thus, by combining 1.4 with Lemmas 1.4 and 1.7, we can reduce the evaluation of L A χ to that of L B χ see [8, Corollary 4.3] The difference sequence spaces c λ 0 and lλ Throughout this paper, let λ = λ k be a strictly increasing sequence of positive reals tending to infinity, that is 0 <λ 0 <λ 1 < and λ k as k. Then, we define the infinite matrix = λ nk n, by λ nk = λ k λ k 1 λ k+1 λ k λ n ; k < n, λ n λ n 1 ; k = n, λ n 0 ; k > n, where, here and in what follows, we shall use the convention that any term with a negative subscript is equal to zero. Recently, Mursaleen and Noman [10] introduced the difference sequence spaces c λ 0 and lλ as the matrix domains of the triangle in the spaces c 0 and l, respectively. 2.1

5 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications It is obvious that c λ 0 and lλ are BK spaces with the same norm given by x l λ = x l = sup n x. n 2.2 Throughout, for any sequence x = x k w, wedefinetheassociated sequence y = y k, which will frequently be used, as the -transform of x, that is y = x. Then, it can easily be shown that k λj λ j 1 y k = x j x j 1 ; λ j=0 k k N 2.3 and hence k λj y j λ j 1 y j 1 x k = ; λ j=0 j λ j 1 k N. 2.4 Further, let X be any of the spaces c λ 0 or lλ and Y be the respective one of the spaces c 0 or l.ifthesequencesx and y are connected by the relation 2.3, then x X if and only if y Y, furthermore if x X then x l λ = y l. In fact, since is a triangle, the linear operator L : X Y, which maps every sequence in X to its associated sequence in Y, is bijective and norm preserving. The β-duals of the spaces c λ 0 and lλ have been determined and characterized some related matrix transformations. We refer the reader to [10] for relevant terminology. Moreover, the following results will be needed in the sequel and we may begin with the following lemma which is immediate by [4, Lemma 3.1 a] and [10, Theorem 5.6]. Lemma 2.1. LetXdenoteanyofthespacesc λ 0 or lλ.ifa= a k X β,then = k l 1 and the equality a k x k = k y k 2.5 holds for every x = x k X, where y = x is the associated sequence defined by 2.3 and a k = λ k k a j ; k N. λ k λ k 1 λ k λ k 1 λ k+1 λ k j=k+1 Lemma 2.2. Let X denote any of the spaces c λ 0 or lλ. Then, we have a X = l 1 = k < for all a = a k X β,where = k is as in Lemma 2.1. Proof. Let Y be the respective one of the spaces c 0 or l, and take any a = a k X β. Then, we have by Lemma 2.1 that = k l 1 and the equality 2.5 holds for all sequences x = x k X and y = y k Y which are connected by the relation 2.3. Further, it follows by 2.2 that x S X if and only if y S Y. Therefore, we derive from 1.2 and 2.5 that a = X sup a k x k = x S X sup k y k y S Y = Y and since l 1, we obtain from Lemma 1.1 that a X = = Y l 1 <

6 46 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications which concludes the proof. Throughout this paper, if A = a nk is an infinite matrix, we define the associated matrix Ā = nk by a nk = λ k nk a nj ; n, k N 2.6 λ k λ k 1 λ k λ k 1 λ k+1 λ k j=k+1 provided the series on the right converge for all n, k N. Then, we have Lemma 2.3. Let X be any of the spaces c λ 0 or lλ, Y the respective one of the spaces c 0 or l,za sequence space and A = a nk an infinite matrix. If A X, Z, thenā Y, Z such that Ax = Āy for all sequences x Xandy Y which are connected by the relation 2.3, whereā = nk is the associated matrix defined by 2.6. Proof. Let x X and y Y be connected by the relation 2.3 and suppose that A X, Z. Then A n X β for all n N. Thus, it follows by Lemma 2.1 that Ā n l 1 = Y β for all n N and the equality Ax = Āy holds, hence Āy Z. Further, we have by2.4 that every y Y is the associated sequence of somex X. Hence, we deduce thatā Y, Z. This completes the proof. Finally, we conclude this section by the following result on operator norms. Lemma 2.4. Let X be any of the spaces c λ 0 or lλ,a= a nk an infinite matrix and Ā = nk the associated matrix. If A is in any of the classes X, c 0, X, c or X,l,then L A = A X,l = sup nk <. n Proof. This is immediate by combining Lemmas 1.3 and 2.2. Remark 2.5. The characterization of matrix classes considered in this paper can be found in [10]. Thus, we shall omit it and only deal with the operator norms and the Hausdorff measures of noncompactness of operators given by infinite matrices in such classes. 3. Compact operators on the spaces c λ 0 and lλ In this section, we establish some identities or estimates for the Hausdorff measures of noncompactness of certain matrix operators on the spaces c λ 0 and lλ. Further, we apply our results to characterize some classes of compact operators on those spaces. We may begin with the following lemma which is immediate by [11, Lemma 3.1]. Lemma 3.1. Let X denote any of the spaces c 0 or l.ifa X, c, thenwehave α k = lim n a nk exists for every k N, α = α k l 1, sup a nk α k <, n lim n A nx = α k x k for all x = x k X.

7 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications Now, let A = a nk be an infinite matrix and Ā = nk the associated matrix defined by 2.6. Then, we have the following result on the Hausdorff measures of noncompactness. Theorem 3.2. Let X denote any of the spaces c λ 0 or lλ. Then, we have a If A X, c 0,then L A χ = lim sup n nk. 3.1 b If A X, c, then 1 lim sup nk ᾱ k L A χ lim sup nk ᾱ k, n n where ᾱ k = lim n nk for all k N. c If A X,l,then 0 L A χ lim sup nk. 3.3 n Proof. Let us remark that the expressions in 3.1 and 3.3 exist by Lemma 2.4. Also, by combining Lemmas 2.3 and 3.1, wededucethattheexpressionin3.2exists. We write S = S X, for short. Then, we obtain by 1.4 and Lemma 1.2 that L A χ = χas. For a, we have AS M c0. Thus, it follows by applying Lemma 1.5 that χas = lim r 3.4 sup I P r Ax l, 3.5 x S where P r : c 0 c 0 r N is the operator defined by P r x = x 0, x 1,...,x r, 0, 0,... for all x = x k c 0. This yields that I P r Ax l = sup A n x for all x X and every r N. Therefore, by using 1.1, 1.2 and Lemma 2.2, we have for every r N that sup x S I P r Ax l = sup This and 3.5 imply that χas = lim sup r A n = X sup Ā n l1. Ā n l1 = lim sup n Ā n l1. Hence, we get 3.1by3.4. Toproveb,wehaveAS M c. Thus, we are going to apply Lemma 1.6 to get an estimate for the value of χas in 3.4. For this, let P r : c c r N be the projectors defined by 1.3. Then, we have for every r N that I P r z = n=r+1 z n ze n and hence I P r z l = sup z n z 3.6 for all z = z n c and every r N, where z = lim n z n and I is the identity operator on c. Now, by using 3.4, we obtain by applying Lemma 1.6 that 1 2 r lim sup I P r Ax l LA χ lim x S r sup I P r Ax l. 3.7 x S

8 48 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications On the other hand, it is given that X = c λ 0 or X = lλ, and let Y be the respective one of the spaces c 0 or l. Also, for every given x X,lety Y be the associated sequence defined by 2.3. Since A X, c,wehavebylemma2.3 that Ā Y, c and Ax = Āy. Further, it follows from Lemma 3.1 that the limits ᾱ k = lim n nk exist for all k, ᾱ = ᾱ k l 1 = Y β and lim n Ā n y = ᾱky k. Consequently, we derive from 3.6 that I P r Ax l = I P r Āy l = sup Ā n y ᾱ k y k = sup nk ᾱ k y k for all r N. Moreover, since x S = S X if and only if y S Y,weobtainby1.2 and Lemma 1.1 that I P r Ax l = sup nk ᾱ k y k sup x S sup y S Y = sup Ā n ᾱ Y = sup Ā n ᾱ l1 for all r N. Hence,from3.7 weget3.2. Finally, to prove c we define the operators P r : l l r N as in the proof of part a for all x = x k l. Then, we have AS P r AS + I P r AS; r N. Thus, it follows by the elementary properties of the function χ that 0 χas χp r AS + χi P r AS for all r N and hence = χi P r AS sup I P r Ax l x S = sup Ā n l1 0 χas lim r sup Ā n l1 = lim sup n Ā n l1. This and 3.4 togetherimply3.3 and complete the proof. Corollary 3.3. Let X denote any of the spaces c λ 0 or lλ. Then, we have a If A X, c 0,then L A is compact if and only if lim n nk = 0.

9 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications b If A X, c, then L A is compact if and only if lim n nk ᾱ k = 0, where ᾱ k = lim n nk for all k N. c If A X,l,then L A is compact if lim nk = n Proof. This result follows from Theorem 3.2 by using 1.5. It is worth mentioning that the condition in 3.8 is only a sufficient condition for the operator L A to be compact, where A X,l and X isanyofthespacesc λ 0 or lλ. More precisely, the following example will show that it is possible for L A to be compact while lim n nk = 0. Hence, in general, we have just if in 3.8 of Corollary 3.3 c. Example 3.4. Let X be any of the spaces c λ 0 or lλ, and define the matrix A = a nk by a n0 = 1 and a nk = 0fork 1 n N. Then, we have Ax = x 0 e for all x = x k X, hence A X,l. Also, since L A is of finite rank, L A is compact. On the other hand, by using 2.6, it can easily be seen that Ā = A. ThusĀ n = e 0 and so Ā n l1 = 1foralln N. This implies that lim n Ā n l1 = 1. Finally, by using the notations of Lemma 1.2, we end this section with the following corollary: Corollary 3.5. We have l λ, c 0 Cl λ, c 0 and l λ, c Clλ, c, thatis,for every matrix A l λ, c 0 or A l λ, c, the operator L A is compact. Proof. Let A l λ, c 0. Then, we have by Lemma 2.3 that Ā l, c 0 which implies that lim n nk = 0[12, p. 4]. This leads us with Corollary 3.3 a to the consequence that L A is compact. Similarly, if A l λ, c then Ā l, c and so lim n nk ᾱ k = 0, where ᾱ k = lim n nk for all k. Hence, we deduce from Corollary 3.3 b that L A Cl λ, c. 4. Some applications In this final section, we apply our previous results to establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators that map any of the spaces c λ 0 or lλ into the matrix domains of triangles in the spaces c 0, c and l. Further, we deduce the necessary and sufficient or only sufficient conditions for such operators to be compact. We assume throughout that A = a nk is an infinite matrix and T = t nk is a triangle, and we define the matrix B = b nk by b nk = t nm a mk ; n, k N, 4.1 that is B = TA and hence B n = t nm A m = t nm a mk ; n N.

10 50 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications Further, let Ā = nk and B = bnk be the associated matrices of A and B, respectively. Then, it can easily be seen that b nk = t nm mk ; n, k N, 4.2 hence B n = t nm Ā m = t nm mk ; n N. Moreover, we define the sequence = k by t nm mk ; k N 4.3 k = lim n provided the limits in 4.3 existforallk N which is the case whenever A c λ, 0 c T or A l λ, c T by Lemmas 1.4 a, 2.3 and 3.1. Now, by using , we have the following results: Theorem 4.1. Let X be any of the spaces c λ 0 or lλ, T a triangle and A an infinite matrix. If A is in any of the classes X,c 0 T, X, c T or X,l T,then L A = A X,l T = sup n t nm mk <. Proof. This is immediate by combining Lemmas 1.4 b and 2.4. Theorem 4.2. Let T be a triangle. If either A l λ, c 0 T or A l λ, c T,thenL A is compact. Proof. This result can be proved similarly as Corollary 3.5 by means of Lemmas 1.4 a and 1.7. Theorem 4.3. Let T be a triangle. Then, we have a If A c λ, 0 c 0 T,then L A χ = lim sup t n nm mk and L A is compact if and only if b If A c λ, 0 c T,then 1 lim sup 2 n and L A is compact if and only if lim t n nm mk = 0. t nm mk k L A χ lim sup n lim t n nm mk k = 0. t nm mk k

11 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications c If either A c λ, l 0 T or A l λ, l T,then 0 L A χ lim sup t n nm mk and L A is compact if lim t n nm mk = Proof. This is obtained from Theorem 3.2 and Corollary 3.3 by using Lemmas 1.4 a and 1.7. Remark 4.4. As we have seen in Example 3.4, it can be shown that the equivalence in 4.4ofTheorem 4.3 c does not hold. Now, it is obvious that Theorems have several consequences with anyparticular triangle T. For instance, let λ = λ k be a strictly increasing sequence of positive reals tending to infinity and = λ nk be the triangle defined by 2.1withthesequenceλ instead of λ. Also, let c0 λ, cλ and l λ be the matrix domains of the triangle in the spaces c 0, c and l, respectively. Then, the following corollaries are immediate by Theorems 4.1 and 4.2. Corollary 4.5. Let X be any of the spaces c λ 0 or lλ and A an infinite matrix. If A is in any of the classes X, c0 λ, X, cλ or X,l λ, then L A = A X,l λ = sup λ n nmmk <. Corollary 4.6. If either A l λ, cλ 0 or A lλ, cλ, thenl A is compact. Similarly, by using Theorem 4.3, we get some identities or estimates for the Hausdorff measures of noncompactness of operators given by matrices in the classes c λ 0, cλ 0, cλ, 0 cλ, c λ 0, lλ and lλ, lλ, and deduce the necessary and sufficient or only sufficient conditions for such operators to be compact. Finally, let bs, cs and cs 0 be the spaces of all sequences associated with bounded, convergent and null series, respectively. Then, we conclude our work with the following consequences of Theorems Corollary 4.7. Let X be any of the spaces c λ 0 or lλ and A an infinite matrix. If A is in any of the classes X, cs 0, X, cs or X, bs, then L A = A X,bs = sup n mk <. Corollary 4.8. If either A l λ, cs 0 or A l λ, cs, thenl A is compact. Corollary 4.9. We have a If A c λ, 0 cs 0,then L A χ = lim sup n mk

12 52 M. Mursaleen, A.K. Noman / Linear Algebra and its Applications and L A is compact if and only if lim n mk = 0. b If A c λ 0, cs, then 1 lim sup 2 n mk k L A χ lim sup n mk k and L A is compact if and only if lim n mk k = 0, where k = lim mk n n for all k N. c If either A c λ, bs 0 or A lλ, bs, then 0 L A χ lim sup n mk and L A is compact if lim n mk = 0. References [1] F. Başar, E. Malkowsky, B. Altay, Matrix transformations on the matrix domains of triangles in the spaces of strongly C 1 -summable and bounded sequences, Publ. Math. Debrecen [2] I. Djolović, Compact operators on the spaces a r 0 and ar c, J. Math. Anal. Appl [3] I. Djolović, E. Malkowsky, A note on compact operators on matrix domains, J. Math. Anal. Appl [4] I. Djolović, E. Malkowsky, Matrix transformations and compact operators on some new mth-order difference sequences, Appl. Math. Comput [5] I. Djolović, E. Malkowsky, A note on Fredholm operators on c 0 T, Appl. Math. Lett [6] E. Malkowsky, Compact matrix operators between some BK spaces, in: M. Mursaleen Ed., Modern Methods of Analysis and Its Applications, Anamaya Publ., New Delhi, 2010, pp [7] E. Malkowsky, V. Rakočević, An introduction into the theory of sequence spaces and measures of noncompactness, Zbornik radova 9 17, Mat. institut SANU Beograd, 2000, pp [8] E.Malkowsky,V.Rakočević, On matrix domains of triangles, Appl. Math. Comput [9] E. Malkowsky, V. Rakočević, S. Živković, Matrix transformations between the sequence spaces w p 0, vp 0, cp 0 1 < p < and certain BK spaces, Appl. Math. Comput [10] M. Mursaleen, A.K. Noman, On some new difference sequence spaces of non-absolute type, Math. Comput. Modelling [11] M. Mursaleen, A.K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal.: TMA [12] M. Stieglitz, H. Tietz, Matrixtransformationen von folgenräumen eine ergebnisübersicht, Math. Z