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Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 DOI 10.1186/s13660-016-1129-6 RESEARCH Open Access Compact matrix operators on a new sequence space related to p spaces Abdullah Alotaibi1 , Mohammad Mursaleen2* and Syed Abdul Mohiuddine1 * Correspondence: [email protected] 2 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India Full list of author information is available at the end of the article Abstract In this paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence space p (r, s, t; B(m) ) which is related to p spaces. By applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact. Further, we study some geometric properties of this space. MSC: 46B15; 46B45; 46B50 Keywords: BK space; matrix transformation; Hausdorff measure of noncompactness; compact operator; geometric properties 1 Background, notation, and preliminaries Let w denote the space of all complex sequences x = (xk )∞ k= . By ∞ , c, c , and φ we denote the sets of all bounded, convergent, null, and finite sequences, respectively. We write cs for the set of all convergent series and p = x ∈ ω : ∞ |xk |p < ∞ for ≤ p < ∞. k= By e and e(n) (n = , , . . .), we denote the sequences with ek = for all k, and e(n) n = and ∞ = for k = n, respectively. For any sequence x = (x ) , let e(n) k k= k x [m] = m xk e(k) k= be its m-section. Let x and y be sequences, X and Y be subsets of ω and A = (ank )∞ n,k= be an infinite matrix of complex numbers. We write xy = (xk yk )∞ , k= Z = x– ∗ Y = {x ∈ ω : xz ∈ Y }, xβ = x– ∗ cs, ∞ β – X = x ∗ cs = a ∈ ω : ak xk converges for all x ∈ X x∈X k= β β β β for the β-dual of X. Note that ∞ = cβ = c = , = ∞ and p = q . © 2016 Alotaibi et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 2 of 19 ∞ k By An = (ank )∞ k= and A = (ank )n= we denote the sequences in the nth row and the kth column of A, and we write An (x) = ∞ ank xk (n = , , . . .) (.) k= β and A(x) = (An (x))∞ n= , provided An ∈ X for all n. The set XA = X(A) = {z ∈ ω : A(z) ∈ X} is called the matrix domain of A in X. Finally (X, Y ) denotes the class of all matrices A that map X into Y , that is for which An ∈ X β for all n and A(x) ∈ Y for all x ∈ X, or equivalently A ∈ (X, Y ) if and only if X ⊂ XA . 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 pn : X → C (n ∈ N), where C denotes the complex field and pn (x) = xn for all x = (xk ) ∈ X and every n ∈ N. A BK space X ⊃ φ is said to have AK if every sequence (k) x = (xk ) ∈ X has a unique representation x = ∞ k= xk e . The sequence spaces ∞ , c, and c are BK spaces with the same sup-norm given by x ∞ = supk |xk |, where the supremum is taken over all k ∈ N. Further, the space p is p /p a BK space with the usual p -norm defined by x p = ( ∞ k= |xk | ) , where ≤ p < ∞. Moreover, the BK spaces c and p ( ≤ p < ∞) have AK [], Examples ., .. Let X and Y be Banach spaces. Then B (X, Y ) is the set of all bounded linear operators L : X → Y , a Banach space with the operator norm defined as usual by L = sup L(x)/ x ≤ L ∈ B (X, Y ) . If Y = C then we write X ∗ for the space of all continuous linear functionals on X with the norm defined by f = sup f (x) : x ≤ f ∈ X∗ . If X ⊂ w is a normed space and a ∈ w then we write ∗ a = a ∗X ∞ = sup ak xk : x ≤ , (.) k= provided the expression on the right-hand side exists and is finite which is the case whenever X ⊃ φ is a BK space and a ∈ X β [], p.. For any subset X of w, the matrix domain of an infinite matrix A in X is defined by XA = {x ∈ w : Ax ∈ X}. An infinite matrix T = (tnk ) is called a triangle if tnn = and tnk = for all k > 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 XT is also a BK space with the norm given by x XT = Tx X for all x ∈ XT [], Theorem ... The following known results are fundamental for our investigation [], Theorem ... Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 3 of 19 Lemma . (a) Let X denote any of the spaces c , c, ∞ , or p . Then we have a ∗X = a X β for all a ∈ X β , where · X β is the natural norm on the dual space X β . (b) Let X and Y be BK spaces. Then every matrix A ∈ (X, Y ) defines an operator LA ∈ B(X, Y ) by LA (x) = Ax for all x ∈ X; we denote this by (X, Y ) ⊂ B(X, Y ). (c) Let X ⊃ φ be a BK space and Y be any of the spaces c , c or ∞ . If A ∈ (X, Y ), then LA = A (X,∞ ) = sup An ∗X < ∞. n Also, let F be the collection of all non-empty and finite subsets of N = {, , , . . .}, throughout. Then we have the following result. Lemma . Let X ⊃ φ be a BK space. If A ∈ (X, ), then A (X, ) ≤ LA ≤ · A (X, ) , where A (X, ) = supN∈F n∈N An ∗X < ∞. For the reader’s convenience, we list a few well-known definitions and results concerning the Hausdorff measure of noncompactness which can be found in [, ], and []. Let S and M be subsets of a metric space (X, d) and ε > . Then S is called an ε-net of M in X if for every x ∈ M there exists s ∈ S such that d(x, 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 M of a metric space X is said to be totally bounded if it has a finite ε-net for every ε > . If X is complete, then M is totally bounded if and only if M is relatively compact (its closure M̄ is a compact set). Let X and Y be Banach spaces. A linear operator L : X → Y is called compact if D(L) = X for the domain of L and, for every bounded sequence (xn )∞ n= in X, the sequence (L(xn ))∞ n= has a convergent subsequence in Y . By MX , we denote the collection of all bounded subsets of a metric space (X, d). If Q ∈ MX , then the Hausdorff measure of noncompactness of Q, is defined by χ(Q) = inf{ε > : Q has a finite ε-net in X}. The function χ : MX → [, ∞) is called the Hausdorff measure of noncompactness. It is well known that if Q, Q , and Q are bounded subsets of a metric space X, then we have χ(Q) = if and only if Q is totally bounded, Q ⊂ Q implies χ(Q ) ≤ χ(Q ), χ(Q) = χ(Q) for the closure Q of Q, χ(Q ∪ Q ) = max χ(Q ), χ(Q ) , χ(Q ∩ Q ) ≤ min χ(Q ), χ(Q ) . Further, if X is a normed space then we also have χ(Q + Q ) ≤ χ(Q ) + χ(Q ), Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 4 of 19 χ(αQ) = |α|χ(Q) for all α ∈ C. Let X and Y be Banach spaces and L ∈ B (X, Y ). Then the Hausdorff measure of noncompactness of the operator L, denoted by L χ , is defined by L χ = χ L(SX ) , (.) where S = {x ∈ X : x ≤ } is the unit ball in X. Also we have L is compact if and only if L χ = . (.) Now we shall point out the well-known result of Goldenštein, Gohberg, and Markus [], Theorem , concerning the Hausdorff measure of noncompactness in Banach spaces. The Hausdorff measure of noncompactness of a bounded subset of the BK space p ( ≤ p < ∞) is given by the following result. Lemma . Let X be a BK space with AK and monotone norm, Q ∈ MX , and Pn : X → X (n ∈ N) be the operator (projection) defined by Pn (x , x , . . .) = x[n] = (x , x , . . . , xn , , , . . .) for all x = (x , x , . . .) ∈ X. Then χ(Q) = lim sup(I – Pn )x . n→∞ x∈Q For some recent related work on this topic, we refer to [–], and []. For some applications in differential and integral equations, we refer to [, –], and []. 2 Results and discussion Throughout this paper, let r, t ∈ U and s ∈ Uo , where U = u = (uk ) ∈ w : uk = for all k and Uo = u = (uk ) ∈ w : u = . For any sequence x = (xn ) ∈ w, we define the sequence x̄ = (x̄n ) of generalized means of x by x̄n = n sn–k tk xk rn (n ∈ N). (.) k= Further, we define the infinite matrix Ā(r, s, t) of generalized means by Ā(r, s, t) nk = ⎧ ⎨s n–k tk /rn ⎩ ( ≤ k ≤ n), (.) (k > n) for all n, k ∈ N. Then, by using the notation of (.), it follows by (.) that x̄ is the Ā(r, s, t)transform of x, that is, x̄ = (Ā(r, s, t))x for all x ∈ w. Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 5 of 19 Moreover, it is obvious by (.) that Ā(r, s, t) is a triangle. Thus, it has a unique inverse (Ā(r, s, t))– which is also a triangle. More precisely, we put D(s) = /s and s s · · · s s ··· s s s s · · · s (s) Dn = n+ . .. .. .. .. (n = , , , . . .). s .. . . . . sn– sn– sn– sn– · · · s sn sn– sn– sn– · · · s Then the entries of (Ā(r, s, t))– are given by ⎧ – ⎨(–)n–k D(s) ( ≤ k ≤ n), n–k rk /tn Ā(r, s, t) nk = ⎩ (k > n) (.) for all n, k ∈ N, that is, (Ā(r, s, t))– = Ā(t, s , r), where s = (sn ) such that sn = (–)n D(s) n for all n ∈ N [], p.. Therefore, we have by (.) that xn = n (–)n–k D(s) n–k rk x̄k tn (n ∈ N). (.) k= For an arbitrary subset X of w, the set X(r, s, t) has recently been introduced in [] as the matrix domain of the triangle Ā(r, s, t) in X, that is, ∞ n sn–k tk xk ∈X . X(r, s, t) = x = (xk ) ∈ w : y = rn k= n= It is obvious that X(r, s, t) is a sequence space whenever X is a sequence space, and we call it the sequence space of generalized means. Further, if X is a BK space then X̄ = X(r, s, t) is also a BK space with the norm given by x X̄ = y X (x ∈ X̄). (.) Recently, Maji and Srivastava [] have defined and studied the sequence space X(r, s, t; B(m) ) for X ∈ {∞ , c, c } which is obtained by combining the generalized means and the mth order generalized difference operator B(m) (u, v). They characterized some compact operators on the spaces X(r, s, t; B(m) ) for X ∈ {∞ , c, c } by using the Hausdorff measure of noncompactness. In this paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence space p (r, s, t; B(m) ). By applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact. Further, we study some geometric properties of this space. The generalized difference matrix of order m denoted as B(m) = B(m) (u, v) = (b(m) nk ), u, v = (see []) is defined as ⎧ m m–n+k n–k ⎪ v (max{, n – m} ≤ k ≤ n), ⎪ ⎨ n–k u b(m) nk = ⎪ ⎪ ⎩ ( ≤ k < max{, n – m}), (k > n). Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 6 of 19 The sequence spaces X(r, s, t; B(m) ) for X ∈ {∞ , c, c } are defined as follows: ∞ X r, s, t; B(m) = x = (xk ) ∈ w : Ā(r, s, t)·B(m) x n= ∈ X . By using the matrix domain, we can write X r, s, t; B(m) = XĀ(r,s,t;B(m) ) = x = (xk ) ∈ w : Ā r, s, t; B(m) x ∈ X , where Ā(r, s, t; B(m) ) = Ā(r, s, t)·B(m) . The sequence x̄ = (x̄n ) is Ā(r, s, t)·B(m) -transform of x = (xn ), i.e. n n m sn–i ti m+j–i i–j x̄n = xj u v i – j rn j= i=j (n ∈ N). In this paper, we are interested in the study of p (r, s, t; B(m) ). We have the following lemma which is immediate by Theorem . in []. β Lemma . If a = (ak ) ∈ (p (r, s, t; B(m) ))β , then ã = (ãk ) ∈ p (= q ) and we have ∞ ak xk = k= ∞ (.) ãk x̄k k= for all x = (xk ) ∈ ¯p with x̄ = (Ā(r, s, t; B(m) ))x, where ãk = rk k+ ∞ (s) ak m + j – i – (–v)j–i i–k Di–k + (–) aj s tk um ti uj–i+m j–i i=k j=k+ ∞ D(s) m + j – i – (–v)j–i + (–)i–k i–k aj j–i ti j=i uj–i+m i=k+ ∞ . (.) The following results will be needed in our study. Lemma . We have a ∗¯p = ã ∗p for all a = (ak ) ∈ ¯p , where ã = (ãk ) is the sequence defined by (.). β β β Proof Let a = (ak ) ∈ ¯p . Then it follows by Lemma . that ã = (ãk ) ∈ p and the equality (.) holds for all sequences x = (xk ) ∈ ¯p and x̄ = (x̄k ) ∈ p which are connected by the relation x̄ = (Ā(r, s, t; B(m) ))x. Further, we see that x ∈ S¯p if and only if x̄ ∈ Sp . Therefore, we derive from (.) and (.) that a ∗¯p ∞ ∞ ãk x̄k = ã ∗p . = sup ak xk = sup x̄∈Sp x∈S¯ p k= This concludes the proof. k= Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 7 of 19 Remark . By combining Lemmas . and ., we have the following: (a) If a ∈ ( (r, s, t; B(m) ))β , then a ∗¯ = supk |ãk | < ∞. q /q (b) If a ∈ (p (r, s, t; B(m) ))β , then a ∗¯ = ( ∞ < ∞, where q = p/(p – ) and k= |ãk | ) p < p < ∞. Throughout this paper, if A = (ank ) is an infinite matrix, we define the associated matrix à = (ãnk ) by ãnk = rk k+ ∞ (s) ank m + j – i – (–v)j–i i–k Di–k + (–) anj s tk um ti uj–i+m j–i i=k j=k+ ∞ D(s) m + j – i – (–v)j–i + (–)i–k i–k anj j–i ti j=i uj–i+m i=k+ ∞ (.) provided the series on the right converge for all n, k ∈ N, which is the case whenever An ∈ (p (r, s, t))β for all n ∈ N [], Theorem .. Then we have the following. Lemma . Let Y be a sequence space and A = (ank ) be an infinite matrix. If A ∈ (p (r, s, t; B(m) ), Y ), then à ∈ (p , Y ) such that Ax = Ãx̄ for all x ∈ p (r, s, t) with x̄ = (Ā(r, s, t))x, where à = (ãnk ) is the associated matrix defined by (.). Proof Suppose that A ∈ (p (r, s, t; B(m) ), Y ) and let x ∈ p (r, s, t; B(m) ). Then An ∈ (p (r, s, t; β B(m) ))β for all n ∈ N. Thus, it follows by Lemma . that Ãn ∈ p for all n ∈ N and the equality Ax = Ãx̄ holds which yields that Ãx̄ ∈ Y , where x̄ is the sequence of generalized means of x, i.e., x̄ = (Ā(r, s, t; B(m) ))x. Further, it is obvious by (.) and Remark . that every x̄ ∈ p is the sequence of generalized means of some x ∈ p (r, s, t; B(m) ). Hence, we deduce that à ∈ (p , Y ). This completes the proof. Finally, we conclude this section by the following results on operator norms. Theorem . Let A = (ank ) an infinite matrix and à = (ãnk ) the associated matrix. If A is in any of the classes (p (r, s, t; B(m) ), ∞ ), (p (r, s, t; B(m) ), c) or (p (r, s, t; B(m) ), c ), then LA = A (p (r,s,t;B(m) ),∞ ) = sup Ãn ∗p < ∞. n Proof This is immediate by combining Lemmas . and .. Theorem . If A ∈ (p (r, s, t; B(m) ), ), then A (p (r,s,t;B(m) ), ) ≤ LA ≤ · A (p (r,s,t;B(m) ), ) , where ∗ Ãn A (p (r,s,t;B(m) ), ) = sup < ∞. N∈F n∈N p Proof This result follows from Lemmas . and .. Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 8 of 19 Theorem . If A ∈ ( (r, s, t; B(m) ), p ), then LA = A ( (r,s,t;B(m) ),p ) = sup ∞ k /p |ãnk | p < ∞. n= Proof The proof is elementary and left to the reader. Remark . The characterizations of matrix classes considered in this paper can easily be obtained as in Corollaries . and . of []. Thus, we shall omit these characterizations and only deal with the operator norms and the Hausdorff measures of noncompactness of some matrix operators which are given by infinite matrices in such classes. 3 Main results In this section, we derive some identities or estimates for the Hausdorff measures of noncompactness of certain matrix operators on the spaces of generalized means. Further, we apply our results to obtain the necessary and sufficient (or only sufficient) conditions for such operators to be compact. We may begin with quoting the following lemma [], Theorem .. Lemma . Let X ⊃ φ be a BK space. Then we have: (a) If A ∈ (X, ∞ ), then ≤ LA χ ≤ lim sup An ∗X . n→∞ (b) If A ∈ (X, c ), then LA χ = lim sup An ∗X . n→∞ (c) If X has AK or X = ∞ and A ∈ (X, c), then · lim sup An – α ∗X ≤ LA χ ≤ lim sup An – α ∗X , n→∞ n→∞ where α = (αk ) with αk = limn→∞ ank for all k ∈ N. Now, let A = (ank ) be an infinite matrix and à = (ãnk ) the associated matrix defined by (.). Then, by combining Lemmas ., . and ., we have the following result. Theorem . We have: (a) If A ∈ (p (r, s, t; B(m) ), ∞ ), then ≤ LA χ ≤ lim sup Ãn ∗p (.) n→∞ and LA is compact if lim Ãn ∗p = . n→∞ (.) Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 9 of 19 (b) If A ∈ (p (r, s, t; B(m) ), c ), then LA χ = lim sup Ãn ∗p (.) n→∞ and LA is compact if and only if lim Ãn ∗p = . n→∞ (.) (c) If A ∈ (p (r, s, t; B(m) ), c), then · lim sup Ãn – α̃ ∗p ≤ LA χ ≤ lim sup Ãn – α̃ ∗p n→∞ n→∞ (.) lim Ãn – α̃ ∗p = , (.) and LA is compact if and only if n→∞ where α̃ = (α̃k ) with α̃k = limn→∞ ãnk for all k ∈ N. Proof Note that parts (a) and (b) are proved in []. Further it is obvious that (.), (.) and (.) are, respectively, obtained from (.), (.), and (.) by using (.). Thus, we have to prove (.), (.) and (.). Since p (r, s, t; B(m) ) is a BK space, we deduce by means of Lemma . that (.) and (.) are immediate by parts (a) and (b) of Lemma ., respectively. To prove (.), we have A ∈ (p (r, s, t; B(m) ), c) and hence à ∈ (X, c) by Lemma .. Therefore, it follows by part (c) of Lemma . that · lim sup Ãn – α̃ ∗X ≤ Là χ ≤ lim sup Ãn – α̃ ∗X , n→∞ n→∞ (.) where α̃ = (α̃k ) and α̃k = limn→∞ ãnk for all k ∈ N. Now, let us write S = SX and S̄ = Sp (r,s,t;B(m) ) , for short. Then we obtain by (.) and Lemma . LA χ = χ LA (S̄) = χ(AS̄) (.) Là χ = χ Là (S) = χ(ÃS). (.) and Further, we see that x ∈ S̄ if and only if x̄ ∈ S, and since Ax = Ãx̄ by Lemma ., we deduce that AS̄ = ÃS. This leads us with (.) and (.) to the consequence that LA χ = Là χ . Hence, we get (.) from (.). This completes the proof. It is worth mentioning that the condition in (.) is only a sufficient condition for the operator LA to be compact, where A ∈ (p (r, s, t; B(m) ), ∞ ) and X is a BK space with AK or X = ∞ . More precisely, the following example will show that it is possible for LA to be compact while limn→∞ Ãn ∗X = . Hence, in general, we have just ‘if ’ in (.) of Theorem .(a). Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 10 of 19 Example . Let us define the matrix A = (ank ) by an = s t /r and ank = for k ≥ (n ∈ N). Let B(m) = I, the identity matrix. Then we have for every x = (xk ) ∈ p (r, s, t; B(m) ), Ax = (s t x /r )e and hence A ∈ (p (r, s, t; B(m) ), ∞ ). Further, it is obvious that LA is of finite rank and so LA is compact. On the other hand, by using (.), it can easily be seen that Ãn = e() for all n ∈ N. Thus, we obtain by Lemma . that Ãn ∗X = for all n ∈ N, which implies that limn→∞ Ãn ∗X = . Moreover, as an immediate consequence of Theorem ., we have the following corollary. Corollary . If either A ∈ (∞ (r, s, t; B(m) ), c ) or A ∈ (∞ (r, s, t; B(m) ), c), then the operator LA is compact. Proof Let A ∈ (¯∞ , c ). Then we have by Lemma . that à ∈ (∞ , c ) which implies that ∗ limn→∞ ( ∞ k= |ãnk |) = , that is, limn→∞ Ãn ∞ = by Lemma .. This leads us with Theorem .(b) to the consequence that LA is compact. Similarly, if A ∈ (∞ (r, s, t; B(m) ), c) then à ∈ (∞ , c) and hence limn→∞ ( ∞ k= |ãnk – α̃k |) = , which can be written as limn→∞ Ãn – α̃ ∗∞ = , where α̃ = (α̃k ) and α̃k = limn→∞ ãnk for all k ∈ N. Therefore, we deduce from Theorem .(c) that LA is compact. Throughout, let Fr (r ∈ N) be the subcollection of F consisting of all non-empty and finite subsets of N with elements that are greater than r, that is, Fr = {N ∈ F : n > r for all n ∈ N} (r ∈ N). Then we have the following. Lemma . (Theorem . in []) Let X ⊃ φ be a BK space. If A ∈ (X, ), then lim ∗ ∗ sup An ≤ LA χ ≤ · lim sup An . r→∞ r→∞ N∈F r n∈N N∈Fr n∈N X X Theorem . If A ∈ (p (r, s, t; B(m) ), ), then ≤ LA χ ≤ · lim A (r) lim A (r) ( (r,s,t;B(m) ), ) ( (r,s,t;B(m) ), r→∞ p r→∞ p ) (.) and LA is compact if and only if lim A (r) = , ( (r,s,t;B(m) ), ) r→∞ p (.) where A (r) (p (r,s,t;B(m) ), ) ∗ Ãn = sup N∈Fr n∈N (r ∈ N). X Proof It is obvious that (.) is obtained by combining Lemmas . and .. Also, by using (.), we get (.) from (.). Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 11 of 19 Corollary . Let < p < ∞, q = p/(p – ). If A ∈ (p (r, s, t; B(m) ), ), then ≤ LA χ ≤ · lim A (r) lim A (r) ( (r,s,t;B(m) ), ) ( (r,s,t;B(m) ), r→∞ p r→∞ p ) and LA is compact if and only if lim A (r) = , ( (r,s,t;B(m) ), ) r→∞ p where q /q ∞ ãnk = sup A (r) (p (r,s,t;B(m) ), ) N∈Fr (r ∈ N). k= n∈N Now, we prove the following result. Theorem . Let ≤ p < ∞. If A ∈ ( (r, s, t; B(m) ), p ), then LA χ = lim sup r→∞ ∞ k /p p |ãnk | (.) n=r and LA is compact if and only if lim sup r→∞ k ∞ p |ãnk | = . (.) n=r Proof We write S̄ = S¯ . Then we see by Lemma . that LA (S̄) = AS̄ ∈ Mp . Thus, it follows from (.) and Lemma . that LA χ = χ(AS̄) = lim sup(I – Pr )(Ax)p , r→∞ (.) x∈S̄ where Pr : p → p (r ∈ N) is the operator defined by Pr (x) = (x , x , . . . , xr , , , . . .) for all x = (xk ) ∈ p and I is the identity operator on p . On the other hand, let x ∈ (r, s, t; B(m) ) be given. Then x̄ ∈ and since A ∈ ( (r, s, t; (m) B ), p ), we obtain from Lemma . that à ∈ ( , p ) and Ax = Ãx̄. Thus, we have for every r ∈ N, (I – Pr )(Ax) = (I – Pr )(Ãx̄) p p = = ∞ Ãn (x̄)p /p n=r+ ∞ p /p ∞ ãnk x̄k n=r+ k= ≤ ∞ ∞ k= n=r+ /p |ãnk x̄k | p Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 k /p ∞ ≤ x̄ sup Page 12 of 19 |ãnk | p n=r+ = x (r,s,t) sup k ∞ /p |ãnk |p . n=r+ This yields ∞ /p p |ãnk | sup (I – Pr )(Ax) p ≤ sup k x∈S̄ (r ∈ N). n=r+ Therefore, we deduce from (.) that LA χ ≤ lim sup r→∞ k /p ∞ |ãnk | p . (.) n=r+ To prove the converse inequality, let b(k) ∈ (r, s, t; B(m) ) be such that (Ā(r, s, t; B(m) ))b(k) = e (k ∈ N), that is, e(k) is the sequence of generalized means of b(k) for each k ∈ N (see Corollary . in []). Then we have by Lemma . that Ab(k) = Ãe(k) = (ãnk )∞ n= for every k ∈ N. Now, let B = {b(k) : k ∈ N}. Then B ⊂ S̄ and hence AB ⊂ AS̄, which implies that χ(AB) ≤ χ(AS̄) = LA χ . Further, it follows by applying Lemma . that (k) ∞ (k) p An b χ(AB) = lim sup r→∞ k = lim sup r→∞ k n=r+ ∞ /p /p |ãnk | p . n=r+ Thus, we obtain lim sup r→∞ k ∞ /p |ãnk | p ≤ LA χ . (.) n=r+ Hence, we get (.) by combining (.) and (.). This completes the proof, since (.) is immediate by (.) and (.). Finally, we end this section with the following example, which shows that the limit in (.) may not be zero, that is, there exist matrix operators in the class B (¯ , p ) which are not compact, where ≤ p < ∞. Example . Let A = (ank ) be the infinite matrix defined by A = Ā(r, s, t; B(m) ). Let B(m) = I, the identity matrix. Since (r, s, t; B(m) ) is the matrix domain of A in , we have A ∈ ( (r, s, t; B(m) ), ) and hence A ∈ (¯ , p ) for ≤ p < ∞. Further, it is trivial to see that the associated matrix à is the identity matrix, that is, ãnn = and ãnk = for k = n (n ∈ N). Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 13 of 19 Now, let r ∈ N be given. Then we have, for every k ∈ N, ∞ |ãnk |p = n=r ⎧ ⎨ (k ≥ r), ⎩ (k < r). This implies that sup k ∞ /p |ãnk | p = (r ∈ N) n=r which leads us with (.) of Theorem . to the conclusion that LA χ = and hence LA is not compact. 4 Geometric properties Recently there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In the literature, there are many papers concerning geometric properties of various Banach sequence spaces. A Banach space X is said to be a Köthe sequence space (see [, ], and []) if X is a subspace of w such that (i) if x ∈ w, y ∈ X, and |x(i)| ≤ |y(i)| for all i ∈ N, then x ∈ X and x ≤ y ; (ii) there exists an element x ∈ X such that x(i) > for all i ∈ N. A Köthe sequence space X is said to have the Fatou property if for any real sequence x and any {xn } in X such that xn → x coordinatewise and supn xn < ∞, we have x ∈ X and xn → x . A Banach space X is said to have the Banach-Saks property if every bounded sequence {xn } in X admits a subsequence {zn } such that the sequence {tk (z)} is convergent in X with respect to the norm, where tk (z) = (z + z + · · · + zk ) k ∀k ∈ N. A Banach space X is said to have the weak Banach-Saks property whenever given any weakly null sequence {xn } in X there exists its subsequence {zn } such that the sequence {tk (z)} converges to zero strongly. Given any p ∈ (, ∞), we say that a Banach space (X, · ) has the Banach-Saks property of type p if there exists a constant c > such that every weakly null sequence {xk } has a subsequence {xk } such that (see []) n xk ≤ cn p ∀n ∈ N. = The Banach-Saks property of type p ∈ (, ∞) and weak Banach-Saks property for Cesàro sequence spaces have been considered in []. We say that a Banach space X has the weak fixed point property if every nonexpansive self-mapping defined on a non-empty weakly compact convex subset of X has a fixed point. Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 14 of 19 In [], García-Falset introduced the following coefficient for a Banach space (X, · ): R(X) = sup lim inf xn – x : x ∈ B(X), {xn } ⊂ B(X) and xn → weakly n→∞ and he proved (see [, ]) that a Banach space X with R(X) < has the weak fixed point property. A Köthe sequence space X is said to be order continuous if for any sequence {xn } and any x in X+ (the positive cone in X) such that xn (i) ≤ x(i) as n → ∞ for all i, n ∈ N and xn (i) → for any i ∈ N, we have xn → as n → ∞. Clarkson modulus of convexity of a normed space (X, · ) is defined (see work by Clarkson [] and Day []) by the formula x + y ; x, y ∈ S(X), x – y = ε δX (ε) = inf – for any ε ∈ [, ]. The inequality δX (ε) > for all ε ∈ (, ] characterizes the uniform convexity of X and the equality δX () = characterizes strict convexity (= rotundity) of X. The Gurariı̆ modulus of convexity of a normed space X is defined (see []) by βX (ε) = inf – inf αx + ( – α)y; x, y ∈ δ(X), x – y = ε α∈[,] for any ε ∈ [, ]. It is obvious that δX (ε) ≤ βX (ε) for any Banach space X and any ε ∈ [, ]. It is also known that βX (ε) ≤ δX (ε) for any ε ∈ [, ] and that X is rotund if and only if βX (ε) = and X is uniformly convex if and only if βX (ε) > for any ε ∈ [, ]. Theorem . The Banach-Saks type of the space p (r, s, t; B(m) ) is equal to p. Proof Let (εn ) be a sequence of positive numbers for which ∞ n= εn ≤ . Let {xn } be a weakly null sequence in p (r, s, t; B(m) ). Set t = x = and t = xn = x . Then there exists r ∈ N such that ∞ (i) t (i)e < ε . p (r,s,t;B(m) ) i=r + Since the fact that {xn } is a weakly null sequence implies that xn → , coordinatewise, there is an n ∈ N such that r xn (i)e(i) i= < ε p (r,s,t;B(m) ) for all n ≥ n . Set t = xn . Then there exists an r > r such that ∞ (i) t (i)e i=r + p (r,s,t;B(m) ) < ε . Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 15 of 19 By using the fact that xn → coordinatewise, there exists an n > n such that r (i) xn (i)e < ε p (r,s,t;B(m) ) i= for all n ≥ n . Continuing this process, we can find by induction two increasing subsequences (ri ) and (ni ) such that rj (i) xn (i)e < εj p (r,s,t;B(m) ) i= for all n ≥ nj+ and ∞ (i) tj (i)e < εj , p (r,s,t;B(m) ) i=rj + where tj = xnj . Hence, n tj j= p (r,s,t;B(m) ) n rj– rj ∞ (i) (i) (i) = tj (i)e + tj (i)e + tj (i)e j= i=rj– + i= n rj (i) ≤ tj (i)e j= p (r,s,t;B(m) ) i=rj– + j= n ∞ (i) + tj (i)e j= i=rj + n rj ≤ tj (i)e(i) j= + n j= and n rj p (i) tj (i)e j= i=rj– + p (r,s,t;B(m) ) p rj i n = ui vk tj (k) j= i=rj– + k= i p n ∞ ≤ ui vk tj (k) j= i= k= ≤ (n + ). Hence we obtain n rj n p tj (i)e(i) ≤ = (n + ) p . j= i=rj– + j= i= p (r,s,t;B(m) ) p (r,s,t;B(m) ) i=rj– + p (r,s,t;B(m) ) i=rj + n rj– (i) + tj (i)e εj p (u,v) Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 16 of 19 By using the inequality ≤ (n + ) p for all n ∈ N and ≤ p < ∞, we have n tj j= ≤ (n + ) p + ≤ (n + ) p . p (r,s,t;B(m) ) Therefore, the space p (r, s, t; B(m) ) has the Banach-Saks type p, which completes the proof of the theorem. Theorem . The Gurariı̆ of modulus of convexity for the normed space p (r, s, t; B(m) ) satisfies the inequality p p ε βp (r,s,t;B(m) ) (ε) ≤ – – for any ≤ ε ≤ . Proof Let x ∈ p (r, s, t; B(m) ). By using (.), we have p p (m) (m) x p (r,s,t;B(m) ) = Ā(r, s, t)·B x p = . Ā(r, s, t)·B x n n Let ≤ ε ≤ and using (.), let us consider the following sequences: p p ε ε x = (xn ) = H – , , , . . . , ,H p p ε ε , H – , , , . . . , t = (tn ) = H – (m) where H = (Ā(r, s, t) · B(m) )– )x)n and zn = ((Ā(r, s, t) · B(m) )t)n , we nk . Since yn = ((Ā(r, s, t) · B have p p ε ε , , , . . . , – , p p ε ε z = (zn ) = – , – , , , . . . . y = (yn ) = By using the sequences given above, we obtain the following equalities: p x (r,s,t;B(m) ) p p p p p ε (m) p + ε = Ā(r, s, t) · B x p = – p p ε ε =– + = , p p p p (m) p – ε + – ε t = Ā(r, s, t) · B = (m) ) (r,s,t;B p p p t Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 17 of 19 p p ε ε =– + = , x – t p (r,s,t;B(m) ) = Ā(r, s, t) · B(m) x – Ā(r, s, t)·B(m) t p p p p p p p p ε ε ε + – –ε = – – – = ε. To complete the estimate of the Gurariı̆ on the modulus of convexity, it remains to calcup late the infimum of αx + ( – α)t (r,s,t;B(m) ) for ≤ α ≤ . We have p inf αx + ( – α)t p (r,s,t;B(m) ) ≤α≤ = inf α Ā(r, s, t) · B(m) x + ( – α) Ā(r, s, t) · B(m) t p ≤α≤ p p p p p ε ε = inf α – + ( – α) – ≤α≤ p p ε ε + α + ( – α) – p p p ε ε = inf – + |α – |p ≤α≤ p p ε = – . Consequently, we get for p ≥ the inequality p p ε , βlp (u,v) (ε) ≤ – – which is the desired result. 5 Conclusion Recently the sequence space X(r, s, t; B(m) ) for X ∈ {∞ , c, c } has been studied by Maji and Srivastava [] which is obtained by combining the generalized means and the mth order generalized difference operator B(m) (u, v). They characterized some compact operators on the spaces X(r, s, t; B(m) ) for X ∈ {∞ , c, c } by using the Hausdorff measure of noncompactness. In this paper, we have derived some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence space p (r, s, t; B(m) ). Further, by applying the Hausdorff measure of noncompactness, we obtained the necessary and sufficient conditions for such operators to be compact. In the last section, we have studied some geometric properties of this space, e.g. the property Banach-Saks type and Gurariı̆ modulus of convexity. Competing interests The authors declare that they have no competing interests. Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 18 of 19 Authors’ contributions All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript. Author details 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2 Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India. Acknowledgements The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia. Received: 22 December 2015 Accepted: 11 July 2016 References 1. Wilansky, A: Summability through Functional Analysis. North-Holland Mathematics Studies, vol. 85. Elsevier, Amsterdam (1984) 2. Malkowsky, E, Rakočević, V: An introduction into the theory of sequence spaces and measures of noncompactness. In: Zbornik Radova, vol. 9, pp. 143-234. Mat. Institut SANU, Beograd (2000) 3. Banaś, J, Goebel, K: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Appl. Math., vol. 60. Dekker, New York (1980) 4. Banaś, J, Mursaleen, M: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, New Delhi (2014) 5. Goldenštein, LS, Gohberg, IT, Markus, AS: Investigations of some properties of bounded linear operators with their q-norms. Učen. Zap. Kishinevsk. Univ. 29, 29-36 (1957) 6. Abyar, E, Ghaemi, MB: Hausdorff measure of noncompactness of matrix operators on some sequence spaces of a double sequential band matrix. J. Inequal. Appl. 2015, 406 (2015) 7. Alotaibi, A, Mursaleen, M, Alamri, BAS, Mohiuddine, SA: Compact operators on some Fibonacci difference sequence spaces. J. Inequal. Appl. 2015, 203 (2015) 8. Başar, F, Malkowsky, E: The characterization of compact operators on spaces of strongly summable and bounded sequences. Appl. Math. Comput. 217, 5199-5207 (2011) 9. Başarir, M, Kara, EE: On some difference sequence spaces of weighted means and compact operators. Ann. Funct. Anal. 2(2), 116-131 (2011) 10. Başarir, M, Kara, EE: On compact operators on the Riesz B(m) difference sequence space. Iran. J. Sci. Technol. 35(A4), 279-285 (2011) 11. Başarir, M, Kara, EE: On the B-difference sequence space derived by generalized weighted mean and compact operators. J. Math. Anal. Appl. 391(1), 67-81 (2012) 12. Başarir, M, Kara, EE: On compact operators on the Riesz B(m) difference sequence spaces II. Iran. J. Sci. Technol. 36(A3), 371-376 (2012) 13. Başarir, M, Kara, EE: On the mth order difference sequence space of generalized weighted mean and compact operators. Acta Math. Sci. 33(3), 797-813 (2013) 14. Clarkson, JA: Uniformly convex spaces. Trans. Am. Math. Soc. 40, 396-414 (1936) 15. Cui, YA, Hudzik, H: On the Banach-Saks and weak Banach-Saks properties of some Banach sequence spaces. Acta Sci. Math. 65, 179-187 (1999) 16. Day, MM: Uniform convexity in factor and conjugate spaces. Ann. Math. 45(2), 375-385 (1944) 17. Diestel, J: Sequences and Series in Banach Spaces. Graduate Texts in Math., vol. 92. Springer, Berlin (1984) 18. García-Falset, J: Stability and fixed points for nonexpansive mappings. Houst. J. Math. 20, 495-505 (1994) 19. García-Falset, J: The fixed point property in Banach spaces with NUS-property. J. Math. Anal. Appl. 215(2), 532-542 (1997) 20. Gurariı̆, VI: On differential properties of the convexity moduli of Banach spaces. Mat. Issled. 2, 141-148 (1969) 21. Hudzik, H, Karakaya, V, Mursaleen, M, Şimşek, N: Banach-Saks type and Gurariı̆ modulus of convexity of some Banach sequence spaces. Abstr. Appl. Anal. 2014, Article ID 427382 (2014) 22. Knaust, H: Orlicz sequence spaces of Banach-Saks type. Arch. Math. 59, 562-565 (1992) 23. Kara, EE, Başarir, M: On some Euler B(m) difference sequence spaces and compact operators. J. Math. Anal. Appl. 379, 499-511 (2011) 24. Kara, EE, Başarir, M, Konca, Ş: On some new weighted Euler sequence spaces and compact operators. Math. Inequal. Appl. 17(2), 649-664 (2014) p 25. de Malafosse, B, Rakočević, V: Applications of measure of noncompactness in operators on the spaces sα , s0α , s(c) α , α . J. Math. Anal. Appl. 323(1), 131-145 (2006) 26. Malkowsky, E: Characterization of compact operators between certain BK spaces. Filomat 27(3), 447-457 (2013) 27. Mursaleen, M, Karakaya, V, Polat, H, Simşek, N: Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means. Comput. Math. Appl. 62, 814-820 (2011) 28. Mursaleen, M, Noman, AK: Compactness by the Hausdorff measure of noncompactness. Nonlinear Anal. 73(8), 2541-2557 (2010) 29. Mursaleen, M, Noman, AK: Hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means. J. Math. Anal. Appl. 417, 96-111 (2014) 30. Aghajania, A, Allahyari, R, Mursaleen, M: A generalization of Darbo’s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68-77 (2014) 31. Banaś, J: Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Cent. Eur. J. Math. 10(6), 2003-2011 (2012) 32. Banaś, J, Lecko, M: Solvability of infinite systems of differential equations in Banach sequence spaces. J. Comput. Appl. Math. 137, 363-375 (2001) 33. Mursaleen, M, Mohiuddine, SA: Applications of measures of noncompactness to the infinite system of differential equations in p spaces. Nonlinear Anal. 75, 2111-2115 (2012) Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193 Page 19 of 19 34. Alotaibi, A, Mursaleen, M, Mohiuddine, SA: Application of measures of noncompactness to infinite system of linear equations in sequence spaces. Bull. Iran. Math. Soc. 41(2), 519-527 (2015) 35. Mursaleen, M, Noman, AK: On generalized means and some related sequence spaces. Comput. Math. Appl. 61(4), 988-999 (2011) 36. Maji, A, Srivastava, PD: On B(m) -difference sequence spaces using generalized means and compact operators. Analysis 34(3), 257-281 (2014) 37. Maji, A, Srivastava, PD: Applications of the Hausdorff measure of noncompactness on the space p (r, s, t; B(m) ), 1 ≤ p < ∞. In: Mathematics and Computing 2013. Springer Proc. Math. Stat., vol. 91, pp. 271-281. Springer, New Delhi (2014)