Measures of noncompactness in some new lacunary difference sequence spaces
aa r X i v : . [ m a t h . F A ] J un Measures of noncompactness in some new lacunary di ff erence sequence spaces Ekrem Savas¸Istanbul Commerce University, 34840 Istanbul, TurkeyE-mail : [email protected] Borgohain ∗ Department of MathematicsIndian Institute of Technology, BombayPowai:400076, Mumbai, Maharashtra; INDIA.E-mail:[email protected]
Abstract:
In this research article, we establish some identities and estimates for the operator normsand the Hausdor ff measures of noncompactness of certain operators on some lacunary di ff erencesequence spaces defined by Orlicz function. Moreover, we apply our results to characterize someclasses of compact operators on those spaces by using the Hausdor ff measure of noncompactness. Key Words:
BK-space; Matrix transformation; Compact operator; Hausdor ff measure of noncom-pactness; lacunary operator, Orlicz function; Di ff erence operator. AMS Classification No: 40A05; 40A25; 40A30; 46B15; 46B45; 46B50. 40C05.
Measures of noncompactness were first introduced and later on applied in fixedpoint theory by Kuratowski [11] and Darbo [8]. Hausdor ff measure of noncom-pactness was introduced by Goldenstein et. al and later on it was studied inbroad sense by Eberhard Malkowsky et al. [5], Feyzi Basar et al. [7], EberhardMalkowsky and Ekrem Savas [6], Mohammed Mursaleen et al. [12,13] and manyothers. Some identities or estimations for the operator norms and the Hausdor ff measures of noncompactness of certain matrix operators on some sequence spaces The work of the authors was carried under the Post Doctoral Fellow under National Board ofHigher Mathematics, DAE, project No. NBHM / PDF.50 / / ff measure of noncompactness ofbounded linear operators between Banach spaces is the characterization of com-pact matrix transformations between BK spaces. W.L.C. Sargent proved thatthe characterizations of compact matrix operators between the classical sequencespaces in almost all cases.Let S and M be subsets of a metric space ( X , d ) and if for ε > x ∈ M there exists s ∈ S such that d ( x , s ) < ε , then S is called an ε -net of M in X .Let M X be a collection of all bounded subsets of a metric space ( X , d ). TheHausdor ff measure of non compactness of the set Q , denoted by χ ( Q ), is definedby, χ ( Q ) = inf { ε > Q has a finite ε − net in X } , where Q ∈ M X . The function χ : M X → [0 , ∞ ) is called the Hausdor ff measure of noncompactness.If Q , Q and Q are bounded subsets of a metric space ( X , d ), then [see Malkowsky[5]), χ ( Q ) = Q is totally bounded , Q ⊂ Q implies χ ( Q ) ≤ χ ( Q ) . Further, the function χ has some additional properties connected with thelinear structure, e.g. χ ( Q + Q ) ≤ χ ( Q ) + χ ( Q ) ,χ ( α Q ) = | α | χ ( Q ) , for all α ∈ C . Let X and Y be Banach spaces and L ∈ B ( X , Y ). Then, the Hausdor ff measureof noncompactness of L , denoted by k L k χ , can be defined by, k L k χ = χ ( L ( S X )) = χ ( L ( B X )) (1)and we have, L is compact if and only if k L k χ = . (2)2 Some preliminary concepts
By a lacunary sequence, we mean an increasing integer sequence θ = ( k r ) suchthat k = h r = k r − k r − → ∞ as r → ∞ , where the intervals determined by θ will be denoted by J r = ( k r − , k r ] and the ratio k r k r − is defined by φ r .For any lacunary sequence θ = ( k r ), the space N θ is defined as, (Freedman etal. [2]) N θ = ( x k ) : lim r →∞ h − r X k ∈ J r | x k − L | = , for some L . The space N θ is a BK space with the norm, k ( x k ) k θ = sup r h − r X k ∈ J r | x k | . An Orlicz function is defined as a function M : [0 , ∞ ) → [0 , ∞ ), which iscontinuous, non-decreasing and convex with M (0) = , M ( x ) >
0, for x > M ( x ) → ∞ , as x → ∞ .The idea of Orlicz function was used to construct the sequence space, [seeLindenstrauss and Tzafriri [10] ℓ M = ( x k ) ∈ w : ∞ X k = M | x k | ρ ! < ∞ , for some ρ > which is a Banach space with the norm, called as Orlicz sequence space, k x k = inf ρ > ∞ X k = M | x k | ρ ! ≤ . The di ff erence sequence spaces ℓ ∞ ( ∆ ) , c ( ∆ ) and c ( ∆ ) of crisp sets are definedas Z ( ∆ ) = { x = ( x k ) : ( ∆ x k ) ∈ Z } , for Z = ℓ ∞ , c and c , where ∆ x = ( ∆ x k ) = ( x k − x k + ),for all k ∈ N , which can be a Banach space with k x k ∆ = | x | + sup k | ∆ x k | . The generalized di ff erence sequence spaces are defined as, for m ≥ n ≥ Z ( ∆ nm ) = { x = ( x k ) : ( ∆ nm x k ) ∈ Z } , for Z = ℓ ∞ , c and c . X is any subset of w , then a matrix domain of an infinite matrix A in X isdefined by, X A = { x ∈ w : Ax ∈ X } . If x ⊃ φ is a BK -space and a = ( a k ) ∈ w , then wedefine, k a k ∗ X = sup x ∈ S X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k = a k x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3) ff erence sequence spaces c λ ( M , ∆ , s , θ ) , c λ ( M , ∆ , s , θ ) and ℓ λ ∞ ( M , ∆ , s , θ ) Consider λ = ( λ k ) ∞ k = to be a strictly increasing sequence of positive reals such that λ k → ∞ as k → ∞ . We define the infinite matrix Λ = ( λ nk ) ∞ n , k = by, λ nk = ( λ k − λ k − ) − ( λ k + − λ k ) λ n ; ( k < n ) , λ n − λ n − λ n ; ( k = n ) ,
0; ( k > n ) , (4)where, we shall use the convention that any term with a negative subscriptis equal to zero. Mursaleen and Noman [13] introduced the di ff erence sequencespaces c λ ( ∆ ) and ℓ λ ∞ ( ∆ ) as the matrix domains of the triangle Λ in the spaces c and ℓ ∞ respectively.In this paper, we study the sequence spaces c λ ( M , ∆ , s , θ ), c λ ( M , ∆ , s , θ ) and ℓ λ ∞ ( M , ∆ , s , θ ) and try to estimate for the operator norms and the Hausdor ff mea-sures of noncompactness of certain operators on these spaces. The spaces c λ ( M , ∆ , s , θ ), c λ ( M , ∆ , s , θ ) and ℓ λ ∞ ( M , ∆ , s , θ ) are BK-spaces with the norm given by, k x k ℓ λ ∞ ( M , ∆ , s ,θ ) = k Λ ( x ) k ℓ ∞ ( M , s ,θ ) = inf ρ > r h r ∞ X k = M | Λ k ( x ) | ρ s k ≤ . (5)The β − duals of a subset X of w are respectively defined by, X β = { a = ( a k ) ∈ w : ax = ( a k x k ) ∈ cs , for all x = ( x k ) ∈ X } Lemma 3.1.
Let X denote any of the spaces c λ ( M , ∆ , s , θ ) or ℓ λ ∞ ( M , ∆ , s , θ ). Then,we have, 4 a k ∗ X = k a k ℓ = ∞ X k = | a k | < ∞ (6)for all a = ( a k ) ∈ X β , where, a k = λ k a k λ k − λ k − + (cid:18) λ k − λ k − − λ k + − λ k (cid:19) ∞ X j = k + a j ; ( k ∈ N ) (7) Proof:
Let Y be the respective one of the spaces c or ℓ ∞ .Assume a = ( a k ) ∈ X β and y = Λ ( x ) be the associated sequence defined by, y k = k X j = λ j − λ j − λ k ! ( x j − x j − ); ( k ∈ N ) . (8)Taking y = Λ ( x ) as the associated sequence, we have a = ( a k ) ∈ ℓ such that forevery x = ( x k ) ∈ X , ∞ X k = a k x k = ∞ X k = a k y k , (9)Since x ∈ S X if and only if y ∈ S Y , (followed by (7)), we can derive that, (by (1)and (9)) k a k ∗ X = inf ρ > r h r ∞ X k = M | a k x k | ρ !! s k ≤ , x ∈ S X = inf ρ > r h r ∞ X k = M | a k y k | ρ !! s k ≤ , y ∈ S Y = k a k ∗ Y It is known that k . k ∗ X = k . k X β on X β , where k . k X β denotes the natural norm onthe dual space X β and X = c , c , ℓ ∞ or ℓ p (1 ≤ p < ∞ ).So if a ∈ ℓ , we obtain that k a k ∗ X = k a k ∗ Y = k a k ℓ < ∞ , which concludes the proof.5et A = ( a nk ) be an infinite matrix and A = ( a nk ) is the associated matrix definedby, a nk = λ k a nk λ k − λ k − + (cid:18) λ k − λ k − − λ k + − λ k (cid:19) ∞ X j = k + a nj ; ( n , k ∈ N ) (10) Lemma 3.2.
Let X be any of the spaces c λ ( M , ∆ , s , θ ) or ℓ λ ∞ ( M , ∆ )and Z be a se-quence space.. If A ∈ ( X , Z ), then A ∈ ( Y , Z ) such that Ax = Ay for all sequences x ∈ X and y ∈ Y , here Y is the respective one of the spaces c or ℓ ∞ . Proof.
Suppose that A ∈ ( X , Z ).For any sequence x = ( x k ) ∈ w and the associated sequence y = Λ ( x ) definedin (8), we have, x k = k X j = λ j y j − λ j − y j − λ j − λ j − ! ; ( k ∈ N ) . (11)Then, A n ∈ X β for all n ∈ N . Also, A n ∈ ℓ = Y β for all n ∈ N and the equality Ax = Ay , followed by equations (8), (9) and (10). Hence, Ay ∈ Z . Further, by (11),we get that every y ∈ Y is the associated sequence of some x ∈ X . Thus, it can bededuced that A ∈ ( Y , Z ), which completes the proof. Lemma 3.3.
Let X be any of the spaces c λ ( M , ∆ , s , θ ) or ℓ λ ∞ ( M , ∆ , s , θ ), A = ( a nk )an infinite matrix and A = ( a nk ) the associated marix. If A is in any of the classes( X , c ) , ( X , c ) or ( X , ℓ ∞ ), then, k L A k = k A k ( X ,ℓ ∞ ) = sup n inf ρ > r h r ∞ X k = M | a nk | ρ !! s k ≤ < ∞ Compact operators on the spaces c λ ( M , ∆ , s , θ ) and ℓ λ ∞ ( M , ∆ , s , θ ) In this section, we are trying to establish some identities or estimates for theHausdor ff measures of noncompactness of certain matrix operators on the spaces c λ ( M , ∆ , s , θ ) and ℓ λ ∞ ( M , ∆ , s , θ ). Also, the results obtained by examining these se-quence spaces are applied to characterize some classes of compact operators onthose spaces. Remark: 4.1.
Let X denote any of the spaces c or ℓ ∞ . If A ∈ ( X , c ), then wehave, • α k = lim n →∞ a nk exists for every k ∈ N , • α = ( α k ) ∈ ℓ , • sup n ∞ X k = | a nk − α k | < ∞ , • lim n →∞ A n ( x ) = ∞ X k = α k x k for all x = ( x k ) ∈ X . Theorem 4.2.
Assume A = ( a nk ) be an infinite matrix and A = ( a nk ) the associ-ated matrix defined by (10). Then, we have the following results on the Hausdor ff measures of noncompactness on the sequence spaces X = c λ ( M , ∆ , s , θ ) , c λ ( M , ∆ , s , θ )or ℓ λ ∞ ( M , ∆ , s , θ ).(a) If A ∈ ( X , c ), then, k L A k χ = lim n →∞ sup inf ρ > r h r ∞ X k = M | a nk | ρ !! s k ≤ . (12)(b) If A ∈ ( X , c ), then, lim n →∞ sup inf ρ > r h r ∞ X k = M | a nk − α k | ρ !! s k ≤ k L A k χ ≤ lim n →∞ sup inf ρ > r h r ∞ X k = M | a nk − α k | ρ !! s k ≤ , (13)where α k = lim n →∞ a nk for all k ∈ N .(c) If A ∈ ( X , ℓ ∞ ), then,0 ≤ k L A k χ ≤ lim n →∞ sup inf ρ > r h r ∞ X k = M | a nk | ρ !! s k ≤ . (14) Proof:
Following Lemma 3.3, it can be easily proved that the expressions in(12) and (13) exist.Similarly, following Remark 4.1. and Lemma 3.2, we can deduce that the ex-pression in (15) also exists.Let X ⊃ φ and Y be BK -spaces. Then, we have, ( X , Y ) ⊂ B ( X , Y ), that is, everymatrix A ∈ ( X , Y ) defines an operator L A ∈ B ( X , Y ) by L A ( x ) = Ax for all x ∈ X .So, k L A k χ = χ ( AS ) , where S = S X , by(1) (15)Let P r : c → c ( r ∈ N ) be the operator defined by P r ( x ) = ( x , x , ... x r , , , .. ) forall x = ( x k ) ∈ c . Then, we have, for Q ∈ M c , χ ( Q ) = lim r →∞ (sup x ∈ Q k ( I − P r )( x ) k ℓ ∞ ) . where I is the identity operator on c .Further, every z = ( z n ) ∈ c has a unique representation as z = ze + ∞ X n = ( z n − z ) e ( n ) ,where z = lim n →∞ z n . The projectors P r : c → c ( r ∈ N ) are obtained by, P r ( z ) = ze + r X n = ( z n − z ) e ( n ) ; ( r ∈ N ) (16)for all z = ( z n ) ∈ c with z = lim n →∞ z n . 8et AS ∈ M c . Then, χ ( AS ) = lim r →∞ (sup x ∈ S k ( I − P r )( A x ) k ℓ ∞ ) , (17)where P r : c → c ( r ∈ N ).This implies that, k ( I − P r )( Ax ) k ℓ ∞ = sup n > r | A n ( x ) | , for all x ∈ X and every r ∈ N . (18)For an infinite matrix A = ( a nk ) ∞ n , k = , we have the A -transform of x as the se-quence Ax = ( A n ( x )) ∞ n = , where A n ( x ) = ∞ X k = a nk x k , for x ∈ w and n ∈ N .Thus, we get, (by (3) and Lemma 3.1)sup x ∈ S k ( I − P r )( Ax ) k ℓ ∞ = sup n > r k A n k ∗ ( ℓ λ ∞ ( M , ∆ , s ,θ )) = sup n > r k A n k ℓ , for every r ∈ N . Which implies that, (using above with(17)) χ ( AS ) = lim r →∞ (sup n > r k A n k ℓ ) = lim n →∞ sup k A n k ℓ . This concludes the proof of (a).To prove (b), let us take Q ∈ M c and P r : c → c ( r ∈ N ) be the projector onto thelinear span of { e , e (0) , e (1) , ... e ( r ) } . Then, we have12 . lim r →∞ sup x ∈ Q k ( I − P r )( x ) k ℓ ∞ ≤ χ ( Q ) ≤ lim r →∞ sup x ∈ Q k ( I − P r )( x ) k ℓ ∞ , where I is the identity operator on c .Since we have AS ∈ M c . We can get an estimate for the value of χ ( AS ) in (15).For this, let P r : c → c ( r ∈ N ) be the projectors defined by (16).Then, we have for every r ∈ N that, 9 I − P r )( z ) = ∞ X n = r + ( z n − z ) e ( n ) and hence, k ( I − P r )( z ) k ℓ ∞ = sup n > r | z n − z | (19)for all z = ( z n ) ∈ c and every r ∈ N , where z = lim n →∞ z n and I is the identityoperator on c .By using (15), we obtain,12 lim r →∞ (sup x ∈ S k ( I − P r )( Ax ) k ℓ ∞ ) ≤ k L A k χ ≤ lim r →∞ (sup x ∈ S k ( I − P r )( Ax ) k ℓ ∞ ) (20)On the other hand, it is given that X = c λ ( M , ∆ , s , θ ) or X = ℓ λ ∞ ( M , ∆ , s , θ ),and let Y be the respective one of the spaces c or ℓ ∞ . Also, let y ∈ Y be theassociated sequence defined by (8). Since A ∈ ( X , c ), we have from Lemma 3.2that A ∈ ( Y , c ) and Ax = Ay . Further, we have the limits α k = lim n →∞ a nk exist for all k , α = ( α k ) ∈ ℓ = Y β and lim n →∞ A n ( y ) = ∞ X k = α k y k . (Remark 4.1)Consequently, k ( I − P r )( Ax ) k ℓ ∞ = k ( I − P r )( Ay ) k ℓ ∞ = sup n > r | A n ( y ) − ∞ X k = α k y k | = sup n > r | ∞ X k = ( a nk − α k ) y k | , for all r ∈ N (by (21)).Moreover, since x ∈ S = S X if and only if y ∈ S Y , we get,sup x ∈ S k ( I − P r )( Ax ) k ℓ ∞ = sup n > r sup y ∈ S Y | ∞ X k = ( a nk − α k ) y k | = sup n > r k A n α k ∗ Y sup n > r k A n α k ℓ for all r ∈ N .This concludes the proof.Finally, to prove (c), let us define the operators P r : ℓ ∞ → ℓ ∞ ( r ∈ N ) as in theproof of part (a) for all x = ( x k ) ∈ ℓ ∞ . Then, we have, AS ⊂ P r ( AS ) + ( I − P r )( AS ); ( r ∈ N ) . Thus, following the elementary properties of the function χ , we have,0 ≤ χ ( AS ) ≤ χ ( P r ( AS )) + χ (( I − P r )( AS )) = χ (( I − P r )( AS )) ≤ sup x ∈ S k ( I − P r )( Ax ) k ℓ ∞ = sup n > r k A n k ℓ , for all r ∈ N .Hence,0 ≤ χ ( AS ) ≤ lim r →∞ (sup n > r k A n k ℓ ) = lim n →∞ sup k A n k ℓ .Combining this together with (15), imply (14) ,which completes the proof. Corollary 4.3.
Let X denote any of the spaces c λ ( M , ∆ , s , θ ) or ℓ λ ∞ ( M , ∆ , s , θ ).Then, we have,(a) If A ∈ ( X , c ), then, L A is compact i ff lim n →∞ inf ρ > r h r ∞ X k = M | a nk | ρ ! s k ≤ = A ∈ ( X , c ), then, L A is compact i ff lim n →∞ inf ρ > r h r ∞ X k = M | a nk − α k | ρ !! s k ≤ = α k = lim n →∞ a nk for all k ∈ N .(c) If A ∈ ( X , ℓ ∞ ), then, L A is compact if lim n →∞ inf ρ > r h r ∞ X k = M | a nk | ρ !! s k ≤ = By applying the previous results, in this section, we are trying to establishsome identities or estimates for the operator norms and the Hausdor ff mea-sure of non-compactness of certain matrix operators that map any of the spaces c λ ( M , ∆ , s , θ ) , c λ ( M , ∆ , s , θ ) and ℓ λ ∞ ( M , ∆ , s , θ ) into the matrix domains of triangles inthe spaces c , c and ℓ ∞ . Further, we deduce the necessary and su ffi cient conditionsfor such operators to be compact. Lemma 5.1.
Let T be a triangle. Then, we have, • For arbitrary subsets X and Y of w , A ∈ ( X , Y T ) if and only if B = TA ∈ ( X , Y ). • Further, if X and Y are BK spaces and A ∈ ( X , Y T ), then k L A k = k L B k .Throughout, we assume that A = ( a nk ) is an infinite matrix and T = ( t nk ) is atriangle, and we define the matrix B = ( b nk ) by b nk = n X m = t nm a mk ; ( n , k ∈ N ), that is B = TA and hence, B n = n X m = t nm A m = n X m = t nm a mk ∞ k = ; ( n ∈ N ) . Consider A = ( a nk ) and B = ( b nk ) be the associated matrices of A and B ,respectively. Then it can easily be seen that,12 nk = n X m = t nm a mk ; ( n , k ∈ N ) . Hence, B n = n X m = t nm A m = n X m = t nm a mk ∞ k = ; ( n ∈ N ).Moreover, we define the sequence a = ( a k ) ∞ k = by, a k = lim n →∞ n X m = t nm a mk ; ( k ∈ N )provided the above limits exist for all k ∈ N which is the case whenever A ∈ ( c λ ( M , ∆ , s , θ ) , c T ) or A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , c T ) by lemmas 5.1, 3.2 and Remark 4.1.Now using the above results, we have the following results: Theorem 5.2.
Let X be any of the spaces c λ ( M , ∆ , s , θ ) or ℓ λ ∞ ( M , ∆ , s , θ ), T atraingle and A an infinite matrix. If A is in any of the classes ( X , ( c ) T ) , ( X , c T ) or( X , ( ℓ ∞ ) T ), then k L A k = k A k ( X , ( ℓ ∞ ) T ) = sup n inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ < ∞ . Theorem 5.3.
Let T be a triangle. If either A ∈ ( ℓ λ ∞ ( M , ∆ , s θ ) , ( c ) T ) or A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , c T ) then L A is compact. Theorem 5.4.
Let T be a triangle. Then, we have,1. If A ∈ ( c λ ( M , ∆ , s , θ ) , ( c ) T ), then, k L A k χ = lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . and L A is compact if and only iflim n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ = .
13. If A ∈ ( c λ ( M , ∆ , s , θ ) , c T ), then . lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk − a k | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ≤ k L A k χ ≤ lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk − a k | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ and L A is compact if and only iflim n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk − a k | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ = .
3. If either A ∈ ( c λ ( M , ∆ , s , θ ) , ( ℓ ∞ ) T ) or A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , ( ℓ ∞ ) T ), then0 ≤ k L A k χ ≤ lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ and L A is compact iflim n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | t nm a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ = . Particular cases: Let λ ′ = ( λ ′ k ) ∞ k = be a strictly increasing sequence of positivereals tending to infinity and Λ ′ = ( λ ′ nk ) be the triangle defined by (4), with thesequence λ ′ instead of λ . Also, let c λ ′ ( M , ∆ , s , θ ) , c λ ′ ( M , ∆ , s , θ ) and ℓ λ ′ ∞ ( M , ∆ , s , θ ) bethe matrix domains of the triangle Λ ′ in the spaces c , c and ℓ ∞ respectively. Particular Case 5.5.
Let X be any of the spaces c λ ( M , ∆ , s , θ ) or ℓ λ ∞ ( M , ∆ , s , θ ) and A an infinite matrix. If A is in any of the classes ( X , c λ ′ ( M , ∆ , s , θ )) , ( X , c λ ′ ( M , ∆ , s , θ ))or ( X , ℓ λ ′ ∞ ( M , ∆ , s , θ )), then k L A k = k A k ( X ,ℓ λ ′∞ ( M , ∆ , s ,θ )) = sup n inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | λ ′ nm a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . Particular Case 5.6.
If either A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , c λ ′ ( M , ∆ , s , θ )) or A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , c λ ′ ( M , ∆ , s , θ )),then L A is compact. 14imilarly, we get some identities or estimates for the Hausdor ff measures ofnoncompactness of operators given by matrices in the classes ( c λ ( M , ∆ , s , θ ) , c λ ′ ( M , ∆ , s , θ )) , ( c λ ( M , ∆ , s , θ ) , c λ ′ ( M , ∆ , s , θ )) , ( c λ ( M , ∆ , s , θ ) , ℓ λ ′ ∞ ( M , ∆ , s , θ ))and ( ℓ λ ∞ ( M , ∆ , s , θ ) , ℓ λ ′ ∞ ( M , ∆ , s , θ )),and deduce the necessary and su ff ucient (or onlysu ffi cient ) conditions for such operators to be compact.Let bs , cs and cs be the spaces of all sequences associated with bounded,convergent and null series,respectively. Then, we have the following results as-sociated with the these sequence spaces, Corollary 5.7.
Let X be any of the spaces c λ ( M , ∆ , s , θ ) or ℓ λ ∞ ( M , ∆ , s , θ ) and A an infinite matrix. If A is in any of the classes ( X , cs ) , ( X , cs ) or ( X , bs ), then, k L A k = k A k ( X , bs ) = sup n inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ < ∞ . Corollary 5.8.
If either A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , cs ) or A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , cs ),then L A is compact. Corollary 5.9.
We have1. If A ∈ ( c λ ( M , ∆ , s , θ ) , cs ), then k L A k χ = lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ and L A is compact if and only iflim n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ = .
2. If A ∈ ( c λ ( M , ∆ , s , θ ) , cs ), then . lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk − a k | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k L A k χ ≤ lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk − a k | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ and L A is compact if and only iflim n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk − a k | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ = , where a k = lim n →∞ ( n X m = a mk ) for all k ∈ N .
3. If either A ∈ ( c λ ( M , ∆ , s , θ ) , bs ) or A ∈ ( ℓ λ ∞ ( M , ∆ , s , θ ) , bs ), then0 ≤ k L A k χ ≤ lim sup n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ and L A is compact iflim n →∞ inf ρ > r h r ∞ X k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X m = M | a mk | ρ !! s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ = . References [1] A. Alotaibi, B. Hazarika and S. A. Mohiuddine (2014). On lacunary statisticalconvergence of double sequences in locally solid Riesz spaces.
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