Characterizations of compact operators on l p − type fractional sets of sequences

Abstract: Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some lp−type fractional difference sets via the gamma function. Although, we characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness,we can only obtain sufficient conditionswhen the final space is l∞. However, we use some recent results to exactly characterize the classes of compactmatrix operatorswhen thefinal space is the set of bounded sequences.


Introduction
The gamma function of a real number x (except zero and the negative integers) is defined by an improper integral: It is known that for any natural number n, Γ(n + 1) = n!, and Γ(n + 1) = nΓ(n) hold for any real number n / ∈ {0, −1, −2, ...}. The fractional difference operator for a fractionα was defined in [1] as It is assumed that the series defined in Eq. (1.1) is convergent for x ∈ ω. The infinite sum in Eq. (1.1) becomes a finite sum ifα is a nonnegative integer. We use the usual convention that any term with a negative subscript is equal to naught, throughout the paper. Let m be a positive integer, then recall the difference operators ∆ (1) and ∆ (m) are defined by: We write ∆ and ∆ (m) for the matrices with ∆ nk = (∆ (1) e (k) )n and ∆ (m) nk = (∆ (m) e (k) )n for all n and k. The topological properties of some spaces that are constructed by the matrix operator ∆ (m) were studied in the paper [2]. Some identities and estimates for the Hausdorff measure of noncompactness of matrix operators from ∆ (m) −type spaces into the sets of bounded ℓ∞, convergent c, null sequences c 0 and also absolutely convergent series were established in [3].
We can write the fractional difference operator defined in Eq. (1.1) as an infinite matrix: Remark 1.1. [1] The inverse of fractional difference matrix is given by For some values ofα, we have The idea of constructing new sets of sequences via infinite matrices started with [4] and then it has been developed by numerous researchers using different triangles [5][6][7][8][9][10]. In the papers [11][12][13][14][15][16][17][18][19] different difference sets of sequences have been studied based on some newly defined infinite matrices. Some new results on the visualization and animations of the topologies of certain sets were illustrated in [20][21][22]. The authors applied their software package for this purpose. Those results have an interesting and important applications in crystallography. The fractional Banach sets of difference sequences ℓ(∆ (α) , p) were geometrically characterized and the modular structure of those sets were investigated in [23]. Many authors have made efforts to apply Hausdorff measure of noncompactness to find compactness conditions of certain sets of sequences during the past decade [24][25][26][27]. Note that, necessary and sufficient compactness conditions for a matrix operator from fractional sets of sequences c 0 (∆ (α) ), c(∆ (α) ) and ℓ∞(∆ (α) ) to the classical sets of sequences have been very recently determined in [28].
Fractional difference sequence spaces have been studied in the literature recently [1,29,30]. The authors of those papers especially studied the properties of fractional operators in their research in addition to focusing on certain fractional sequence spaces.
In this work, we consider the fractional sets of sequences ℓp(∆ (α) ) for 1 ≤ p ≤ ∞ and determine operator norms of our spaces. We establish some identities and estimates for the Hausdorff measures of noncompactness of certain operators on the sets ℓp(∆ (α) ) of fractional orders. We characterize some classes of compact operators on these sets. Note that, we can only obtain sufficient conditions when the final space is ℓ∞. However, we use the results of [31,32] to obtain necessary and sufficient conditions for the classes of compact matrix operators from ℓp(∆ (α) ) spaces into the sets of bounded sequences and for the classes (ℓ 1 (∆ (α) ), ℓ∞) and (ℓ∞(∆ (α) ), ℓ∞).

Preliminaries
The β dual of a set X is defined by Note that c β 0 = c β = ℓ β ∞ = ℓ 1 and ℓ β p = ℓq. Here q is the conjugate of p, it means q = p/(p − 1) for 1 < p < ∞, q = ∞ for p = 1 and q = 1 for p = ∞.
Given any infinite matrix A = (a nk ) ∞ n,k=0 of complex numbers and any sequence x, we write An = (a nk ) ∞ k=0 for the sequence in the n th row of A, An x = ∑︀ ∞ k=0 a nk x k (n = 0, 1, . . . ) and Ax = (An x) ∞ n=0 , provided An ∈ X β for all n.
If X and Y are subsets of ω, then is called a triangle if tnn ≠ 0 and t nk = 0 for all k > n. We denote its inverse by S. A BK space is a Banach space with continuous coordinates. A BK space X is said to have AK if every sequence x ∈ X has the unique representation x = ∑︀ k x k e (k) . Consider the following fractional difference sets of sequences for 1 ≤ p < ∞: Let us define the sequence y = (y k ) which will be used, by the ∆ (α) -transform of a sequence x = (x k ), that is, It seems those spaces can be considered as the matrix domains of the triangle ∆ (α) in the classical sequence spaces ℓp, where 1 ≤ p < ∞. We also have the following relation between the sequences x = (x k ) and y = (y k ): By Lemma 2.1, the defined fractional difference sequence spaces are complete, linear, BK spaces with the following norm: Let X be a normed space. Then S X = {x ∈ X : ‖x‖ = 1} andB X = {x ∈ X : ‖x‖ ≤ 1} denote the unit sphere and closed unit ball in X, where X is a normed space.
By Fr (r = 0, 1, . . . ), we denote the subcollection of Fr consisting of all nonempty and finite subsets of N with terms that are greater than r, that is Given a ∈ ω, we write provided the expression on the right hand side is defined and finite which is the case whenever X is a BK space and a ∈ X β . Lemma 2.2. Let X and Y be BK spaces. [34,Theorem 1.23]).

, Remark 3.5(a)]).
We have the following results for operator norms of bounded operators following [36,Theorem 2.8]. where where Y is any of the spaces c 0 , c, ℓ∞.
We obtain the following result as an immediate consequence of Lemma 2.2(ii).

Main results related to compact operators
The following notations are needed to establish estimates and identities for the Hausdorff measure of noncompactness of matrix operators and characterize the classes of compact operators. We also use the results in Katarina's paper [31] to prove our results.
We recall the definitions of Hausdorff measure of noncompactness of bounded subsets of a metric space, and Hausdorff measure of noncompactness of operators between Banach spaces.
If X and Y are infinite-dimensional complex Banach spaces then 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 (xn) in X, the sequence (L(xn)) has a convergent subsequence. We denote the class of such operators by C(X, Y).
Let X and Y be Banach spaces and χ 1 and χ 2 be measures of noncompactness on X and Y. Then the operator L : X → Y is called (χ 1 , χ 2 )-bounded if L(Q) ∈ M Y for every Q ∈ M X and there exists a positive constant C such that χ 2 (L(Q)) ≤ Cχ 1 (Q) (3.1) for every Q ∈ M X . If an operator L is (χ 1 , χ 2 )-bounded then the number is called the (χ 1 , χ 2 )-measure of noncompactness of L. In particular, if χ 1 = χ 2 = χ, then we write ‖L‖χ instead of ‖L‖ (χ,χ) .

Lemma 3.2. [34, Corollary 2.26] Let X and Y be Banach spaces and L ∈ B (X, Y). Then, we have
and L ∈ C(X, Y) if and only if ‖L‖ χ = 0.
When the first space is ℓ 1 , a problem takes place since β dual of ℓ 1 is ℓ∞, which has no AK. In this case, the following study of Sargent [32] is used in order to characterize compact operators. |a n,k1 − a n,k2 | = sup n |a n,k1 − a n,k2 | uniformly in k 1 and k 2 , (1 ≤ k 1 , k 2 < ∞). (ii) If A ∈ (X, c), then we have We are now ready to state the main results related to compact operators. We start with establishing some estimates for the norms of bounded linear operators L A on the given fractional sequence spaces ℓp(∆ (α) ).
(i) If A ∈ (X, c 0) , then we have (ii) If A ∈ (X, c), then we have (iİi) If A ∈ (X, ℓ 1) , then we have Proof. This is an immediate consequence of Theorem 2.9, Lemma 3.6 and Theorem 3.7.

Conclusion
The main objective of this paper is to consider fractional sequence spaces ℓp(∆ (α) ) for 1 ≤ p ≤ ∞ via the gamma function. We derive some estimates and identities for the norms of bounded linear operators on fractional sequence spaces ℓp(∆ (α) ). In general, Hausdorff measure of noncompactness is used to find necessary and sufficient conditions for a matrix operator on a given sequence space to be a compact operator. However, we can only obtain sufficient conditions when the final space is ℓ∞, that is for a matrix class (X, ℓ∞). This is why we use the results of [31] and [32] to exactly obtain necessary and sufficient conditions for the classes of compact matrix operators in (ℓp(∆ (α) ), ℓ∞) for 1 ≤ p ≤ ∞. In addition to these characterizations, we apply Hausdorff measure of noncompactness to establish necessary and sufficient conditions for a matrix operator to be a compact operator from fractional sequence spaces (X, Y), where X = ℓp(∆ (α) ) or X = ℓ 1 (∆ (α) ) (1 < p < ∞) and Y is any of the spaces c 0 , c and ℓ 1 . All these results are given with the following table: 1.  ℓp(∆ (α) ) ℓ 1 (∆ (α) ) c 0 1. 5. c 2. 6. |â n,k1 −â n,k2 | = sup n |â n,k1 −â n,k2 |, uniformly in k 1 and k 2 , (1 ≤ k 1 , k 2 < ∞) .