On sequence spaces generated by binomial difference operator of fractional order

Abstract

In this article we introduce binomial difference sequence spaces of fractional order α, $\begin{array}{} b_p^{r,s} \end{array}$ (Δ(α)) (1 ≤ p ≤ ∞) by the composition of binomial matrix, Br,s and fractional difference operator Δ(α), defined by (Δ(α)x)k = $\begin{array}{} \displaystyle \sum\limits_{i=0}^{\infty}(-1)^i\frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)}x_{k-i} \end{array}$. We give some topological properties, obtain the Schauder basis and determine the α, β and γ-duals of the spaces. We characterize the matrix classes ( $\begin{array}{} b_p^{r,s} \end{array}$(Δ(α)), Y), where Y ∈ {ℓ∞, c, c0, ℓ1} and certain classes of compact operators on the space $\begin{array}{} b_p^{r,s} \end{array}$(Δ(α)) using Hausdorff measure of non-compactness. Finally, we give some geometric properties of the space $\begin{array}{} b_p^{r,s} \end{array}$(Δ(α)) (1 < p < ∞).

Sequence spaces derived by the triple band generalized Fibonacci difference operator

“…For relevant literature, one may refer to [2,13,40,[49][50][51][52]. The reader may also consult the recent publications [22,24,25,53,62], which are related to compact operators and Hmnc in BK -spaces.…”

“…The set X is a sequence space and is known as the domain of matrix in the space X. Additionally, if X is BK -space and is a triangle, then X is also BK -space endowed with the norm s X = s X [27], where the matrix = (ψ rv ) is called a triangle if ψ rr = 0 for all r ∈ N and ψ rv = 0 for v > r. Using this famous result several authors [4,29,35,41,48] in the literature constructed new BK -spaces. We also mention [22,23,26,[53][54][55][62][63][64] for some recent publications and textbooks [6,47,61] in this field.…”

A new regular infinite matrix defined by Jordan totient function and its matrix domain in ℓp

“…The spaces are and In a recent paper by İlkhan, 4 matrix domains of the matrix Φ in the spaces c and c 0 have been introduced. By using the domains of some special summability matrices, several new spaces have been introduced by Sarıgöl, 5 Et, 6 Et and Çolak, 7 Altay et al, 8 Altay and Başar, 9 Kirişçi and Başar, 10 Mursaleen, 11 Mursaleen and Noman, 12,13 Demiriz and Çakan, 14 Kara, 15 Kara and İlkhan, 16 Candan, 17,18 Kirişçi, 19 Başarır and Kara, 20,21 Meng and Mei, 22 Das and Hazarika, 23 Yaying and Hazarika, 24 İlkhan et al, 25 Zengin Alp and İlkhan, 26 İlkhan, 27 Erdem and Demiriz, 28 and Demiriz et al 29 …”

Norm of matrix operator on Orlicz-binomial spaces and related operator ideal

“…For some recent papers on sequence spaces, one may refer to [5,8,16,20,36,39,40]. Let X and Y be two sequence spaces and let A = (a nk ) be an infinite matrix of real entries.…”