On compact operators and some EulerB(m)-difference sequence spaces

Abstract

Keywords: B (m) -difference sequence spaces Schauder basis α-, β-and γ -duals Matrix transformations Compact operators Hausdorff measure of noncompactness Altay and Ba¸sar (2005) [1] and Altay, Ba¸sar and Mursaleen (2006) [2] introduced the Euler sequence spaces e t 0 , e t c and e t ∞ . Başarır and Kayıkçı (2009) [3] defined the B (m) -difference matrix and studied some topological and geometric properties of some generalized Riesz B (m) -difference sequence space. In this paper, we introduce the Euler B (m) -difference sequence spaces e t 0 (B (m) ), e t c (B (m) ) and e t ∞ (B (m) ) consisting of all sequences whose B (m) -transforms are in the Euler spaces e t 0 , e t c and e t ∞ , respectively. Moreover, we determine the α-, β-and γ -duals of these spaces and construct the Schauder basis of the spaces e t 0 (B (m) )and e t c (B (m) ). Finally, we characterize some matrix classes concerning the spaces e t 0 (B (m) ) and e t c (B (m) ) and give the characterization of some classes of compact operators on the sequence spaces e t 0 (B (m) ) and e t ∞ (B (m) ).

On some binomial B ( m ) $B^{(m)}$ -difference sequence spaces

“…These spaces are the natural continuation of [18, 23–25]. Our results are the generalization of the matrix domain of the Euler matrix.…”

“…For instance, if we take , then we obtain the spaces , and , defined by Kara and Başarir [23]. If we take , and , then we obtain the spaces and , defined by Polat and Başar [18].…”

“…Also, Başarir and Kayikçi [22] defined the matrix by which is reduced to the matrix in the case , . Kara and Başarir [23] introduced the spaces , and of -difference sequences.…”

“…These new sequence spaces are the generalization of the sequence spaces defined in [24, 25] and [23]. Also, we give some inclusion relations and compute the bases and α -, β - and γ -duals of these sequence spaces.…”

Some normed binomial difference sequence spaces related to the ℓ p $\ell_{p}$ spaces

“…Here any term with negative subscript is equal to naught. Many authors have used especially the Euler matrix for defining new sequence spaces, for instance, Kara and Başarir [17], Karakaya and Polat [18] and Polat and Başar [15]. …”