On Absolute Euler Spaces and Related Matrix Operators
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RESEARCH ARTICLE
On Absolute Euler Spaces and Related Matrix Operators Fadime Go¨kc¸e1
•
Mehmet Ali Sarıgo¨l1
Received: 28 February 2017 / Revised: 13 January 2019 / Accepted: 16 May 2019 Ó The National Academy of Sciences, India 2019
Abstract In the present paper, we extend Euler sequence spaces erp and er1 by using the absolute Euler method in place of p-summable, which include the spaces lp , l1 , erp and er1 , investigate some topological structures, and determine a-, b-, c-duals and base. Further, we characterize certain matrix and compact operators on those spaces, and also obtain their norms and Hausdorff measures of noncompactness. Keywords Absolute Euler summability Matrix transformations Sequence spaces Bounded operators Hausdorff measures of noncompactness Mathematics Subject Classification 40C05 40D25 40F05 46A45
1 Introduction By x, we denote the space of all real- or complex-valued sequences. Any vector subspace of x is called a sequence space. We write c and l1 for the sequence space of all convergent and bounded sequences, respectively. Also by bs ; cs and lp ðp 1Þ, we denote the space of all bounded, convergent, p-absolutely convergent series, respectively. A complete sequence space X with a norm is called a BK& Fadime Go¨kc¸e [email protected] Mehmet Ali Sarıgo¨l [email protected]
space provided each of the maps pn : X ! C defined by pn ðxÞ ¼ xn is continuous for all n 0, where C denotes the complex field. A sequence ðbn Þ in a normed space X is called a Schauder base for X if for every x 2 XPthere is a unique ðxnÞ of scalars such that x ¼ 1 n¼0 xn bn , sequence P m or, x n¼0 xn bn ! 0 as m ! 1: For example, the sequence ðeðkÞ Þ is a Schauder base of lp with respect to its natural norm, 1 p\1; where eðkÞ is the sequence whose only nonzero term is 1 in kth place for each k 2 N, but the space l1 hasn’t the Schauder base [1]. Let X and Y be two subspaces of x and A ¼ ðanv Þ be an arbitrary infinite matrix of complex numbers. By AðxÞ ¼ ðAn ðxÞÞ, we denote P1the A-transform of the sequence x ¼ ðxv Þ, i.e., An ðxÞ ¼ v¼0 anv xv ; provided that the series converges for n 0. Then, A defines a matrix transformation from X into Y, and it is denoted by A 2 ðX; YÞ or A : X ! Y if Ax ¼ ðAn ðxÞÞ 2 Y for every x 2 X. The a-, b-, c-duals of X and domain of the matrix A in X are defined by X a ¼ f 2 x : ðn xn Þ 2 l for all x 2 X g; X b ¼ f 2 x : ðn xn Þ 2 cs for all x 2 X g; X c ¼ f 2 x : ðn xn Þ 2 bs for all x 2 X g and XA ¼ fx ¼ ðxn Þ 2 x : AðxÞ 2 X g;
ð1Þ
respectively. P Now, take av as an infinite series with nth partial sum sn and let ðhn Þ be a sequence of nonnegative terms. Then, P we say that the series av is summable j A; hn jp and j A; hn j1 , 1 p\1, if (see [2]) 1 X p hp1 n jAn ðsÞ An1 ðsÞj \1 n¼0
1
Department of Mathematics, University of Pamukkale, 20070 Denizli, Turkey
and
123
F. Go¨kc¸e, M. A. Sarıgo¨l
sup h1=p n jAn ðsÞ An1 ðsÞj\1; n
respectively, where p is the conjugate of p, i.e., 1=p þ 1=p ¼ 1 for p [ 1, and 1=p ¼ 0 for p ¼ 1.
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