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On the Riesz almost convergent sequences space. (English) Zbl 1250.46005

Summary: The purpose of this paper is to introduce new spaces \(\widehat{f}\) and \(\widehat{f}_0\) that consist of all sequences whose Riesz transforms of order one are in the spaces \(f\) and \(f_0\), respectively. We also show that \(\widehat{f}\) and \(\widehat{f}_0\) are linearly isomorphic to the spaces \(f\) and \(f_0\), respectively. The \(\beta\)- and \(\gamma\)-duals of the spaces \(\widehat{f}\) and \(\widehat{f}_0\) are computed. Furthermore, the classes \((\widehat{f} : \mu)\) and \((\mu : \widehat{f})\) of infinite matrices are characterized for any given sequence space \(\mu\) and determine the necessary and sufficient conditions on a matrix \(A\) to satisfy \(B_R - \text{core}(Ax) \subseteq K - \text{core}(x)\), \(B_R - \text{core}(A_R) \subseteq st - \text{core}(x)\) for all \(x \in \ell_\infty\).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
40C05 Matrix methods for summability
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References:

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