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Some new sequence spaces derived by the domain of generalized difference matrix. (English) Zbl 1201.40001

Summary: Let \(\lambda \) denote any one of the classical spaces \(\ell \infty , c, c_{0}\) and \(\ell p\) of bounded, convergent, null and absolutely \(p\)-summable sequences, respectively, and \(\hat{\lambda}\) also be the domain of the generalized difference matrix \(B(r,s)\) in the sequence space \(\lambda \), where \(1\leq p<\infty \). The present paper is devoted to studying on the sequence space \(\hat{\lambda}\). Furthermore, the \(\beta\)- and \(\gamma\)-duals of the space \(\hat{\lambda}\) are determined, and the Schauder bases for the spaces \(\hat{c_0}\), \(\hat{\ell_1}\) and \(\hat{\ell_p}\) are given, and some topological properties of the spaces \(\hat{c_0}\), \(\hat{\ell_1}\) and \(\hat{\ell_p}\) are examined. Finally, the classes \(\hat{\lambda_1:\lambda_2}\) and \(\hat{\lambda}_1:\hat{\lambda}_2\) of infinite matrices are characterized, where \(\lambda _{1}\in\{\ell_{\infty} ,c,c_{0},\ell p,\ell _{1}\}\) and \(\lambda _{2}\in\{\ell_{\infty},c,c_{0},\ell _{1}\}\).

MSC:

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