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Some new sequence spaces derived by the domain of the triple band matrix. (English) Zbl 1228.40006

Summary: Let \(\lambda \) denote any of the classical spaces \(\ell_{\infty },c,c_{0}\), and \(\ell _{p}\) of bounded, convergent, null, and absolutely \(p\)-summable sequences, respectively, and let \(\lambda (B)\) also be the domain of the triple band matrix \(B(r,s,t)\) in the sequence space \(\lambda \), where \(1<p<\infty\). The present paper is devoted to studying the sequence space \(\lambda (B)\). Furthermore, the \(\beta \)- and \(\gamma \)-duals of the space \(\lambda (B)\) are determined, the Schauder bases for the spaces \(c(B), c_{0}(B)\), and \(\ell _{p}(B)\) are given, and some topological properties of the spaces \(c_{0}(B), \ell _{1}(B)\), and \(\ell _{p}(B)\) are examined. Finally, the classes \((\lambda _{1}(B):\lambda _{2})\) and \((\lambda _{1}(B):\lambda _{2}(B))\) 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
40H05 Functional analytic methods in summability
46B45 Banach sequence spaces
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