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Matrix transformations and sequence spaces equations. (English) Zbl 1270.40008

Summary: We study sequence spaces equations (SSE) with operators, which are determined by an identity, each term of which is a sum or a sum of products of sets of the form \(\chi _{a}\left( T\right)\) and \(\chi _{f\left( x\right) }\left( T\right)\) where \(f\) maps \(U^{+}\) to itself, \(\chi \) is either of the symbols \(s, s^{0}\), or \(s^{\left( c\right) }\). Then we solve five (SSE) of the form \(\chi _{a}+\chi _{x}^{\prime }=\chi _{b}^{\prime }\), where \(\chi , \chi ^{\prime }\) are either \(s^{{{}0}}, s^{\left( c\right) }\), or \(s\). We apply the previous results to the solvability of the systems \(s_{a}^{0}+s_{x}\left( \Delta \right) =s_{b}\), \(s_{x}\supset s_{b}\), and \(s_{a}+s_{x}^{\left( c\right) }\left( \Delta \right) =s_{b}^{\left( c\right) }\), \(s_{x}^{\left( c\right) }\supset s_{b}^{\left( c\right) }\). Finally, we solve the (SSE) with operators defined by \(\chi _{a}\left( C\left( \lambda \right) D_{\tau }\right) +s_{x}^{\left( c\right) }\left( C\left( \mu \right) D_{\tau }\right) =s_{b}^{\left( c\right) }\) where \(\chi \) is either \(s^{0}\), or \(s\).

MSC:

40H05 Functional analytic methods in summability
46A45 Sequence spaces (including Köthe sequence spaces)
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References:

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