## 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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