Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SUMMING AND DOMINATED OPERATORS ON A CARTESIAN PRODUCT OF c0 (𝓧) SPACES

  • Badea, Gabriela (Department of Mathematics, Ovidius University of Constanta) ;
  • Popa, Dumitru (Department of Mathematics, Ovidius University of Constanta)
  • Received : 2016.05.04
  • Published : 2017.05.01
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Abstract

We give the necessary condition for an operator defined on a cartesian product of $c_0(\mathcal{X})$ spaces to be summing or dominated and we show that for the multiplication operators this condition is also sufficient. By using these results, we show that ${\Pi}_s(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_s(l^m_2{\mid}m{\in}\mathbb{N})$ for s > 2 or a copy of $1_s(l^m_1{\mid}{\in}\mathbb{N})$, for any $l{\leq}S$ < ${\infty}$. Also ${\Delta}_{s_1,{\ldots},s_n}(c_0,{\ldots},c_0;c_0)$ contains a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}$ if ${\upsilon}_n(s_1,{\ldots},s_n){\leq}2$ or a copy of $l_{{\upsilon}_n(s_1,{\ldots},s_n)}(l^m_2{\mid}m{\in}\mathbb{N})$ if 2 < ${\upsilon}_n(s_1,{\ldots},s_n)$, where ${\frac{1}{{\upsilon}_n(s_1,{\ldots},s_n})}={\frac{1}{s_1}}+{\cdots}+{\frac{1}{s_n}}$. We find also the necessary and sufficient conditions for bilinear operators induced by some method of summability to be 1-summing or 2-dominated.

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Acknowledgement

Supported by : Romanian National Authority for Scientific Research

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