http://chineseinput.net/에서 pinyin(병음)방식으로 중국어를 변환할 수 있습니다.
변환된 중국어를 복사하여 사용하시면 됩니다.
SUMMING AND DOMINATED OPERATORS ON A CARTESIAN PRODUCT OF c0 (X) SPACES
Gabriela Badea,Dumitru Popa 대한수학회 2017 대한수학회지 Vol.54 No.3
We give the necessary condition for an operator defined on a cartesian product of $c_{0}\left( \mathcal{X}\right) $ 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}\left( c_{0},\ldots,c_{0};c_{0}\right) $ contains a copy of $l_{s}\left( l_{2}^{m}\mid m\in \mathbb{N}\right) $ for $s>2$ or a copy of $l_{s}\left( l_{1}^{m}\mid m\in \mathbb{N}\right) $, for any $1\leq s<\infty $. Also, $\Delta _{s_{1},\ldots,s_{n}}\left( c_{0},\ldots,c_{0};c_{0}\right) $ contains a copy of $ l_{v_{n}\left( s_{1},\ldots,s_{n}\right) }$ if $v_{n}\left( s_{1},\ldots,s_{n}\right) \leq 2$ or a copy of $l_{v_{n}\left( s_{1},\ldots,s_{n}\right) }\left( l_{2}^{m}\mid m\in \mathbb{N} \right) $ if $2<v_{n}\left( s_{1},\ldots,s_{n}\right) $, where $\frac{1}{ v_{n}\left( s_{1},\ldots,s_{n}\right) }$ $=\frac{1}{s_{1}}+\cdots +\frac{1}{ s_{n}}$. We find also the necessary and sufficient conditions for bilinear operators induced by some method of summability to be $1$-summing or $2$ -dominated.
SUMMING AND DOMINATED OPERATORS ON A CARTESIAN PRODUCT OF c<sub>0</sub> (𝓧) SPACES
Badea, Gabriela,Popa, Dumitru Korean Mathematical Society 2017 대한수학회지 Vol.54 No.3
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.