The Study of Matrix Properties and Primary Banach Space Problem

• Main Author: 徐淑真
• Other Authors: 林英雄
• Format: Others
• Language: en_US
• Published: 2001
• Online Access: http://ndltd.ncl.edu.tw/handle/96175955643240285192

碩士 === 國立彰化師範大學 === 數學系 === 89 === Let $P$ be a projection of a reflexive Banach space $X$ with a symmetric basis $\{e_n;f_n\}$. Then there exists a symmetric basis $\{e^{\prime}_n;f^{\prime}_{n}\}$ of $X$ and one of $P$ and $I-P$, say $Q$, such that the matrix of $Q$ with respect to $\{e^{\prime}_n\}$ has the properties: \begin{description} \item{(i)} Each column and each row contain at most finitely many nonzero entries; \item{(ii)} There exist infinitely many $\{v_n\}$ of positive integers such that $Q(e^{\prime}_{v_n})=0$, $Q^*(f^{\prime}_{v_n})=0$ $(n\in {\Bbb N})$; \item{(iii)} There exist infinitely many $\{w_n\}$ of positive integers such that $Q^*(f^{\prime}_{w_n})=f^{\prime}_{w_n}$ $(n\in {\Bbb N})$; \end{description} or $(I-Q)(X)\approx X$. This matrix properties is closely related to the answer of the problem that every reflexive Banach space with a symmetric basis is primary, which is unsolved for a lengthy time.