The Study of Matrix Properties and Primary Banach Space Problem
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...
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| Format: | Others |
| Language: | en_US |
| Published: |
2001
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| Online Access: | http://ndltd.ncl.edu.tw/handle/96175955643240285192 |
| Summary: | 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.
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