AI 中文总结
本文研究了算子上的球覆盖性质与卡林代数，证明了特定条件下卡林代数不满足BCP，并展示了在某些条件下B(X)具有UBCP。
AI 中文摘要
一个巴拿赫空间X被称为具有球覆盖性质（BCP），如果X的单位球可以被可数多个开球B(x_i, r_i)覆盖，其中每个i∈N都有r_i≤‖x_i‖。如果存在R, δ>0使得对于所有i∈N，r_i≤R且‖x_i‖-r_i>δ，则称X具有统一球覆盖性质（UBCP）。本文证明，如果X具有1-无条件基或X是具有收缩1-无条件基的巴拿赫空间的1-补集子空间，则卡林代数B(X)/K(X)不满足BCP。同时证明，如果X具有收缩无条件基且无条件常数小于2，则B(X)具有UBCP。
英文摘要
A Banach space $X$ is said to have the ball covering property (BCP) if the unit sphere of $X$ can be covered by countably many open balls $B(x_i, r_i)$ with $r_i\leq \|x_i\|$ for each $i\in\mathbb{N}$. If there are $R, δ>0$ so that $r_i\leq R$ and $\|x_i\|-r_i>δ$ for all $i\in\mathbb{N}$, then we say that $X$ has the uniform ball covering property (UBCP). In this paper, we show that if $X$ has an $1$-unconditional basis or $X$ is an $1$-complemented subspace of a Banach space with a shrinking $1$-unconditional basis, then the Calkin algebra $\mathcal{B}(X)/\mathcal{K}(X)$ fails the BCP. It is also shown that if $X$ has a shrinking unconditional basis with unconditional constant less than 2, then $\mathcal{B}(X)$ has the UBCP.
Commentsaccepted for publication in Studia Mathematica
