Ball covering property on operators and Calkin algebras
Abstract
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 $R$, $\delta \gt 0$ such that $r_i\leq R$ and $\|x_i\|-r_i \gt \delta $ 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 a $1$-unconditional basis or $X$ is a $1$-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.
