Ball Covering Property on Operators and Calkin Algebra

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 $r_i\le \Vert x_i\Vert$ for each $i\in \mathbb{N}$. If there are $R,\delta >0$ so that $r_i\le R$ and $\Vert x_i\Vert -r_i>\delta$ 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 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.

accepted for publication in Studia Mathematica