Ball Covering Property on Operators and Calkin Algebra

2026/07/16 by Sreejith Siju, Bentuo Zheng
Mathematics · #math.FA

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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≤ ‖x_i‖ for each i∈ℕ. If there are R, δ>0 so that r_i≤ R and ‖x_i‖-r_i>δ for all i∈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 B(X)/K(X) fails the BCP. It is also shown that if X has a shrinking unconditional basis with unconditional constant less than 2, then B(X) has the UBCP.

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