Renorming of ℓ1 and the fixed point property
Abstract
For any k ∈ N, let Pk denote the natural projections on ℓ1. Let | | | ṡ | | | be an equivalent norm of ℓ1 that satisfies all of the following four conditions: (1)There are α > 4 and a positive (decreasing) sequence (αn) in (0, 1) such that for any normalized block basis {fn} of (ℓ1, | | | ṡ | | |) and x ∈ ℓ1 with Pk - 1 (x) = x and | | | x | | | < αk,under(lim sup, n → ∞) | | | fn + x | | | ≤ 1 + frac(| | | x | | |, α) .(2)There are two strictly decreasing sequences {βk} and {γk} withunder(lim, k → ∞) βk = 0 and under(lim, k → ∞) γk = 1 such that for any normalized block basis {fn} of (ℓ1, | | | ṡ | | |) and x with (I - Pk) (x) = x,under(lim inf, n → ∞) | | | fn + x | | | ≥ 1 - βk + γk- 1 | | | x | | | .(3)For any k ∈ N, {norm of matrix} I - Pk {norm of matrix} = 1.(4)The unit ball of (ℓ1, | | | ṡ | | |) is σ (ℓ1, c0)-closed. In this article, we prove that the space (ℓ1, | | | ṡ | | |) has the fixed point property for the nonexpansive mapping. This improves a previous result of the author. © 2009 Elsevier Inc. All rights reserved.
Publication Title
Journal of Mathematical Analysis and Applications
Recommended Citation
Lin, P. (2010). Renorming of ℓ1 and the fixed point property. Journal of Mathematical Analysis and Applications, 362 (2), 534-541. https://doi.org/10.1016/j.jmaa.2009.09.064
