On approximation properties related to unconditionally p‐compact operators and Sinha–Karn p‐compact operators

Abstract

We establish new results on the -approximation property for the Banach operator ideal  =  𝑢𝑝 of the unconditionally 𝑝-compact operators in the case of 1 ≤ 𝑝 < 2. As a consequence of our results, we provide a negative answer for the case 𝑝 = 1 of a problem posed by Kim. Namely, the  𝑢1 -approximation property implies neither the  1 -approximation property nor the (classical) approximation property; and the  1 -approximation property implies neither the  𝑢1 -approximation property nor the approximation property. Here,  𝑝 denotes the 𝑝-compact operators of Sinha and Karn for 𝑝 ≥ 1. We also show for all 2 < 𝑝, 𝑞 < ∞ that there is a closed subspace 𝑋 ⊂ 𝓁 𝑞 that fails the  𝑟 -approximation property for all 𝑟 ≥ 𝑝.