Weak precompactness in projective tensor products

• Published in: Indagationes Mathematicae
• Main Authors: Rodríguez Ruiz, José, Rueda Zoca, Abraham
• Format: Article in Journal/Newspaper
• Language: English
• Published: Elsevier 2023
• Subjects: Projective tensor product; ℓ1-sequence; Weakly compact set; Weakly precompact set; Coarse p-limited set; tylli
• Online Access: https://hdl.handle.net/10481/84925; https://doi.org/10.1016/j.indag.2023.08.003

The research was supported by grants PID2021-122126NB-C32 (J. Rodríguez) and PID2021-122126NB-C31 (A. Rueda Zoca), funded by MCIN/ AEI /10.13039/501100011033 and “ERDF A way of making Europe”, and also by grant 21955/PI/22 (funded by Fundación Séneca - ACyT Región de Murcia, Spain). The research of A. Rueda Zoca was also supported by grants FQM-0185 and PY20_00255 (funded by Junta de Andalucía, Spain). We give a sufficient condition for a pair of Banach spaces (X,Y) to have the following property: whenever W1⊆X and W2⊆Y are sets such that {x⊗y:x∈W1,y∈W2} is weakly precompact in the projective tensor product X⊗̂πY, then either W1 or W2 is relatively norm compact. For instance, such a property holds for the pair (ℓp,ℓq) if 1<p,q<∞ satisfy 1/p+1/q≥1. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ1 into X⊗̂πY for arbitrary Banach spaces X and Y, in connection with the compactness of all operators from X to Y∗. PID2021-122126NB-C32 PID2021-122126NB-C31 MCIN/ AEI /10.13039/501100011033 “ERDF A way of making Europe” Fundación Séneca - ACyT Región de Murcia, Spain 21955/PI/22 Junta de Andalucía FQM-0185, PY20_00255