Some applications of Rademacher sequences in Banach lattices
It has been already known for some time that (generalized) Rademacher sequences can be defined and successfully used in arbitrary Banach lattices. The paper under review contains several interesting applications of such sequences. It is worth mentioning that quite often the Rademacher sequences do not show up in the statements of results but they play a crucial role in the proofs. Two sample results follow. Theorem 3.1. For a Banach lattice \(E\) the following three assertions are equivalent. (1) The conjugate space \(E^*\) is atomic. (2) For each sequence \(\{f_n\}\) in \(E^*\) that converges to zero in the topology \(\sigma(E^*,E)\), it follows that \(\inf_n|f_n|= 0\). (3) There is no sequence \(\{f_n\}\) in \(E^*\) such that \(f_n\to 0\) in \(\sigma(E^*,E)\), and \(|f_n|= f\) for all \(n\) and some \(f\in E^*\). Theorem 4.7. Consider two infinite-dimensional spaces \(L_p(\mu)\) and \(L_q(\nu)\) with \(1\leq \min p\), \(q<\infty\). Then the following two statements are equivalent. (1) Each continuous operator \(T: L_p(\mu)\to L_q(\nu)\) is compact. (2) \(p>q\) and either (a) or (b) holds. (a) \(p>2\) and \((L_p(\mu))^*\) is atomic. (b) \(q< 2\) and \(L_q(\nu)\) is atomic. The second theorem is a nice and, to some extent, a final generalization of Pitt's theorem asserting that \({\mathcal L}(\ell_p,\ell_q)= {\mathcal K}(\ell_p,\ell_q)\) for \(p>q\) and of its generalization by H. Rosenthal to \(L_p\)-spaces.
scientific article; zbMATH DE number 1203808
scientific article