Analytic functions with values in lattices and symmetric spaces of measurable operators
Abstract
Abstract Let 0 < p , p i ≤ ∞, 0 < q , q i < ∞ ( i = 1, 2) such that Let E be a quasi-Banach lattice which fails to contain c 0 and whose α-convexity constant is equal to 1 for some 0 < α < ∞. Then for every f ∈ H ( E ( q ) ) there exist g ∈ H p , 0 ( E ( q 0 ) ), h ∈ H p 1 ( E ( q 1 ) ) such that Consequently, E is q-concave for some finite q if and only if E is uniformly H 1 -convexifiable in the sense of [24]. Analogous results are also obtained for symmetric spaces of measurable operators. Another result proved in the paper says that if E is a symmetric quasi-Banach function space on (0, ∞) having the analytic Radon–Nikodym property then L E ( M , τ) also possesses this property for any semifinite von Neumann algebra ( M , τ).