New Convexity and Fixed Point Properties in Hardy and Lebesgue-Bochner Spaces

Abstract

We show that for the Hardy class of functions H 1 with domain the ball or polydisc in C N , a certain type of uniform convexity property (the uniform Kadec-Klee-Huff property) holds with respect to the topology of pointwise convergence on the interior; which coincides with both the topology of uniform convergence on compacta and the weak * topology on bounded subsets of H 1 .Also, we show that if a Banach space X has a uniform Kadec-Klee-Huff property, then the Lebesgue-Bochner space L p (µ, X) 1 ≤ p < ∞ must have a related uniform Kadec-Klee-Huff property. Consequently, by known results, the above spaces have normal structure properties and fixed point properties for non-expansive mappings.

The uniform Kadec-Klee property for the Lorentz spaces Lw,1

“…In the literature these are called the regular weights (see for example [4,8] 3. We say that L B]1 has property P if whenever we are given two sequences (/") and (g n ) such that ||/ n || = 1, ||/ B + g n \\ -> 1 as n -> oo, and /", g n are disjointly supported for each n, then ||g n || -> 0 as « -> oo.…”

“…The main result implies a fixed point theorem for non-expansive mappings (Corollary 3.3). See for example [3,5,9,12,13] for further results about the uniform Kadec-Klee property in classical spaces.…”

The fixed point property via dual space properties

“…Since H 1 has the weak * uniform Kadec-Klee property [2], its predual C(T )/A 0 has the weak fixed point property by this theorem. In the same manner, since C 1 (H ), the ideal of trace class operators on a Hilbert space H , has the weak * uniform Kadec-Klee property [12], its predual C ∞ (H ), the ideal of compact operators in B(H ), has the weak fixed point property.…”

Geometrical Background of Metric Fixed Point Theory

“…Next, fix n and consider the function g(t) = Φ( tx + x n ) where t ∈ R. By our assumption and Theorem 8, g is differentiable and g (t) = J φ (tx + x n )(x) for every t ∈ R. Since Φ is convex and increasing on [0, ∞), g is also convex. Hence g is continuous and consequently, (13) Φ…”

“…It should be noticed that independently of Huff a property equivalent to NUC was introduced in [46] under the name of noncompact uniform convexity. Lennard [69] extended Huff's concepts to an abstract topology τ (see also [13]). He used them to obtain a fixed point theorem in the space L 1 (Ω) with the clm topology.…”