Smoothness, asymptotic smoothness and the Blum-Hanson property

Abstract

Abstract. We isolate various sufficient conditions for a Banach space X to have the so-called Blum-Hanson property. In particular, we show that X has the Blum-Hanson property if either the modulus of asymptotic smoothness of X has an extremal behaviour at infinity, or if X is uniformly Gâteaux smooth and embeds isometrically into a Banach space with a 1-unconditional finite-dimensional decomposition.

The Blum–Hanson property for 𝒞(K) spaces

“…As shown in [1], the same is true for the space CpT 2 q. On the other hand, it is observed in [7] that this is not so in the space Cpr0, 1sq, for the following reason: if θ : r0, 1s Ñ r0, 1s is a continuous map and if the iterates θ n converge pointwise to some continuous map α : r0, 1s Ñ r0, 1s, then the convergence is in fact uniform.…”

“…For the "if" part of the proof, we will make use of a result from [7] which is stated as Lemma 2.1 below.…”

“…On the other hand, it is known since [1] that CpT 2 q, the space of all continuous real-valued functions on the torus T 2 , does not have this property. Further results and references can be found in [7].…”

The Blum-Hanson Property

“…Once this fact is observed, there are several ways of proving Theorem 1. Probably the most elegant argument is the one presented in [15,Sec. 6.1], which relies on the existence of spectral measures for contractions on complex Hilbert spaces.…”

“…We will also prove Theorem 1 and give some examples of spaces which have BH. During the second lecture (Section 3), we will present and prove a recent criterion, due to Lefèvre-Matheron-Primot [15], proving that certain contractions (sometimes all contractions) on certain Banach spaces have BH. We will present some of its applications, as well as its limits.…”