An Isomorphic Characterization of Property (β) of Rolewicz

In the paper it is shown that in a Banach space with a basis there is a norm with property (β) of S. Rolewicz if and only if there is a norm which is simultaneously nearly uniformly convex and nearly uniformly smooth. The Kuratowski measure of noncompactness of a set A in a Banach space is the infimum α(A) of those ǫ > 0 for which there is a covering of A by a finite number of sets Ai with diam(Ai) < ǫ. Let X be a Banach space with closed unit ball B. By the drop D(x,B) defined by an element x ∈ X \B, we mean conv({x}∪B) and we let R(x,B) = D(x,B)\B. Rolewicz [16] has proved that X is uniformly convex if and only if for each ǫ > 0 there is a δ > 0 such that 1 < ‖x‖ < 1 + δ implies diam(R(x, b)) < ǫ. In connection with this he has introduced [17] the following property. A Banach space X is said to have property (β) if for each ǫ > 0 there is a δ > 0 such that 1 < ‖x‖ < 1 + δ implies α(R(x,B)) < ǫ. The notation of nearly uniform convexity (NUC) has been introduced by Huff [4]. Rolewicz [17] has given the following equivalent definition. A Banach space X is said to be (NUC) if for each ǫ > 0 there is a δ, 0 < δ < 1, such that the measure of non-compactness of the slice S(f, δ) = {x ∈ X : ‖x‖ ≤ 1, f(x) ≥ 1 − δ} is smaller than ǫ for each continuous linear functional f with ‖f‖ = 1. A Banach space X is uniformly Kadec-Klee (UKK) if for every ǫ > 0 there is a δ > 0 such that ‖x‖ ≤ 1− δ whenever x is a weak limit of some sequence {xn} in B with sep(xn) = inf{‖xn − xm‖ : n 6= m} > ǫ. Huff [4] has proved that X is NUC if and only if X is reflexive and UKK. Rolewicz [17] has shown that UC ⇒ (β) ⇒ NUC. The class of Banach spaces with an equivalent norm with property (β) coincides neither with that of superreflexive spaces (independently proved by Montesinos and Torregrosa [13] and the author [5]), nor with the class of nearly uniformly convexifiable spaces (cf. [6] and [7]). In [8] and [9] we have defined the notions k-β, k ≥ 1, and k-NUC, k ≥ 2, where 1-β coincides with property (β). All of these properties imply NUC and they are even isomorphically stronger. Moreover, we have shown that Schachermayer’s space [18] is an example of a k-NUC space with k = 8, which fails to have an equivalent 1-β norm (i.e. with property (β)). In [9] we have also given some equivalent formulations of the notations k-β and k-NUC; in particular, we shall use in the sequel the following characterization of property (β). 1 Proposition 1. A Banach space X has the property (β) if and only if for each ǫ > 0 there exists a δ, 0 < δ < 1, such that for every element x ∈ B and every sequence {xn} ⊂ B with sep(xn) > ǫ, there is an index i so that ‖x+ xi‖/2 ≤ 1− δ. Sekowski and Stachura [19] and Prus [15] have independently defined the notion of nearly uniform smoothness (NUS) (see below). They have proved that a Banach space X (resp. X) is NUS if and only if X (resp. X) is NUC. We shall use also the equivalent definition given by Prus [15]. A Banach space X is said to be nearly uniformly smooth (NUS) if for every ǫ > 0 there exists an η > 0 such that for each t ∈ [0, η) and each basic sequence {un} in B there is an i > 1 such that ‖u1 + tui‖ ≤ 1 + ǫt. Prus has investigated finite dimensional decompositions of Banach spaces with (p, q)-estimates [14], and in [15] he has given a nice isomorphic characterization of NUS and NUC for Banach spaces with a countable basis in terms of (p, q)-estimates. (He has also mentioned that, using total biorthogonal systems instead of bases, the isomorphic characterization of NUS can be easily generalized to the case of separable spaces). Let {xn} be a basis of a Banach space X with coefficient functionals x ∗ n ∈ X . An element x ∈ X is said to be a block of {xn} if either x = 0 or the set supp x = {n : x∗n(x) 6= 0} is finite. A family {Xn} of finite dimensional subspaces of X is a blocking of {xn} provided there exists an increasing sequence of integers {nk}, n1 = 1, such that Xk = [xi] nk+1−1 i=nk for each k. We say that blocks y1, . . . , yn are disjoint (with respect to the blocking {Xk}) if min {