Almost contractive retractions in Orlicz spaces

Abstract

ALMOST CONTRACTIVE RETRACTIONS IN ORLICZ SPACESGRZEGORZ LEWICKI AND GIULIO TROMBETTA Let Bk denote the Euclidean unit ball in R* equipped with the A:-dimensional Lebesgue measure and let <f> : K + -* R + be a convex function satisfying 0(0) = 0, <j)(t) > 0 for some t > 0. Denote by E* = E^{B k ) the Orlicz space of finite elements (see (1.6)) generated by <fi. The aim of this paper is to show that there exists a retraction of the closed unit ball in E^, onto the unit sphere in E$ being a (2 + e^-s e t contraction (Theorem 3.6), which generalises [9, Corollary 6] proved for the case of L p [ -l , l ] , 1 ^ p < oo. Here 74, denote the Hausdorff measure of noncompactness. This theorem is proved both for the Amemiya and the Luxemburg norms. Also some related results concerning the case of s-convex (0 < s ^ 1) functions are presented.

Examples of proper k-ball contractive retractions in F-normed spaces

“…We observe that in the particular case α( f ) = 2 1+ f the mapping Q coincides with that introduced in [18] (see also [5,6,12]). …”

“…The space E ϕ equipped with the Luxemburg q-norm is a q-normed ideal space. By [13,Theorem 9.3] it follows that E ϕ is regular, and by [13,Theorem 7.6] we have S ⊆ E ϕ and that E ϕ is the closure of S with respect to the q-norm · Then by [12,Lemma 2.3], for all f ∈ E ϕ , we have β(a) f q ϕ f a q ϕ f q ϕ . Therefore the space E ϕ is a regular q-norm ideal space satisfying conditions (P1) and (P2) of the previous section.…”

Proper 1-ball contractive retractions in Banach spaces of measurable functions

“…Then by Lemma 2.4 the space L* satisfies property (PI). The following lemma proved in [12] shows that (P2) holds in Z,*. Observe that, if $(£) = V where 1 ^ p < 00, then L<& is the Lebesgue space L p := L p [0,1], with the standard norm || • || p .…”

“…In this section we have improved the results in the L p and L$ spaces of [17,12], respectively. Though the mapping Q is the same as the one introduced in those papers, here we construct in both cases a different retraction R and, above all, our proofs are based on different ideas and techniques.…”

“…In [19] it was shown that W{X) ^ 6 for any Banach space, reaching the value 4 and 3 depending on the geometry of the space X. Results in other Banach spaces can be found in [6,12,16,17]. Recently, in [1,Theorem 4] it has been proved that if the Banach space X has a monotone norm, then for any e > 0 there exists a (1 + e)-ball contractive retraction of B(X) onto S(X).…”