The Convexity Characteristic of the Calder(?)n-Lozanovski(?) Sequence Spaces and Some Geometric Problems
|Keywords||the Characteristic of Convexity Orlicz-Lorentz sequence space order isometric copy δ2-condition Orlicz space Banach lattice Calderón-Lozanovskiǐspace uniformly monotone|
In this paper,we were devoted to some geometric problems of the Calderon—Loza novskii sequence spaces eΦ,where e was a symmetric Banach sequence space with the Fatou property. Some problems about embeddings of l∞in eΦunder different assumptions were studied. Based on the above study,we achieved some results on order continuous property of eΦand the convexity characteristicε0(eΦ). Then we gave a necessary and sufficient condition of Orlicz—Lorentz sequence spaceλΦ,ωbeing uniformly monotone. By studying on the convexity characteristic of eΦ,the equalitty expression of the convexity characteristic ofλΦ,ωwas given,which generalized Y.Cui’s results and we derived a condition of AΦ,ωbeing uniformly non-square and uniformly rotund.The paper is divided into five chapters.In the first chaper,we listed some basic definitions,signs,the background and main contents.In the second chapter,some problems about embeddings of l∞in eΦand order continuous property of eΦwere studied.We obtained that:1. ifΦ(?)δ2,then eΦcontains an order isometric copy of 200.2.1et e is order continuous,then eΦcontains an order isometric copy of l∞if and only ifΦ(?)δ2.3.ifΦvanishes only at zero,then eΦis order continuous if and only if e is order continuous andΦ∈δ2.In the third chapter,we estimated the convexity characteristicε0(eΦ)of eΦ.Then we got some geometric properties of eΦ,including some conditions about uniform rotundity and uniform non-squareness in eΦ.Main results:1.1etΦis strictly convex on [0,ub], if e is not order continuous orΦ(?)δ2,thenε0(eΦ)=2；if e is uniformly monotone andΦ∈δ2,thenε0(eΦ)≤2(1-p(Φ))/(1+p(Φ)).2.if e is uniformly monotone,Φ∈δ2 andΦis uniformly convex on [0,ub],then eΦis uniformly rotund.3.if e is uniformly monotone,Φis strictly convex on[0,ub],Φ∈δ2 andΨ∈δ2,then eΦis uniformly non-square.In the fourth chapter,we studied uniform monotonity in Orlicz-Lorentz sequence spaceλΦ,ωand a property of Kothe dual MΨ,ω.We obtained that:1.if e is uniformly monotone andΦ∈δ2,then eΦis uniformly monotone.2.λΦ,ωis uniformly monotone if and only ifωis regular,Φ∈δ2.3.ifωis regular andΦ(?)δ2,then MΦ,ωcontains an isomorphic copy of l∞.In the fifth chapter,we applied the results in the above chapters to Orlicz—Lorentz sequence spaceλΦ,ω,and deduced that:1.when u→0lim (Φ(u))/u=0,ifΦ(?)δ2 orΨ(?)δ2,orωis not regular,thenε0(λΦ,ω)=2；ifΦδ2 andωis regular,thenε0(λΦ,ω)≤(2(1-p(Φ))/(1+p(Φ);if Φis strictly convex on[0,ub],Φ∈δ2 andωregular,thenε0(λΦ,ω)=(2(1-p(Φ))/(1+p(Φ)).According to this result,we obtained a sufficient condition about uniform rotundity ofλΦ,ω:ifωis regular,Φ∈δ2 andΦis uniformly convex on[0,ub]i.e.p(Φ)=1,thenλΦ,ωis uniformly rotund.2.ifωis regular,uniform non-squareness, B-convexity, reflexivity andΦ∈δ2,Ψ∈δ2 are pairwise equivalent inλΦ,ω.