Volume 2010, Article ID 962842,10pages doi:10.1155/2010/962842
Research Article
Orlicz Sequence Spaces with a Unique
Spreading Model
Cuixia Hao,
1Linlin L ¨u,
2and Hongping Yin
31Department of Mathematics, Heilongjiang University, Harbin 150080, China
2Department of Information Science, Star College of Harbin Normal University, Harbin 150025, China 3Department of Mathematics, Inner Mongolia University, Tongliao 028000, China
Correspondence should be addressed to Cuixia Hao,[email protected]
Received 24 December 2009; Accepted 23 March 2010
Academic Editor: Shusen Ding
Copyrightq2010 Cuixia Hao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the set of all spreading models generated by weakly null sequences in Orlicz sequence spaces equipped with partial order by domination. A sufficient and necessary condition for the above-mentioned set whose cardinality is equal to one is obtained.
1. Introduction
LetXbe a separable infinite dimensional real Banach space. There are three general types of questions we often ask. In general, not much can be said in regard to this question “what can be said about the structure ofX itself” and not much more can be said about the question “doesX embedded into a nice subspace”. The source of the research on spreading models was mainly from the question “finding a nice subspaceY ⊆ X” 1. The spreading models
usually have a simpler and better structure than the class of subspaces ofX 2,3. In this
paper, we study the question concerning the set of all spreading models whose cardinality is equal to one.
2. Preliminaries and Observations
An Orlicz functionMis a real-valued continuous nondecreasing and convex function defined fort ≥0 such thatM0 0 and limt→ ∞Mt ∞.IfMt 0 for somet >0,Mis said to be a degenerate function.Muis said to satisfy theΔ2conditionM∈Δ2if there existK,
u0 >0 such thatM2u≤ KMufor 0 ≤u ≤ u0. We denote the modular of a sequence of numbersx{xi}∞i1byρMx
∞
i1Mxi. It is well known that the space
lM
x{xi}∞i1:ρMλx ∞
i1
Mλxi<∞for someλ >0
2.1
endowed with the Luxemburg norm
xinfλ >0 :ρM
x λ
≤1 2.2
is a Banach sequence space which is called Orlicz sequence space. The space
hM
x{xi}:ρMλx
∞
i1
Mλxi<∞for eachλ >0
2.3
is a closed subspace oflM. It is easy to verify that the spaceslp1 ≤ p < ∞are just Orlicz sequence spaces, and Orlicz sequence spaces are the generalization of the spaceslp1 ≤ p <
∞. Furthermore, ifMis a degenerate Orlicz function, thenlM ∼l∞andhM ∼c0 6. In the context, the Orlicz functions considered are nondegenerate. Let
EM,1
Mλt
Mλ : 0< λ <1
, CM,1 convEM,1. 2.4
They are nonvoid norm compact subsets of C0,1 consisting entirely of Orlicz functions which might be degenerate6, lemma 4.a.6.
Definition 2.1. LetXbe a separable infinite dimensional Banach space. For every normalized basic sequenceyiin a Banach space and for everyεn↓0, there exist a subsequencexiand a normalized basic sequencexisuch that for alln∈N,aini1∈−1,1nandn≤k1<· · ·< kn,
n
i1
aixki −
n
i1
aixi
< εn. 2.5
The sequencexiis called the spreading model ofxiand it is a suppression-1 unconditional basic sequence ifyiis weakly null4.
Theorem 2.2. Let xn be a normalized basic sequence in X and let εn ↓ 0. Then there exists a
subsequenceynofxnso that for alln, aini1 ⊆−1,1and integersn≤k1 < k2 <· · ·kn, n≤
i1< i2 <· · ·in,
n
j1
ajykj −
n
j1
ajyij
< εn. 2.6
In order to proveTheorem 2.2, we should have to recall the following definitions and theorem.
Fork ∈ N, Nk is the set of all subsets ofNof sizek. We may take it as the set of subsequences of lengthk,niki1 withn1 < · · · < nk.Nω denotes all subsequences ofN. Similar definitions apply toMkandMwifM∈Nw.
Definition 2.3see1. LetI1andI2be two disjoint intervals. For anyk1, . . . , kn,i1, . . . , in∈ Nkand scalarsaini1if
n
j1
ajykj ∈Ii,
n
j1
ajyij
∈Ii i1 or 2, 2.7
then we call Ii i 1 or 2 “color” k1, . . . , kn and i1, . . . , in. Meanwhile, we say k1, . . . , knhas the same “color” asi1, i2, . . . , in, whereyiis a sequence of a Banach space. We identify the same “color” subsets ofNk, saying they are 1-colored.
Definition 2.4see1. The family ofNk k∈Nis called finitely colored provided that it only contains finite subsets in “color” sense, and each subset is 1-colored.
Theorem 2.5see1. Letk∈Nand letNkbe finitely colored. Then there existsM∈Nωso thatMkis 1-colored.
Proof ofTheorem 2.2. We accomplish the proof in two steps.
Step 1. We shall prove that for anyn∈Z , there existsyi⊆xisuch that for anyaini1 ⊆ −1,1, n≤k1< k2 <· · ·kn, n≤i1 < i2 <· · ·in,
n
j1
ajykj −
n
j1
ajyij
< εn∗. 2.8
k1, k2, . . . knbyIlif
n
j1
ajykj
∈Il. 2.9
In the same way, we can also “color”i1, i2,· · ·inbyIl.
We can take −1,1n as the unit ball in finite-dimensional space ln1; then −1,1n is sequentially compact; moreover, it is totally bounded and complete. Underln
1-metric, take
N {z1n, z2n, . . . zmn}forεn/4-net of−1,1n. For any element of netN, repeat the above process, and letzkn zknjnj1, k 1,2, . . . m. We partition0, ninto subintervalsIlml1 of length< εn/2 and “color”k1, k2, . . . , knbyIlif
n
j1
zkn j yij
∈Il. 2.10
Since the length ofIl< εn/2, we have
n
j1
zkn j ykj
−
n
j1
zkn j yij
< εn
2 k1,2, . . . , m. 2.11
Secondly, we shall prove that for anyaini1⊆−1,1n, ∗holds. SinceN{z n
1 , z
n
2 , . . . z
n
m } is theεn/4-net of−1,1n, there existszkn0 zkn0
j
n
j1such that
aini1−zkn0
n
j1
aj−zkn0
j < εn
4. 2.12
Therefore, we have
n
j1
ajykj ≤
n
j1
aj−zkn0
j
ykj
n
j1
zkn0 jykj
≤n
j1
aj−zkn0
j ·ykj
n
j1
zkn0 jykj
n j1
aj−zkn0
j n
j1
zkn
0jykj
< εn
4
n
j1
zkn0 jykj
.
Hence, n
j1
ajykj − n
j1
zkn0 jykj
< εn
4. 2.14
Similarly, we obtain
n
j1
ajyij − n
j1
zkn0 jyij
< εn
4 . 2.15
Thus n
j1
ajykj − n
j1
ajyij n
j1
ajykj − n
j1
zkn
0jykj n
j1
zkn
0jykj − n
j1
ajyij n
j1
zkn
0jyij − n
j1
zkn
0jyij ≤ n
j1
ajykj − n
j1
zkn0 jykj
n
j1
zkn0 jykj
− n
j1
ajyij n
j1
zkn
0jykj − n
j1
zkn
0jyij
< εn
4
εn 4
εn 2 εn.
2.16
Step 2. We apply diagonal argument to prove that there existsyi ⊆ xisuch that for any
n∈Z , aini1 ⊆−1,1, n≤k1< k2<· · ·kn, n≤i1< i2<· · ·in,
n
j1
ajykj − n
j1
ajyij
< εn. 2.17
ByStep 1, in view ofn 1, there existsyi1 ⊆ xisuch that for anya ∈ −1,1, for any
k1∈Z , i1∈Z , n≤k1, n≤i1, we have
ay1
k1
−ay1
i1
Obviously,{yi1}is also a normalized basic sequence. So in view ofn2, there existsyi2⊆
yi1such that for anyai2i1⊆−1,1, n≤k1 < k2, n≤i1< i2,
2
j1
ajyk2j
−
2
j1
ajyij2
< ε2. 2.19
Repeating the above process, for anyn, there existsyin⊆yin−1such that for anyaini1⊆
−1,1, n≤k1< k2 <· · ·kn, n≤i1 < i2 <· · ·in, we have
n
j1
ajyknj
−
n
j1
ajyijn
< εn. 2.20
Finally, we choose the diagonal subsequenceyii⊂xi; for anyn,aini1 ⊆−1,1, n≤k1<
k2<· · ·kn, n≤i1< i2<· · ·in, we obtain that
n
j1
ajykkjj
−
n
j1
ajyijij
< εn. 2.21
Definition 2.6. LetX be a separable infinite-dimensional Banach space. A normalized basic sequencexi⊂Xgenerates a spreading modelxiif for someεn↓0, for alln∈N, n≤k1<
· · ·< kn, andain1 ⊆−1,1,
1 εn−1
n
i1
aixi
≤
n
i1
aixki
≤1 εn
n
i1
aixi
. 2.22
Theme 2.7. Definition 2.6is equivalent toDefinition 2.1.
Proof. We can easily concludeDefinition 2.1fromDefinition 2.6
By theDefinition 2.1, we know thatxiis a spreading model generated byxi. For
any fixedaini1⊆−1,1, we partition0, ninto some subintervalsIjmj1of length< ερand “color”k1, k2, . . . knbyIlif
n
i1
ajyki
∈Il 1≤l≤m. 2.23
Letρ∈Z , ρ≥nandρ≤k1<· · ·< ki0 <· · ·kn; then
n
i1
aixki −
n
i1
aixi
where δρ ↓ 0, δρ > 0. Using the same procedure ofTheorem 2.2, we can get that for any aini1⊆−1,1, εn↓0,
n
i1 1 1 εnaixki
−
n
i1 1 1 εnaixi
< δρ. 2.25
Thus
n
i1 1 1 εnaixki
< δρ
n
i1 1 1 εnaixi
δρ
1 1 εn
n
i1
aixi
≤δρ
n
i1
aixi
. 2.26
Lettingρ → ∞, then
n
i1 1 1 εnaixki
≤
n
i1
aixi
. 2.27
That is,
n
i1
aixki
≤1 εn
n
i1
aixi
. 2.28
Similarly,
1 εn−1
n
i1
aixi
≤
n
i1
aixki
. 2.29
Hence, we obtain that
1 εn−1
n
i1
aixi
≤
n
i1
aixki
≤1 εn
n
i1
aixi
. 2.30
LetSPwXbe the set of all spreading modelsxigenerated by weakly null sequences xiin X endowed with order relation by domination, that is, xi ≤ yiif there exists a constantK≥1 such thataixi ≤K
aiyifor scalarsai; thenSPwX,≤is a partial order set. Ifxi≤yiandyi≤xi, we callxiequivalent toyi, denoted byxi∼yi.
We identifyxiandyiinSPwXifxi∼yi.
Lemma 2.8see5. If an Orlicz sequence spacehMdoes not contain an isomorphic copy ofl1, then
3. Orlicz Sequence Spaces with Equivalent Spreading Models
Definition 3.1see7. Letxnbe a normalized Schauder basis of a Banach spaceX.xnis said to be lowerresp., uppersemihomogeneous if every normalized block basic sequence of the basis dominatesresp., is dominated by xn.
Lemma 3.2see7. LetMbe an Orlicz function withM1 1, M∈Δ2, and leteidenote the
unit vector basis of the spacehM. The basis is
alower semi-homogeneous if and only ifCMst≥MsMtfor alls, t∈0,1and some C≥1,
bupper semi-homogeneous if and only ifMst≤CMsMtfors, t, Cas above.
Lemma 3.3see6. The spacelp, orc0ifp∞, is isomorphic to a subspace of an Orlicz sequence
spacehMif and only ifp∈αM, βM, where
αMsup
⎧ ⎪ ⎨ ⎪ ⎩q: sup0<λ,
t≤1
Mλt Mλtq <∞
⎫ ⎪ ⎬ ⎪
⎭, 3.1
βMinf
⎧ ⎪ ⎨ ⎪ ⎩q: sup0<λ,
t≤1
Mλt Mλtq >0
⎫ ⎪ ⎬ ⎪
⎭. 3.2
Lemma 3.4see5. LetM ∈Δ2,lM be an Orlicz sequence space which is not isomorphic tol1.
Suppose thatSPwlMis countable, up to equivalence. Then
ithe unit vector basis oflMis the upper bound ofSPwlM;
iithe unit vector basis oflpis the lower bound ofSPwlM, wherep∈αM, βM.
Theorem 3.5. Let M ∈ Δ2, and let ei be the unit basis of the spacelM. Ifei is lower
semi-homogeneous, then|SPwlM|1if and only iflMis isomorphic tolp, p∈αM, βM.
Proof. Sufficiency. SinceM ∈Δ2,SPwlMis countable, then byLemma 3.4,lMis the upper bound ofSPwlM, andlp, p∈αM, βMis the lower bound ofSPwlM.SincelMis isomorphic tolp, p∈αM, βM, we get|SPwlM|1.
Necessity. If|SPwlM|1, then|CM,1|1 byLemma 2.8, that is, all the functions inCM,1are equivalent toM.
Forp∈αM, βM, we define the functionMnt 6as follows:
Mnt A−n1
1
un/ωn
where 0 < un < vn < ωn ≤ 1 with ωn → 0, un/vn → 0, An
1
un/ωnMsωns
−p−1ds. Obviously,Mnt∈CM,1; next we shall prove thatMntis equivalent toM
Mnt Mt A
−1 n
1
un/wn
Mtswn Mt s
−p−1ds. 3.4
Since s ≤ 1, swn ≤ wn, and M is nondecreasing convex function, therefore, Mtswn ≤
Mtwn; then
Mnt Mt A
−1 n
1
un/wn
Mtswn Mt s
−p−1ds
≤A−n1
1
un/wn
Mtwn Mt s
−p−1ds
1
pA
−1 n
Mtwn Mt
1−
un
wn
−p
.
3.5
Sincetwn< tandMtwn< Mt, we have
Mnt Mt ≤A
−1 n
Mtwn Mt
1−
un
wn
−p
≤ 1
pA
−1 n
1−
un
wn
−p
. 3.6
Notice that for any fixedn, the right side of the above inequality is a constant; then we obtain
Mn≤M
Mnt Mt A
−1 n
1
un/wn
Mtswn Mt s
−p−1ds. 3.7
Byun/wn≤s≤1, we haves−p−1≥un/wn−p−1andMtswn≥Mtun; hence
Mnt Mt ≥A
−1 n
Mtun Mt
un
wn
−p−1 1− un
wn
. 3.8
Sinceϕt Mt/tp,nϕwn< ϕvn/2, and
nMun wpn
< Mvn/2
vn/2p
. 3.9
Moreover,
wpn
vpn
> n2−pMwn
We obtain that
Mnt Mt ≥A
−1 n
Mtun Mt
un
wn
−p−1 1− un
wn
> A−n1
1− un
wn
wpn
vpn
Mtun Mt
> n·2−pA−n1
1− un
wn
Mwn Mvn/2
Mtun Mt .
3.11
Since 0< t, un ≤ 1, {ei}is lower semihomogeneous; then byLemma 3.2, we have for some
C≥1
CMtun≥MtMun. 3.12
Therefore,
Mnt Mt > n·2
−pC−1A−1 n
1− un
wn
Mwn
Mvn/2Mun. 3.13
Thus we getMn≥M.
So by3.4 and3.7, we can know thatMn is equivalent toM.By Lemma 3.3and its proof6, Theorem 4.a.9, we obtain thatMntuniformly converges totp on0,1/2. SinceCM,1is the closed subset ofC0,1/2, we have thattp∈CM,1,tpis equivalent toM, and thereforelMis isomorphic tolp.
Acknowledgment
The first author was supported by the NSF of Chinano. 10671048and by Haiwai Xueren Research Foundation in Heilongjiang Provinceno. 1055HZ003.
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