Volume 2010, Article ID 962842, 10 pagesdoi:10.1155/2010/962842 Research Article Orlicz Sequence Spaces with a Unique Spreading Model Cuixia Hao,1 Linlin L ¨u,2 and Hongping Yin3 1 Depar
Trang 1Volume 2010, Article ID 962842, 10 pages
doi:10.1155/2010/962842
Research Article
Orlicz Sequence Spaces with a Unique
Spreading Model
Cuixia Hao,1 Linlin L ¨u,2 and Hongping Yin3
1 Department of Mathematics, Heilongjiang University, Harbin 150080, China
2 Department of Information Science, Star College of Harbin Normal University, Harbin 150025, China
3 Department of Mathematics, Inner Mongolia University, Tongliao 028000, China
Correspondence should be addressed to Cuixia Hao,haocuixia@yahoo.com
Received 24 December 2009; Accepted 23 March 2010
Academic Editor: Shusen Ding
Copyrightq 2010 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
Let X be 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 of X itself” and not much more can be said about the question
“does X embedded into a nice subspace” The source of the research on spreading models was mainly from the question “finding a nice subspace Y ⊆ X” 1 The spreading models
paper, we study the question concerning the set of all spreading models whose cardinality is equal to one
The notion of a spreading model is one of the application of Ramsey theory It is a useful tool of digging asymptotic structure of Banach space, and it is a class of asymptotic
model and gave a result that every normalized weakly null sequence contains an asymptotic unconditional subsequence, they call the subsequence spreading model It was not until the last ten years that the theory of spreading models was developed, especially in recent five
and solved some of them Afterwards, Sari et al discussed some problems among them and obtained fruitful results This paper is mainly motivated by some results obtained by Sari et
Trang 22 Preliminaries and Observations
An Orlicz function M is a real-valued continuous nondecreasing and convex function defined
u0 > 0 such that M2u ≤ KMu for 0 ≤ u ≤ u0 We denote the modular of a sequence of
i1 Mxi It is well known that the space
l M
i1
2.1
endowed with the Luxemburg norm
λ
is a Banach sequence space which is called Orlicz sequence space The space
h M
i1
2.3
context, the Orlicz functions considered are nondegenerate Let
E M,1 M λt
They are nonvoid norm compact subsets of C0, 1 consisting entirely of Orlicz functions
Definition 2.1 Let X be a separable infinite dimensional Banach space For every normalized
and n ≤ k1< · · · < k n,
n
i1
a i x k i
−
n
i1
a i x i
The following theorem guarantees the existence of a spreading model of X We shall
give a detailed proof
Trang 3Theorem 2.2 Let x n be a normalized basic sequence in X and let ε n ↓ 0 Then there exists a
subsequence y n of x n so that for all n, a in
i1< i2 < · · · i n ,
n
j1
a j y k j
−
n
j1
a j y i j
In order to prove Theorem 2.2 , we should have to recall the following definitions and theorem.
denotes all subsequences of N.
Definition 2.3see 1 Let I1and I2be two disjoint intervals For anyk1, , k n , i1, , i n ∈
i1if
n
j1
a j y k j
∈ I i ,
n
j1
a j y i j
k1, , k n has the same “color” as i1, i2, , i n , where y i is a sequence of a Banach space
Definition 2.4see 1 The family of Nk k ∈ N is called finitely colored provided that it
only contains finite subsets in “color” sense, and each subset is 1-colored
Theorem 2.5 see 1 Let k ∈ N and let Nk
be finitely colored Then there exists M ∈ N ω so that M k is 1-colored.
Proof of Theorem 2.2 We accomplish the proof in two steps.
Step 1 We shall prove that for any n ∈ Z , there existsy i ⊆ x i such that for any a in
i1 ⊆
−1, 1, n ≤ k1< k2 < · · · k n , n ≤ i1 < i2 < · · · i n ,
n
j1
a j y k j
−
n
j1
a j y i j
Trang 4
k1, k2, k n by I lif
n
j1
a j y k j
N {z n1 , z n2 , z n m } for ε n /4-net of −1, 1 n For any element of net N, repeat the above
j n j1 , k 1, 2, m We partition 0, n into subintervals I lm
l1of
length < ε n /2 and “color” k1, k2, , k n by I lif
n
j1
z n k
j y i j
n
j1
z n k
j y k j
−
n
j1
z n k
j y i j
, ∗ holds Since N {z n1 , z n2 , z n m }
0jn
a in i1 − z n k
0
j1
j − z n k
0j
ε n
Therefore, we have
n
j1
a j y k j
≤
n
j1
a j − z n k
0j
y k j
n
j1
z n k
0j y k j
j1
j − z n k
0j
y k j
n
j1
z n k
0j y k j
j1
j − z n k
0j
n
j1
z n k
0j y k j
< ε n
n
j1
z n k
0j y k j
.
2.13
Trang 5
n
j1
a j y k j
−
n
j1
z n k
0j y k j
Similarly, we obtain
n
j1
a j y i j
−
n
j1
z n k
0j y i j
Thus
n
j1
a j y k j
−
n
j1
a j y i j
n
j1
a j y k j
−
n
j1
z n k
0j y k j
n
j1
z n k
0j y k j
−
n
j1
a j y i j
n
j1
z n k
0j y i j
−
n
j1
z n k
0j y i j
≤
n
j1
a j y k j
−
n
j1
z n k
0j y k j
n
j1
z n k
0j y k j
−
n
j1
a j y i j
n
j1
z n k
0j y k j
−
n
j1
z n k
0j y i j
< ε n
2.16
Step 2 We apply diagonal argument to prove that there exists y i ⊆ x i such that for any
n ∈ Z , a in
i1 ⊆ −1, 1, n ≤ k1< k2< · · · k n , n ≤ i1< i2< · · · i n ,
n
j1
a j y k j
−
n
j1
a j y i j
k1∈ Z , i1∈ Z , n ≤ k1, n ≤ i1, we have
ay k11 −ay1
i1
Trang 6
Obviously,{y1i } is also a normalized basic sequence So in view of n 2, there exists y2i ⊆
i1 ⊆ −1, 1, n ≤ k1 < k2, n ≤ i1< i2,
2
j1
a j y k2
j
−
2
j1
a j y2i j
i1⊆
−1, 1, n ≤ k1< k2 < · · · k n , n ≤ i1 < i2 < · · · i n, we have
n
j1
a j y n k j
−
n
j1
a j y i n j
k2< · · · k n , n ≤ i1< i2< · · · i n, we obtain that
n
j1
a j y k j
k j
−
n
j1
a j y i j
i j
Definition 2.6 Let X be a separable infinite-dimensional Banach space A normalized basic
· · · < k n, anda in
1 ε n−1
n
i1
a i x i
≤
n
i1
a i x k i
≤ 1 ε n
n
i1
a i x i
Theme 2.7. Definition 2.6is equivalent toDefinition 2.1
Proof We can easily concludeDefinition 2.1fromDefinition 2.6
“color”k1, k2, k n by I lif
n
i1
a j y k i
Let ρ ∈ Z , ρ ≥ n and ρ ≤ k1< · · · < k i0 < · · · k n; then
n
i1
a i x k i
−
n
i1
a i x i
Trang 7
where δ ρ ↓ 0, δ ρ > 0 Using the same procedure ofTheorem 2.2, we can get that for any
a in
n
i1
1
1 ε n a i x k i
−
n
i1
1
1 ε n a i x i
Thus
n
i1
1
1 ε n a i x k i
< δ ρ
n
i1
1
1 ε n a i x i
n
i1
a i x i
n
i1
a i x i
Letting ρ → ∞, then
n
i1
1
1 ε n a i x k i
≤
n
i1
a i x i
That is,
n
i1
a i x k i
≤ 1 ε n
n
i1
a i x i
Similarly,
1 ε n−1
n
i1
a i x i
≤
n
i1
a i x k i
Hence, we obtain that
1 ε n−1
n
i1
a i x i
≤
n
i1
a i x k i
≤ 1 ε n
n
i1
a i x i
order set If x i ≤ y i and y i ≤ x i, we call x i equivalent to y i, denoted by x i ∼ y i
We identify x i and y i in SP w X if x i ∼ y i
Lemma 2.8 see 5 If an Orlicz sequence space hM does not contain an isomorphic copy of l1, then the sets SP w h M and C M,1 coincide That is, SP w h M C M,1
Trang 83 Orlicz Sequence Spaces with Equivalent Spreading Models
Definition 3.1see 7 Let xn be a normalized Schauder basis of a Banach space X x n is
Lemma 3.2 see 7 Let M be an Orlicz function with M1 1, M ∈ Δ2, and let e i denote the
unit vector basis of the space h M The basis is
a lower semi-homogeneous if and only if CMst ≥ MsMt for all s, t ∈ 0, 1 and some
C ≥ 1,
b upper semi-homogeneous if and only if Mst ≤ CMsMt for s, t, C as above.
Lemma 3.3 see 6 The space lp , or c0if p ∞, is isomorphic to a subspace of an Orlicz sequence space h M if and only if p ∈ α M , β M , where
⎧
⎪
⎪q : sup 0<λ,
t≤1
M λt
M λt q < ∞
⎫
⎪
⎧
⎪
⎪q : sup 0<λ,
t≤1
M λt
M λt q > 0
⎫
⎪
Lemma 3.4 see 5 Let M ∈ Δ2, l M be an Orlicz sequence space which is not isomorphic to l1 Suppose that SP w l M is countable, up to equivalence Then
Theorem 3.5 Let M ∈ Δ2, and let e i be the unit basis of the space l M If e i is lower
semi-homogeneous, then |SP w l M | 1 if and only if l M is isomorphic to l p , p ∈ α M , β M .
Proof Su fficiency Since M ∈ Δ2, SP w l M is countable, then byLemma 3.4, lMis the upper
to l p , p ∈ α M , β M , we get |SP w l M | 1.
Necessity If |SP w l M | 1, then |C M,1| 1 byLemma 2.8, that is, all the functions in CM,1are
equivalent to M.
n
u /ω
Trang 9where 0 < u n < v n < ω n ≤ 1 with ω n → 0, u n /v n → 0, A n 1
u n /ω n Msω n s −p−1 ds.
M n t
M t A−1n
1
u n /w n
M tsw n
M n t
M t A−1n
1
u n /w n
M tsw n
M t s −p−1 ds
n
1
u n /w n
M tw n
M t s −p−1 ds
p A
−1
n
M tw n
M t
u n
w n
.
3.5
M n t
M t ≤ A−1n
M tw n
M t
u n
w n
p A
−1
n
u n
w n
Notice that for any fixed n, the right side of the above inequality is a constant; then we obtain
M n ≤ M
M n t
M t A−1n
1
u n /w n
M tsw n
M n t
M t ≥ A−1n
M tu n
M t
u n
w n
w n
nM u n
w p n
< M v n /2
Moreover,
w p n
v p n
> n2 −p M w n
Trang 10We obtain that
M n t
M t ≥ A−1n
M tu n
M t
u n
w n
w n
> A−1n
w n
w p n
v p n
M tu n
M t
> n · 2 −p A−1n
w n
M w n
M v n /2
M tu n
M t .
3.11
C ≥ 1
Therefore,
M n t
M t > n · 2 −p C−1A−1n
w n
M w n
Acknowledgment
References
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3 S J Dilworth, E Odell, and B Sari, “Lattice structures and spreading models,” Israel Journal of
Mathematics, vol 161, pp 387–411, 2007.
4 A Brunel and L Sucheston, “On B-convex Banach spaces,” Mathematical Systems Theory, vol 7, no 4,
pp 294–299, 1974
5 B Sari, “On the structure of the set of symmetric sequences in Orlicz sequence spaces,” Canadian
Mathematical Bulletin, vol 50, no 1, pp 138–148, 2007.
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7 M Gonz´alez, B Sari, and M W´ojtowicz, “Semi-homogeneous bases in Orlicz sequence spaces,” in
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Providence, RI, USA, 2007
... class="text_page_counter">Trang 9where < u n < v n < ω n ≤ with ω n → 0, u n... 8
3 Orlicz Sequence Spaces with Equivalent Spreading Models
Definition 3.1see 7 Let xn be a. .. w h M and C M,1 coincide That is, SP w h M C M,1
Trang