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 
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 
itself" and not much more can be said about the question "does 
embedded into a nice subspace". The source of the research on spreading models was
mainly from the question "finding a nice subspace 
" [1]. The spreading models usually have a simpler and better structure than the class
of subspaces of 
[2, 3]. In this 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 unconditional basis. In 1974, Brunel and Sucheston [4] introduced the concept of spreading 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 years. In 2005, Androulakis et al. in [2] put forward several questions on spreading models 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 al. in their papers [3, 5].
2. Preliminaries and Observations
An Orlicz function 
is a real-valued continuous nondecreasing and convex function defined for 
such that 
and 
If 
for some 
, 
is said to be a degenerate function. 
is said to satisfy the 
condition 
if there exist 
, 
such that 
for 
. We denote the modular of a sequence of numbers 
by 
. It is well known that the space
(21)endowed with the Luxemburg norm
(22)is a Banach sequence space which is called Orlicz sequence space. The space
(23)is a closed subspace of 
. It is easy to verify that the spaces 
are just Orlicz sequence spaces, and Orlicz sequence spaces are the generalization
of the spaces 
. Furthermore, if 
is a degenerate Orlicz function, then 
and 
[6]. In the context, the Orlicz functions considered are nondegenerate. Let
(24)They are nonvoid norm compact subsets of 
consisting entirely of Orlicz functions which might be degenerate [6, lemma 
].
Definition 2.1.
Let 
be a separable infinite dimensional Banach space. For every normalized basic sequence
in a Banach space and for every 
, there exist a subsequence 
and a normalized basic sequence 
such that for all 
, 
and 
,
(25)The sequence 
is called the spreading model of 
and it is a suppression-1 unconditional basic sequence if 
is weakly null [4].
The following theorem guarantees the existence of a spreading model of 
. We shall give a detailed proof.
Theorem 2.2.
Let 
be a normalized basic sequence in 
and let 
. Then there exists a subsequence 
of 
so that for all 
and integers 
(26)In order to prove Theorem 2.2, we should have to recall the following definitions and theorem.
For 
is the set of all subsets of 
of size 
. We may take it as the set of subsequences of length 
, 
with 
. 
denotes all subsequences of 
. Similar definitions apply to 
and 
if 
.
Definition 2.3 . (see [1]).
Let 
and 
be two disjoint intervals. For any 
, 
and scalars 
if
(27)then we call 
"color" 
and 
. Meanwhile, we say 
has the same "color" as 
, where 
is a sequence of a Banach space. We identify the same "color" subsets of 
, saying they are 1-colored.
Definition 2.4 (see [1]).
The family of 
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 
and let 
be finitely colored. Then there exists 
so that 
is 1-colored.
Proof of Theorem 2.2.
We accomplish the proof in two steps.
Step 1.
We shall prove that for any 
, there exists 
such that for any 
(28)Firstly, for fixed 
by the Definition 2.4, we can prove that the above inequality holds. In fact, we
partition 
into subintervals 
of length 
and "color" 
by 
if
(29)In the same way, we can also "color" 
by 
.
We can take 
as the unit ball in finite-dimensional space 
; then 
is sequentially compact; moreover, it is totally bounded and complete. Under 
-metric, take 
for 
-net of 
. For any element of net 
, repeat the above process, and let 
. We partition 
into subintervals 
of length 
and "color" 
by 
if
(210)Since the length of 
, we have
(211)Secondly, we shall prove that for any 
holds. Since 
is the 
-net of 
, there exists 
such that
(212)Therefore, we have
(213)Hence,
(214)Similarly, we obtain
(215)Thus
(216)Step 2.
We apply diagonal argument to prove that there exists 
such that for any 
(217)By Step 1, in view of 
, there exists 
such that for any 
, for any 
, we have
(218)Obviously, 
is also a normalized basic sequence. So in view of 
, there exists 
such that for any 
,
(219)Repeating the above process, for any 
, there exists 
such that for any 
, we have
(220)Finally, we choose the diagonal subsequence 
; for any 
, we obtain that
(221)Definition 2.6.
Let 
be a separable infinite-dimensional Banach space. A normalized basic sequence 
generates a spreading model 
if for some 
, for all 
, and 
,
(222)Theme 2.7.
Definition 2.6 is equivalent to Definition 2.1.
Proof.
We can easily conclude Definition 2.1 from Definition 2.6
By the Definition 2.1, we know that 
is a spreading model generated by 
. For any fixed 
, we partition 
into some subintervals 
of length 
and "color" 
by 
if
(223)Let 
and 
; then
(224)where 
. Using the same procedure of Theorem 2.2, we can get that for any 
(225)Thus
(226)Letting 
, then
(227)That is,
(228)Similarly,
(229)Hence, we obtain that
(230)Let 
be the set of all spreading models 
generated by weakly null sequences 
in 
endowed with order relation by domination, that is, 
if there exists a constant 
such that 
for scalars 
; then 
is a partial order set. If 
and 
, we call 
equivalent to 
, denoted by 
. We identify 
and 
in 
if 
.
Lemma 2.8 (see [5]).
If an Orlicz sequence space 
does not contain an isomorphic copy of 
, then the sets 
and 
coincide. That is, 
.
3. Orlicz Sequence Spaces with Equivalent Spreading Models
Definition 3.1 (see [7]).
Let 
be a normalized Schauder basis of a Banach space 
. 
is said to be lower (resp., upper) semihomogeneous if every normalized block basic
sequence of the basis dominates (resp., is dominated by) 
.
Lemma 3.2 (see [7]).
Let 
be an Orlicz function with 
, and let 
denote the unit vector basis of the space 
. The basis is
(a)lower semi-homogeneous if and only if 
for all 
and some 
,
(b)upper semi-homogeneous if and only if 
for 
as above.
Lemma 3.3 (see [6]).
The space 
, or 
if 
, is isomorphic to a subspace of an Orlicz sequence space 
if and only if 
, where
(31)
(32)Lemma 3.4 . (see [5]).
Let 
, 
be an Orlicz sequence space which is not isomorphic to 
. Suppose that 
is countable, up to equivalence. Then
(i)the unit vector basis of 
is the upper bound of 
;
(ii)the unit vector basis of 
is the lower bound of 
, where 
.
Theorem 3.5.
Let 
, and let 
be the unit basis of the space 
. If 
is lower semi-homogeneous, then 
if and only if 
is isomorphic to 
.
Proof.
Sufficiency. Since 
, 
is countable, then by Lemma 3.4, 
is the upper bound of 
, and 
is the lower bound of 
Since 
is isomorphic to 
, we get 
Necessity.
If 
, then 
by Lemma 2.8, that is, all the functions in 
are equivalent to 
.
For 
, we define the function 
[6] as follows:
(33)where 
with 
. Obviously, 
; next we shall prove that 
is equivalent to 
(34)Since 
, 
, and 
is nondecreasing convex function, therefore, 
; then
(35)Since 
and 
, we have
(36)Notice that for any fixed 
, the right side of the above inequality is a constant; then we obtain 
(37)By 
, we have 
and 
; hence
(38)Since 
, 
, and
(39)Moreover,
(310)We obtain that
(311)Since 
is lower semihomogeneous; then by Lemma 3.2, we have for some 
(312)Therefore,
(313)Thus we get 
So by (3.4) and (3.7), we can know that 
is equivalent to 
By Lemma 3.3 and its proof ([6], Theorem 
.a.9), we obtain that 
uniformly converges to 
on 
Since 
is the closed subset of 
, we have that 
, 
is equivalent to 
, and therefore 
is isomorphic to 
Acknowledgment
The first author was supported by the NSF of China (no. 10671048) and by Haiwai Xueren Research Foundation in Heilongjiang Province (no. 1055HZ003).
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