- Research Article
- Open Access
Orlicz Sequence Spaces with a Unique Spreading Model
© Cuixia Hao et al. 2010
- Received: 24 December 2009
- Accepted: 23 March 2010
- Published: 31 May 2010
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.
- Banach Space
- Basic Sequence
- Spreading Model
- Unconditional Basis
- Orlicz Function
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 " . 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  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  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].
They are nonvoid norm compact subsets of consisting entirely of Orlicz functions which might be degenerate [6, lemma ].
The sequence is called the spreading model of and it is a suppression-1 unconditional basic sequence if is weakly null .
The following theorem guarantees the existence of a spreading model of . We shall give a detailed proof.
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 ).
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 ).
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 ).
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.
In the same way, we can also "color" by .
Definition 2.6 is equivalent to Definition 2.1.
We can easily conclude Definition 2.1 from Definition 2.6
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 ).
If an Orlicz sequence space does not contain an isomorphic copy of , then the sets and coincide. That is, .
Definition 3.1 (see ).
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 ).
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 ).
Lemma 3.4 . (see ).
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 .
Let , and let be the unit basis of the space . If is lower semi-homogeneous, then if and only if is isomorphic to .
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
If , then by Lemma 2.8, that is, all the functions in are equivalent to .
Thus we get
So by (3.4) and (3.7), we can know that is equivalent to By Lemma 3.3 and its proof (, 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
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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