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.
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].
2. Preliminaries and Observations
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 .
In order to prove Theorem 2.2, we should have to recall the following definitions and theorem.
Definition 2.3 . (see ).
Definition 2.4 (see ).
Theorem 2.5 . (see ).
Proof of Theorem 2.2.
We accomplish the proof in two steps.
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 ).
3. Orlicz Sequence Spaces with Equivalent Spreading Models
Definition 3.1 (see ).
Lemma 3.2 (see ).
Lemma 3.3 (see ).
Lemma 3.4 . (see ).
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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