Báo cáo toán học: Some Results on the Properties D3 (f ) and D4 (f )

Mục đích của bài viết này là để cho characterizations subspaces và thương số của ∞ (I) ⊗ Π LF (α, ∞) và 1 (I) ⊗ Π LF (α, ∞) không gian là một phần mở rộng của kết quả Apiola [1] cho các trường hợp phi hạt nhân.
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Báo cáo toán học: "Some Results on the Properties D3 (f ) and D4 (f )"Vietnam Journal of Mathematics 34:2 (2006) 139–147 9LHWQD P -RXUQDO RI 0$ 7+ (0$ 7, &6 ‹ 9$67 Some Results on the Properties D3 (f ) and D4 (f ) Pham Hien Bang Department of Mathematics Thai Nguyen University of Education, Thai Nguyen, Vietnam Received February 17, 2005 Revised April 10, 2006Abstract. The aim of this paper is to give characterizations of subspaces and quo-tients of ∞ (I )⊗Π Lf (α, ∞) and 1 (I )⊗Π Lf (α, ∞)-spaces which are an extension ofresults of Apiola [1] for the non-nuclear case.2000 Mathematics subject classification: 46A04, 46A11, 46A32, 46A45Keywords: nuclear space, D3 (f ) property, D4 (f ) property1. IntroductionIn a series of important papers (see [1- 5, 9]) Vogt and Wagner studied char-acterizations of subspaces and quotients of nuclear power series spaces. LaterApiola in [1] has given a characterization of subspaces and quotients of nuclearLf (α, ∞)-spaces. Namely, he proved that a Frechet space E is isomorphic to asubspace (resp. quotient) of a stable nuclear Lf (α, ∞)-space if and only if Eis Λ(f, α, N)-nuclear in the sense of Ramanujan and Rosenberger (see [3]) andE ∈ D3 (f ) (see Theorem 3.2 in [1]) (resp E ∈ D4 (f ), see Theorem 3.4 in [1]). Inthis paper we investigate the Apiola’s results for the non-nuclear case. Namelywe prove the following result.Main theorem. Let E be a Frechet space. Then (i) E has D3 (f ) property if and only if there exists an index set I such that E is isomorphic to a subspace of ∞ (I )⊗Π Lf (α, ∞)-space for every stable nuclear exponent sequence α = (αj ).(ii) E has D4 (f ) property if and only if there exists an index set I such that140 Pham Hien Bang E is a quotient of 1 (I )⊗Π Lf (α, ∞)-space for every stable nuclear exponent sequence α = (αj ).Notice that when f (t) = t for t 0 and α = (log(j + 1))j the above theoremhas been proved by Vogt [5]. This paper is organized as follows. Beside theintroduction the paper contains three sections. In the second section we recallsome backgrounds concerning Lf (α, ∞)-spaces and D3 (f ) and D4 (f ) proper-ties. Some results of Apiola in [1] are presented also in this section. The thirdone is devoted to prove some auxiliary results which are used for the proof ofMain Theorem. The proof of Main Theorem is in the fourth section.2. Backgrounds2.1. Recall that a real function f on [0, +∞) is called a Dragilev function if fis rapidly increasing and logarithmically convex. This means that f (at) = ∞ for all a > 1 and t → log f (et ) lim t→+∞ f (t)is convex.Since f is rapidly increasing then there exists R > 0 such that f −1 (M t) RM f −1 (t) ∀t 0; ∀M 1(see [1]).For each Dragilev function f and each exponent sequence α = (αj ), i.e 0 <αj αj +1 for j 1 and lim αj = +∞ we define t→+∞ |ξj |ef (kαj ) } < ∞ ∀k Lf (α, ∞) = {ξ = (ξj ) ⊂ C : ξ = 1. k j12.2. Let E be a Frechet space with a fundamental system of semi-norms . 1 . 2 . . . and f a Dragilev function. We say that E has the property D3 (f ) if there exists p such that for everyM 1 and every q p, there exists k q such that M f −1 log( x f −1 log( x x p) x q) q kfor all x ∈ E {0}. We say that E has the property D4 (f ) if for every p there exists q p, andfor every k q there exists M 1 such that f −1 log( u ∗ u ∗) M f −1 log( u ∗ u ∗) q k p qfor all u∗ ∈ E {0}, where ∗ = sup |u(x)| : x u 1. q q2.3. ...