ON X-KOTHE¨ ECHELON SPACES AND APPLICATIONS By Fernando Blasco U. D. Matematicas,´ ETSI Montes, Universidad Politecnica´ de Madrid (Communicated by S. Dineen, m.r.i.a.) [Received 18 June 1997. Read 29 January 1998. Published 30 December 1998.] Abstract Properties such as Heinrich’s density condition or being Montel for X-Kothe¨ echelon spaces are considered and applications to the study of reflexivity of spaces of n-homogeneous continuous polynomials on Frechet´ spaces are given. Introduction In this article we study a class of Frechet´ spaces, the X-Kothe¨ echelon spaces, that generalises the class of Kothe¨ echelon spaces λp(A). The spaces λp(A), 1 ≤ p ≤∞, and many of their properties have been widely studied (see, for instance, [3], [4], [22], [30]) because they are a good source of examples and counterexamples in functional analysis. The spaces studied here are more general than Kothe¨ echelon spaces and are also a good source of getting examples, as we shall see in § 4. Bellenot defines X-Kothe¨ echelon spaces and obtains some of their properties in [2]. This ‘new’ class of echelon spaces has already been used to give examples and counterexamples [27]. We describe properties of X-Kothe¨ echelon spaces, point out similarities with ‘classical’ Kothe¨ echelon spaces and give applications. The paper consists of four sections: in § 1 we study basic properties of λX (A) and describe the dual space. In § 2 we characterise X-Kothe¨ echelon spaces that have Heinrich’s density condition. Section 3 describes λX (A) spaces that are Montel in terms of their Kothe¨ matrix. Finally, in § 4, examples and applications are given, which show the importance of these spaces in constructing new examples in functional analysis. 1. Basic properties ∞ Let X be a Banach space with a basis {en}n=1 and let a =(an)n∈N be a sequence of positive numbers. Suppose that k·kdenotes the norm in X. We define ( ) X∞ N Xa = (xn)n∈N ∈ C : anxnen ∈ X n=1 AMS Subject Class. (1991): 46G20,46A03 Mathematical Proceedings of the Royal Irish Academy, 98A (2), 191–208 (1998) c Royal Irish Academy 192 Mathematical Proceedings of the Royal Irish Academy P∞ and, for (xn)n∈N ∈ Xa, we define k(xn)n∈Nka = k n=1 anxnenk. It is easy to check that (Xa, k·ka) is a Banach space. Moreover, the mapping X∞ ja:(xn)n∈N 3 Xa 7→ anxnen ∈ X n=1 is an isometry between Xa and X. ∞ Definition 1.1 [2]. Let X be a Banach space, {en}n=1 a basis for X and A aKothe¨ matrix defined over N (i.e. A =(am)m∈N, where am is a sequence am =(am,k)k∈N, satisfying that am+1,k ≥ am,k > 0 for each pair (m, k) ∈ N × N). Then ( ) X∞ N λX (A)= x =(xn)n∈N ∈ C : am,nxnen ∈ X for all m ∈ N n=1 is called an X-Kothe¨ echelon space (with X as the associated Banach space). We give a Frechet´ space structure to λX (A) when we endowP it with the topology defined by the seminorms {k·k } , where k(x ) k = k ∞ a x e k . am m∈N n n∈N am n=1 m,n n n Classical Kothe¨ echelon spaces are a particular case of X-Kothe¨ echelon spaces. In the following, X will always be a Banach space with 1-unconditional Schauder basis, which sometimes will be shrinking and/or boundedly complete. We recall that an unconditionalP basis {en}n∈N for a Banach space X over C is called 1-unconditional ∞ ∞ if, for every x = n=1 xnen ∈ X and every complex sequence (θn)n=1 with |θn|≤1 P∞ P∞ for each n ∈ N, k n=1 θnxnenk ≤ k n=1 xnenk . For the definitions of shrinking and boundedly complete basis see [11]. Proposition 1.2. Let A =(am)m∈N beaKothe¨ matrix over N and let X be a Ba- nach space with a 1-unconditional basis {e }∞ . Then λ (A)=limX is a reduced n n=1 X ←− am projective limit. m Proof. By definition 1.1 and the definition of projective limit λ (A)=limX . To X am ←−m prove that it is reduced, fix m ∈ N and consider, for each n ∈ N, the sequence u =(δ ) ∈ λ (A). Since (u ) is total in X ,λ (A) is dense in X . Thus, the n n,k k∈N X n n∈N am X am projective limit is reduced. (The particular case where X is reflexive can be seen in [21, lemma 1 and theorem 1].) To obtain more information about X-Kothe¨ echelon spaces we need to define co-echelon spaces. For this purpose we adopt the following notations. Definition 1.3 [4, definitions 1.2 and 1.4]. For a Kothe¨ matrix A =(am)m∈N we de- 1 ¯ note by V the matrix (vm)m∈N, where vm =(vm,n)n∈N, with vm,n = , and by V the am,n set ¯ V = {¯v =(¯vn)n∈N: ¯vn ≥ 0 for each n ∈ N and sup am,n¯vn < ∞ for each m ∈ N}. n Blasco—On X-Kothe¨ echelon spaces 193 Remark 1.4. Every v ∈ V is a sequence of positive numbers, but this is not true for ¯ ¯ ¯v ∈ V. In fact, every sequence µ =(µn)n∈N, where µn =0forn large belongs to V. It is also important to note that, in the case we are considering, it is always possible to find positive elements in V.¯ We construct some of those elements by fixing a sequence (Rk)k∈N of positive numbers and defining ¯vn = min{Rkvk,n:1≤ k ≤ n}. By the unconditionality of the basis, it is easy to verify that ¯vn > 0 for all n and that ¯ ¯v =(¯vn)n∈N ∈ V. Definition 1.5. Let λX (A)beanX-Kothe¨ echelon space. The co-echelon spaces are defined as k (V )=∪ X and K (V¯ )=∩ X with the natural topologies X m∈N vm X ¯v∈V¯ ¯v (i) k (V )=limX , X vm −→m ¯ (ii) KX (V )=limX¯v. −→¯v It is well known that the inductive limit used in the definition of classical co- echelon spaces is a regular inductive limit [4, proof of theorem 2.7]. Results due to Komatsu ([21]) show that this is also true in our case when X is a reflexive Banach space: Proposition 1.6 [21, lemma 1 and theorem 6]. Let (Xj )j∈N be a sequence of reflexive Banach spaces. Then limXj is a complete reflexive space. Moreover, for every bounded −→j subset B in limXj there exists an index k such that B is bounded as a subset of Xk, −→j that is the inductive limit is regular. ¯ Both kX (V ) and KX (V ), which often coincide, can be considered as ‘dual spaces’ of λX (A). The following proposition extends the result due to Bierstedt, Meise and Summers for classical Kothe¨ echelon spaces [4, theorem 2.3]. Proposition 1.7. Let X be a Banach space with 1-unconditional boundedly ∞ ¯ complete basis {en}n=1. Then kX (V ) and KX (V ) are algebraically and topologically isomorphic. ¯ ¯ Proof. First we prove KX (V ) ⊂ kX (V ). If not, then there is x ∈KX (V ) such that x 6∈ X for every m ∈ N. Since the basis {e }∞ is boundedly complete, this is vm n n=1 n o PN equivalent to saying that, for each m ∈ N, the set n=1 vm,nxnen : N ∈ N is not bounded. Since {e }∞ is 1-unconditional basis there exists a strictly increasing n n=1 P Nm sequence (Nk)k∈N of natural numbers such that n=1 vm,nxnen >mfor every m ∈ N. 194 Mathematical Proceedings of the Royal Irish Academy Next we define the sequence ¯v =(¯vn)n∈N by ¯vn = vk,n, for Nk−1 <n≤ Nk (we assume N0 = 0). Since (vj )j∈N is decreasing we have am,n¯vn ≤ max {1, max{am,nvk,n: k<mand Nk−1 <n≤ Nk}} for each m and n ∈ N. Hence ¯v ∈ V.¯ Note that, for n ≤ N , we have ¯v ≥ v and, since {e }∞ is 1-unconditional, we P m P n m,n n n=1 Nm Nm ¯ finally obtain n=1 ¯vnxnen ≥ n=1 vm,nxnen >m,which contradicts x ∈KX (V ). So there must be m ∈ N for which x ∈ Xv . Since kX (V )=∪ Xv we obtain x ∈ kX (V ) m j j ¯ and hence KX (V ) ⊂ kX (V ). To prove the reverse inclusion, take m ∈ N,x∈ X and ¯v ∈ V.¯ Then vm X∞ X∞ X∞ ¯vn xn¯vnen = xn vm,nen ≤ sup am,n¯vn xnvm,nen < ∞, vm,n n=1 n=1 n∈N n=1 ¯ which implies that x ∈ X¯v and then kX (V ) ⊂ X¯v for every ¯v ∈ V. Hence we have the ¯ inclusion kX (V ) ⊂KX (V ). ¯ Finally we prove that kX (V ) and KX (V ) are topologically isomorphic. The ¯ above inequality gives the continuity of the inclusion kX (V ) ⊂KX (V ). To see the continuity of the opposite inclusionS let us take a neighbourhood U of0inkX (V ). We can assume that U = Γ¯ ( ρ B ), where (ρ ) is a sequence of positive m Xvm m m∈N m∈N numbers. For each m ∈ N, let 2m αm = max max{am,n:1≤ n ≤ m}, . ρm For each n ∈ N, choose j ∈ N, 1 ≤ j ≤ n, such that α v = min{α v :1≤ n n jn jn,n m m,n m ≤ n}. The sequence w¯ =(w¯ ) , where w¯ = α v is also in V¯ and w¯ > 0for n n∈N n jn jn,n n all n ∈ N (see Remark 1.4).