GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 1, 1997, 27-40

Λ_{0}-NUCLEAR OPERATORS AND Λ_{0}-NUCLEAR SPACES IN p-ADIC ANALYSIS

A. K. KATSARAS AND C. PEREZ-GARCIA

Abstract. For a Köthe sequence space, the classes of Λ_{0}-nuclear spaces and spaces with the Λ_{0}-property are introduced and studied and the relation between them is investigated. Also, we show that, for Λ_{0} ≠ c_{0}, these classes of spaces are in general different from the corresponding ones for Λ_{0} = c_{0}, which have been extensively studied in the non-archimedean literature (see, for example, [1]–[6]).

Introduction

Throughout this paper K will be a complete non-archimedean valued field whose valuation | · | is non-trivial, and E, F, … will be locally convex spaces over K. We always assume that E, F, … are Hausdorff.

It is well known (see [5]) that a locally convex space E is nuclear if and only if

(1) For every Banach space F, every continuous linear map (or operator) from E into F is compact.

Nuclear spaces are closely related to the locally convex spaces E satisfying the following property:

(2) Every operator from E into c_{0} is compact (see [5]).

On the other hand, it is well known that if F is a normed space, then an operator T from E into F is compact if and only if there exist an equicontinuous sequence (f_{n}) in E′, a bounded sequence (y_{n}) in F, and an element (λ_{n}) of c_{0} such that

T (x) = ∑_{n=1}^{∞} λ_{n}f_{n}(x)y_{n} ∀x ∈ E  (∗)

(an operator satisfying this condition is called a nuclear operator, see [7]). 27

1991 Mathematics Subject Classification. 46S10.

Key words and phrases. p-adic functional analysis, Köthe sequence spaces, nuclearity, compactoidity.

1072-947X/97/0100-0027$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

In [7] and [8] the authors studied several properties of operators T which can be represented as in (*) where (λ_{n}) belongs to some Köthe sequence space Λ_{0}. They are called Λ_{0}-nuclear operators.

Let us introduce for a locally convex space E the following properties: (1′) For every Banach space F, every operator from E into F is Λ_{0}-

nuclear.

(2′) Every operator from E into c_{0} is Λ_{0}-nuclear.

In this paper we study property (1′) as related to (2′). We show that if Λ_{0} ≠ c_{0}, the class of spaces satisfying property (1′) (resp. (2′)) is in general different from the corresponding one for Λ_{0} = c_{0}.

In the classical case of spaces over the real or complex field, analogous problems have been studied by several authors (see, for example, [9]–[14]).

§ 1. Preliminaries

Let E be a locally convex space over K. We will denote by cs(E) the collection of all continuous non-archimedean seminorms on E. For p ∈ cs(E), E_{p} will be the associated normed space E/ ker p endowed with the usual norm, and π_{p} : E ⟶ E_{p} will be the canonical surjection. E is said to be of countable type if for every p ∈ cs(E), E_{p} is a normed space of countable type (i.e., E_{p} is the closed linear hull of a countable set). For p ∈ cs(E) and r > 0, B_{p}(0, r) will be the set {x ∈ E : p(x) ≤ r}. Also, for each continuous linear functional f ∈ E′, we define ∥f∥_{p} = sup{|f(x)|/p(x) : x ∈ E, p(x) ≠ 0}.

Next, we will recall the definition of a non-archimedean Köthe space Λ(P). By a Köthe set we will mean a collection P of sequences α = (α_{n}) of non-negative real numbers with the following properties:

(1) For each n ∈ N there exists α ∈ P with α_{n} ≠ 0.

(2) If α, α′ ∈ P, then there exists β ∈ P with α, α′ ≪ β, where α ≪ β means that there exists d > 0 such that α_{n} ≤ dβ_{n} for all n.

For α ∈ P and a sequence ξ = (ξ_{n}) in K, we define p_{α}(ξ) = sup_{n} α_{n}|ξ_{n}|. The non-archimedean Köthe sequence space Λ(P) is the space of all sequences ξ in K for which p_{α}(ξ) < ∞ for all α ∈ P. On Λ(P) we consider the locally convex topology generated by the family {p_{α} : α ∈ P } of nonarchimedean seminorms. Under this topology Λ(P) is a complete Hausdorff locally convex space over K. The set |Λ| = {|x| : x ∈ Λ(P)} is a Köthe set. By Λ̄ we will denote the Köthe space Λ(|Λ|). Also, by Λ_{0} = Λ_{0}(P) we will denote the closed subspace of Λ(P) consisting of all ξ = (ξ_{n}) for which α_{n}|ξ_{n}| → 0 for each α = (α_{n}) ∈ P. In case P consists of a single constant sequence (1, 1, …), we have Λ(P) = ℓ^{∞} and Λ_{0}(P) = c_{0}. Also, we give the following interesting example:

p-ADIC ANALYSIS: Λ_{0}-NUCLEAR OPERATORS, Λ_{0}-NUCLEAR SPACES 29

Let B = (b_{n}^{k}) be an infinite matrix of strictly positive real numbers and satisfying the conditions b_{n}^{k} ≤ b_{n}^{k+1} for all k, n. For each k, let α^{(k)} = (b_{1}^{k}, b_{2}^{k}, …). Then, P = {α^{(k)} : k = 1, 2, … } is a Köthe set for which Λ_{0}(P) coincides with the Köthe space K(B) = {(λ_{n}) : λ_{n} ∈ K, ∀n and lim_{n} |λ_{n}|b_{n}^{k} = 0, k = 1, 2, 3, … } associated with the matrix B (see [4]). Also, the topology on Λ_{0}(P) for this P coincides with the normal topology on K(B) considered in [4]. This kind of spaces play an important role in padic analysis, since every non-archimedean countably normed Fréchet space E with a Schauder basis can be identified with K(B), for some infinite matrix B ([4], Proposition 2.4).

We will say that the Köthe set P is a power set of infinite type if (i): For each α ∈ P we have 0 < α_{n} ≤ α_{n+1} for all n, and (ii): For each α ∈ P there exists β ∈ P with α^{2} ≪ β. We will say that P is stable if for each α ∈ P there exists β ∈ P such that sup_{n} α_{2n}/β_{n} < ∞ . By [7], Proposition 2.11, P is stable if and only if Λ(P) (or Λ_{0}(P)) is stable. (Recall that a locally convex space E is called stable if E × E is topologically isomorphic to E.)

Finally, we will recall the concepts of Λ_{0}-compactoid sets and Λ_{0}-nuclear operators (see [7]). For a bounded subset A of a locally convex space E, p ∈ cs(E) and a non-negative integer n, the nth Kolmogorov diameter δ_{n,p}(A) of A with respect to p is the infimum of all |μ|, μ ∈ K, for which there exists a subspace F of E with dim(F) ≤ n such that A ⊂ F +μB_{p}(0, 1). The set A is called Λ_{0}-compactoid if for each p ∈ cs(E) there exists ξ = (ξ_{n}) ∈ Λ_{0} such that δ_{n,p}(A) ≤ |ξ_{n+1}| for all n (or equivalently α_{n}δ_{n−1,p}(A) → 0 for all α ∈ P). An operator (continuous linear map) T ∈ L(E, F) between two locally convex spaces E,F over K is called:

(1) Λ_{0}-nuclear if there exist an equicontinuous sequence (f_{n}) in E′, a bounded sequence (y_{n}) in F, and an element (λ_{n}) of Λ_{0} such that

T x = ∑_{n=1}^{∞} λ_{n}f_{n}(x)y_{n} ∀x ∈ E;

(2) Λ_{0}-compactoid if there exists a neighborhood V of zero in E such that T (V) is Λ_{0}-compactoid in F;

(3) Λ_{0}-quasinuclear if for each q ∈ cs(F) there exist a sequence (f_{n}) in E′, a p ∈ cs(E), and an element (λ_{n}) of Λ_{0} such that ∥f_{n}∥_{p} ≤ |λ_{n}| (n ∈ N) and q(T x) ≤ sup_{n} |f_{n}(x)| for all x ∈ E. (For the ideal structure of these classes of operators see [7].)

By Theorem 4.4 of [7], every Λ_{0}-nuclear operator is Λ_{0}-compactoid. Also, every Λ_{0}-compactoid operator is Λ_{0}-quasinuclear. Indeed, if T is Λ_{0}- compactoid and q ∈ cs(F), then π_{q} ∘ T : E ⟶ F_{q} is also Λ_{0}-compactoid ([7], Proposition 3.21) and so π_{q} ∘T is Λ_{0}-nuclear ([7], Theorem 4.7). Hence, T is Λ_{0}-quasinuclear.

It follows from Theorem 4.6 of [7] that if F is a normed space, then T is Λ_{0}-nuclear ⇔ T is Λ_{0}-compactoid ⇔ T is Λ_{0}-quasinuclear.

In case Λ_{0} = c_{0}, the concepts of Λ_{0}-compactoid set, Λ_{0}-compactoid operator, and Λ_{0}-nuclear operator coincide with the concepts of a compactoid set, a compact operator, and a nuclear operator, respectively.

For further information we refer to [15] (for normed spaces) and to [16] (for locally convex spaces).

From now on in this paper we will assume that the Köthe set P is a power set of infinite type.

§ 2. Spaces with the Λ_{0}-Property

Locally convex spaces E for which every T ∈ L(E, c_{0}) is compact have been studied by N. De Grande-De Kimpe in [2] and [3] and more recently by T. Kiyosawa in [6].

A natural extension of this kind of spaces is given by

Definition 2.1. We say that a locally convex space E has the Λ_{0}-pro-perty if every T ∈ L(E, c_{0}) is Λ_{0}-nuclear (or, equivalently, Λ_{0}-compactoid).

In this section, we study several properties of spaces with the Λ_{0}-property. In this way, we extend and complete the results previously obtained by N. De Grande-De Kimpe and T. Kiyosawa.

Proposition 2.2.

(a) If E has the Λ_{0}-property and M is a subspace of E such that every T ∈ L(M, c_{0}) has an extension T̄ ∈ L(E, c_{0}) (e.g., when M is dense or when M is complemented), then M has the Λ_{0}-property.

(b) A locally convex space E has the Λ_{0}-property if and only if its completion Ê has the Λ_{0}-property.

(c) A quotient of a space E with the Λ_{0}-property also has the same property.

(d) If P is stable, then the product of a family of spaces with the Λ_{0}- property has the same property.

Proof. Property (a) is obvious.

(b): It follows by (a) that if Ê has the Λ_{0}-property, then E has also the same property.

Conversely, suppose that E has the Λ_{0}-property. Let T ∈ L(Ê, c_{0}) and let T_{1} be the restriction of T to E. Since T_{1} is Λ_{0}-compactoid, there exists a zero-neighborhood U in E such that T_{1}(U) is Λ_{0}-compactoid in c_{0}. Then V = Ū Ê is a zero-neighborhood in Ê for which T (V) is Λ_{0}-compactoid in c_{0}, and so T is Λ_{0}-compactoid.

(c): Let M be a closed subspace of E and let S ∈ L(E/M, c_{0}). If π : E ⟶ E/M is the quotient map, then T = S ∘ π ∈ L(E, c_{0}) is Λ_{0}- compactoid. If V is a neighborhood of zero in E such that T (V) is Λ_{0}- compactoid in c_{0}, then π(V) is a neighborhood of zero in E/M for which S(π(V)) = T (V) is Λ_{0}-compactoid in c_{0}. Hence S is Λ_{0}-compactoid.

(d): Let E = ∏_{i} E_{i}, where each E_{i} has the Λ_{0}-property, and let T ∈ L(E, c_{0}). Then T is bounded on a neighborhood W of zero in E. This neighborhood can be taken in the form W = ∏_{i} U_{i}, where U_{i} is a zeroneighborhood in E_{i} and the set J = {i ∈ I : U_{i} ≠ E_{i}} is finite. Clearly, T vanishes on the subspace ∏_{i∉J} E_{i} of E and so we may assume that I is finite, i.e., E = E_{1} × E_{2} × … × E_{n} for some n ∈ N. For j = 1, 2, ..., n, let π_{j} : E_{j} ⟶ E be the canonical inclusion. Since T_{j} = T ∘ π_{j} ∈ L(E_{j}, c_{0}) is Λ_{0}-compactoid, there exists a zero-neighborhood V_{j} in E_{j} such that T_{j}(V_{j}) is Λ_{0}-compactoid in c_{0}. Then, V = V_{1} ×V_{2} ×⋯×V_{n} is a zero neighborhood in E for which T (V) = T_{1}(V_{1}) + … + T_{n}(V_{n}) is Λ_{0}-compactoid in c_{0} ([7], Proposition 3.14). Thus T is Λ_{0}-compactoid.

Now, we fix some notation which we will use in the sequel. For each n ∈ N, there are unique k, m ∈ N such that n = (2m − 1)2^{k−1}. In the following lemma π_{1}, π_{2} : N ⟶ N will be defined by π_{1}(n) = k and π_{2}(n) = m when n = (2m − 1)2^{k−1}.

Lemma 2.3. Suppose that P is countable and stable and, for each k ∈ N, let ξ^{k} = (ξ_{n}^{k})_{n} ∈ Λ_{0}. Then there exists a sequence (λ_{k})_{k} of non-zero elements of K such that (λ_{π_{1}(n)}ξ_{π_{2}(n)}^{π_{1}(n)})_{n} ∈ Λ_{0}.

Proof. We may assume that P = (α^{(k)})_{k∈N}, where α^{(k)} ≤ α^{(k+1)} for all k. Since P is stable, we may also assume that for each k ∈ N there exists 0 < d_{k} < ∞ with d_{k} ≤ d_{k+1} such that sup_{m} α_{2^{k}.m}^{(k)}/α_{m}^{(k+1)} ≤ d_{k}. Choose λ_{k} ∈ K, 0 < |λ_{k}| ≤ 1 such that p_{α^{(k+1)}}(λ_{k}ξ^{k}) ≤ k^{−1}d_{k}^{−1} (k ∈ N). We claim that the sequence (λ_{k})_{k} satisfies the requirements.

Indeed, let r ∈ N and let ϵ > 0 be given. Choose k_{0} > max{r, 1/ϵ}. Also, choose η_{n}^{k} ∈ K with |η_{n}^{k}| = max_{m≥n} |ξ_{m}^{k}| (k, n ∈ N). Then, η^{k} = (η_{n}^{k})_{n} ∈ Λ_{0} for all k ∈ N and so there exists m_{0} ∈ N such that d_{k_{0}}α_{m}^{(k_{0})}|η_{m}^{k}| < ϵ for all m ≥ m_{0} and all k ≤ k_{0}. Let n > m_{0}2^{k_{0}}. If k = π_{1}(n) < r, then k < k_{0} and hence m > m_{0}. Thus, for k = π_{1}(n) < r, we have

α_{n}^{(r)}|λ_{π_{1}(n)}ξ_{π_{2}(n)}^{π_{1}(n)}| ≤ α_{n}^{(r)}|η_{m}^{k}| ≤ α_{m2^{r}}^{(r)}|η_{m}^{k}| ≤ d_{k_{0}}α_{m}^{(k_{0})}|η_{m}^{k}| < ϵ.

For r ≤ k = π_{1}(n) < k_{0}, we have

α_{n}^{(r)} ≤ α_{m2^{k}}^{(k)} ≤ d_{k}α_{m}^{(k+1)} ≤ d_{k_{0}}α_{m}^{(k_{0})}

and, since m > m_{0} we obtain that

α_{n}^{(r)}|λ_{k}ξ_{m}^{k}| ≤ d_{k_{0}}α_{m}^{(k_{0})}|ξ_{m}^{k}| < ϵ.

Analogously, we can prove that if π_{1}(n) = k ≥ k_{0} > r, then we have α_{n}^{(r)}|λ_{k}ξ_{m}^{k}| < ϵ.

Hence, for n > m_{0}.2^{k_{0}}, we obtain α_{n}^{(r)}|λ_{π_{1}(n)}ξ_{π_{2}(n)}^{π_{1}(n)}| < ϵ, which clearly completes the proof.

Theorem 2.4. Let P be countable and stable. Then the locally convex direct sum and the inductive limit of a sequence of spaces with the Λ_{0}- property have also the same property.

Proof. Let E = ⨁_{k=1}^{∞} E_{k}, where each E_{k} has the Λ_{0}-property and let T ∈ L(E, c_{0}). If I_{k} : E_{k} ⟶ E is the canonical inclusion, then T ∘I_{k} ∈ L(E_{k}, c_{0}) is Λ_{0}-nuclear (k ∈ N). Therefore, for each k, there exist ξ^{k} = (ξ_{m}^{k})_{m} ∈ Λ_{0}, a sequence (y_{m}^{k})_{m} in the unit ball of c_{0}, and an equicontinuous sequence (h_{m}^{k})_{m} in E′_{k} such that

(T ∘ I_{k})(y) = ∑_{m=1}^{∞} ξ_{m}^{k}h_{m}^{k}(y)y_{m}^{k} (y ∈ E_{k}).

For each k ∈ N let q_{k} ∈ cs(E_{k}) with |h_{m}^{k}| ≤ q_{k} for all m. Also, let π_{1}, π_{2} and (λ_{k})_{k} be as in Lemma 2.3. Then q(x) = max_{k} |λ_{k}|^{−1}q_{k}(x_{k}) (x = (x_{k})_{k} ∈ E) defines a continuous seminorm on E. For each pair (m, k) of positive integers, the function g_{m}^{k} : E ⟶ K, x ⟶ λ_{k}^{−1}h_{m}^{k}(x_{k}) is a continuous linear map on E such that |g_{m}^{k}| ≤ q for all k, m. Also, for each x = (x_{k})_{k} = ∑_{k=1}^{∞} I_{k}(x_{k}) ∈ E we have

T x = ∑_{k=1}^{∞} ∑_{m=1}^{∞} λ_{k}ξ_{m}^{k}g_{m}^{k}(x)y_{m}^{k}.

For n = (2m − 1)2^{k−1}, set f_{n} = g_{m}^{k} ∈ E′, z_{n} = y_{m}^{k} ∈ c_{0}, ξ_{n} = λ_{k}ξ_{m}^{k} ∈ K. By Lemma 2.3 (ξ_{n})_{n} ∈ Λ_{0}. Further, T x = ∑_{n=1}^{∞} ξ_{n}f_{n}(x)z_{n} for all x ∈ E, and so T is Λ_{0}-nuclear.

Finally, we observe that the inductive limit of a sequence of spaces is linearly homeomorphic to a quotient of the corresponding direct sum.

Remark. A subspace of a space with the Λ_{0}-property need not have in general the same property. Indeed, let Λ_{0} = c_{0} and suppose that the valuation on K is dense. Then, ℓ^{∞} has the Λ_{0}-property ([15], Corollary 5.19) but, clearly, c_{0} does not have the same property.

Examples.

1. As we will see in the next section, every Λ_{0}-nuclear space has the Λ_{0}-property.

2. If Λ_{0} = c_{0} and the valuation on K is dense, then ℓ^{∞} has the Λ_{0}- property.

3. If P is countable and K is not spherically complete, then Λ̄ has the Λ_{0}-property ([8], Corollary 4.6).

4. If E is an infinite-dimensional Banach space with a basis, then E does not have the Λ_{0}-property. Indeed, E contains a complemented subspace linearly homeomorphic to c_{0} ([15], Corollary 3.18).

For a locally convex space E over K, we will denote by Λ_{0}{E′} the family of all sequences (g_{n}) in E′ for which there exist p ∈ cs(E) and (λ_{n}) ∈ Λ_{0} such that ∥g_{n}∥_{p} ≤ |λ_{n}| for all n. For a sequence w = (g_{n}) ∈ Λ_{0}{E′}, we define a continuous non-archimedean seminorm p_{w} on E by

p_{w}(x) = sup_{n} |g_{n}(x)| (x ∈ E).

The next Theorem gives several descriptions of spaces with the Λ_{0}- property.

Theorem 2.5. For a locally convex space E, the following properties are equivalent:

(i) E has the Λ_{0}-property.

(ii) For every T ∈ L(E, c_{0}) there exist T_{1} ∈ L(E, Λ_{0}), which is Λ_{0}- nuclear, and T_{2} ∈ L(Λ_{0}, c_{0}) such that T = T_{2} ∘ T_{1}.

(iii) If F is a locally convex space of countable type, then every T ∈ L(E, F) is Λ_{0}-quasinuclear.

(iv) If F is a normed space and T ∈ L(E, F), then T is Λ_{0}-nuclear if and only if its range, R(T), is of countable type.

(v) Let (T_{n}) be an equicontinuous sequence of operators from E into a normed space F such that R(T_{n}) is of countable type for all n and such that (T_{n}) converges pointwise to a T ∈ L(E, F). Then T is Λ_{0}-nuclear.

(vi) For every equicontinuous sequence (f_{n}) in E′, which converges pointwise to zero, there exists w ∈ Λ_{0}{E′} such that ∥f_{n}∥_{p_{w}} ≤ 1 for all n.

(vii) For every equicontinuous sequence (f_{n}) in E′, which converges pointwise to zero, there exist (g_{n}) ∈ Λ_{0}{E′}, α ∈ P, d > 0, and an infinite matrix (ξ_{ik}) of elements of K, with lim_{n→∞} ξ_{in} = 0 for all i and |ξ_{in}| < dα_{i} for all n, such that

f_{n}(x) = ∑_{i=1}^{∞} g_{i}(x)ξ_{in} (x ∈ E).

If, in addition, P is stable, then properties (i) → (vii) are equivalent to: (viii) The topology of uniform convergence on the members of Λ_{0}{E′}

coincides with the topology τ_{0} of countable type which is associated with the topology of E (see [17]).

Proof. For the equivalence of (i) and (ii) see the proof of Theorem 4.6 in [7].

(i) ⇒ (iii): Let F be a locally convex space of countable type. For every p ∈ cs(F), the associated normed space F_{p} is of countable type and so F_{p} is linearly homeomorphic to a subspace of c_{0}. Hence π_{p} ∘ T : E ⟶ F_{p} is Λ_{0}-nuclear ([7], Theorem 4.11). Thus T is Λ_{0}-quasinuclear.

(iii) ⇒ (iv): Observe that, since Λ_{0} ⊂ c_{0}, we have that every Λ_{0}-nuclear operator is also nuclear, and hence its range is of countable type.

(iv) ⇒ (v): Let (T_{n}) and T be as in (v). Since every R(T_{n}) is of countable type, the closed linear hull Z of ⋃_{n} R(T_{n}) is of countable type. Also, since T x ∈ Z for all x ∈ E, (iv) implies that T is Λ_{0}-nuclear.

(i) ⇔ (vi): From Theorem 4.6 of [7] it follows that a map T ∈ L(E, c_{0}) is Λ_{0}-nuclear if and only if there exists w ∈ Λ_{0}{E′} such that ∥T x∥ ≤ p_{w}(x) for all x ∈ E. Now, apply Lemma 2.2 of [3] to get the conclusion.

(ii) ⇔ (vii): By Lemma 2.2 of [3] it follows that a linear map T from Λ_{0} into c_{0} is continuous if and only if there exist an infinite matrix (ξ_{ij}) of elements of K, an α ∈ P and d > 0 such that |ξ_{ij}| ≤ dα_{i} for all i, j, lim_{j→∞} ξ_{ij} = 0 for all i and T x = (∑_{i=1}^{∞} x_{i}ξ_{ij})_{j} for all x = (x_{i}) ∈ Λ_{0}. Also, by Theorem 3.3 of [8], it follows that a linear map S ∈ L(E, Λ_{0}) is Λ_{0}- nuclear if and only if there exists (g_{n}) ∈ Λ_{0}{E′} such that T x = (g_{n}(x))_{n} for all x ∈ E. Now, the conclusion follows again by Lemma 2.2 of [3].

Finally, suppose that P is stable.

(vi) ⇔ (viii): We first observe that, since P is stable, the family of seminorms {p_{w} : w ∈ Λ_{0}{E′}} is upwards directed. Also, we know that τ_{0} is the topology of uniform convergence on the equicontinuous sequences in E′ which converge pointwise to zero. Now, the result follows.

Remark. If a locally convex space E has the Λ_{0}-property, then every T ∈ L(E, c_{0}) is compact, since Λ_{0} ⊂ c_{0}. But the converse is not true in general.

Example. Suppose that the valuation on K is dense. It is well known that every T ∈ L(ℓ^{∞}, c_{0}) is compact. However, if Λ_{0} ≠ c_{0}, there are operators from ℓ^{∞} to c_{0} which are not Λ_{0}-nuclear ([8], Corollary 3.7).

§ 3. Λ_{0}-Nuclear Spaces

Nuclear spaces have been extensively studied in the non-archimedean literature (see, for example, [5] for a collection of the basic properties of these spaces). A natural extension of this kind of spaces is the following:

Definition 3.1. A locally convex space E is called Λ_{0}-nuclear if for each p ∈ cs(E) there exists q ∈ cs(E), p ≤ q, such that the canonical map Φ_{pq} : E_{q} ⟶ E_{p} is Λ_{0}-nuclear (or, equivalently, Λ_{0}-compactoid).

In this section we study the relationship between the Λ_{0}-nuclear spaces and the spaces with the Λ_{0}-property considered in the previous section. We first need some preliminary machinery.

Let m ∈ N and let ξ^{(1)}, …, ξ^{(m)} be m elements of Λ_{0}. For j = (n − 1)m + k, where 1 ≤ k ≤ m, set ξ_{j} = ξ_{n}^{(k)}. If P is stable, then ξ = (ξ_{j}) ∈ Λ_{0} (we will denote ξ by ξ^{(1)} ∗ ξ^{(2)} ∗ … ξ^{(m)}).

Indeed, let α ∈ P and let m_{1} ∈ N be such that m ≤ 2^{m_{1}}. Since P is stable, there exist β ∈ P and d > 0 such that α_{n.2^{m_{1}}} /β_{n} ≤ d for all n. Given ϵ > 0, there exists n_{0} ∈ N such that dβ_{n}|ξ_{n}^{(k)}| < ϵ for k = 1, …, m and n ≥ n_{0}. If j ≥ n_{0}m and j = (n − 1)m + k, then n ≥ n_{0} and so

α_{j}|ξ_{j}| ≤ α_{nm}|ξ_{j}| ≤ α_{n.2^{m_{1}}} |ξ_{j}| ≤ dβ_{n}|ξ_{n}^{(k)}| < ϵ.

Lemma 3.2. Let P be stable. Then, for each positive integer m, the function Ψ_{m} : Λ_{0}^{m} ⟶ Λ_{0}, Ψ_{m}(ξ^{(1)}, …, ξ^{(m)}) = ξ^{(1)} ∗ … ∗ ξ^{(m)} is a linear homeomorphism from Λ_{0}^{m} onto Λ_{0}.

Proof. It is easy to see that Ψ_{m} is a bijection. To prove the continuity of Ψ_{m}, recall that, given α ∈ P, there exist β ∈ P and d > 0 such that α_{nm} ≤ dβ_{n} for all n, and so, p_{α}(Ψ_{m}(ξ)) ≤ d max_{1≤k≤m} p_{β}(ξ^{(k)}) for all ξ = (ξ^{(1)}, …, ξ^{(m)}) ∈ Λ_{0}^{m} which proves that Ψ_{m} is continuous.

Also, Ψ_{m}^{−1} is continuous. In fact, for ξ = (ξ_{n}) ∈ Λ_{0} we have Ψ_{m}^{−1}(ξ) = (ξ^{(1)}, …, ξ^{(m)}), where ξ^{(k)} = (ξ_{k}, ξ_{m+k}, ξ_{2m+k}, …) (k = 1, …, m). Also, for each α ∈ P we get p_{α}(ξ) ≥ max_{1≤k≤m} p_{α}(ξ^{(k)}), and the result follows.

Proposition 3.3. For a locally convex space E consider the following properties:

(i) For every Banach space F and for every T ∈ L(E, F), there are T_{1} ∈ L(E, Λ_{0}) and T_{2} ∈ L(Λ_{0}, F) such that T = T_{2} ∘ T_{1}.

(ii) E is of countable type and for every T ∈ L(E, c_{0}) there exist T_{1} ∈ L(E, Λ_{0}) and T_{2} ∈ L(Λ_{0}, c_{0}) such that T = T_{2} ∘ T_{1}.

(iii) If {p_{i} : i ∈ I} is a generating family of continuous seminorms on E, then E is linearly homeomorphic to a subspace of the product space Λ_{0}^{I}.

(iv) E is linearly homeomorphic to a subspace of Λ_{0}^{J} for some set J. Then, (i) ⇔ (ii) ⇒ (iii) ⇒ (iv). If, in addition, P is stable, then properties (i) → (iv) are equivalent.

Proof. The implication (i) ⇒ (iii) can be proved analogously to (1) ⇒ (2) in Proposition 3.7 of [18].

(i) ⇒ (ii): Since (i) implies (iii) and since Λ_{0} is of countable type, we derive that E is also of countable type ([16], Proposition 4.12).

(ii) ⇒ (i): Let F be a Banach space and let T ∈ L(E, F).

First, assume that the range, R(T), is finite-dimensional. Then, there exists a linear homeomorphism h from R(T) onto a closed subspace M of Λ_{0}. On the other hand, since the dual of Λ_{0} separates the points, there exists a continuous linear projection Q from Λ_{0} onto M. Hence T = T_{2} ∘T_{1}, where T_{1} = h ∘ T ∈ L(E, Λ_{0}) and T_{2} = h^{−1} ∘ Q ∈ L(Λ_{0}, F).

Now, assume that R(T) is infinite-dimensional. Since E is of countable type, the closure of R(T) is an infinite-dimensional Banach space of countable type and so it is linearly homeomorphic to c_{0}. Now, the conclusion follows by (ii).

Now, assume that P is stable. Then, the implication (iv) ⇒ (i) can be proved by using Lemma 3.2 in a similar way as (3) ⇒ (1) in Proposition 3.7 of [18].

As in Theorems 3.2 and 3.4 of [18] we obtain the following

Proposition 3.4. For a locally convex space E, consider the following properties:

(i) E is Λ_{0}-nuclear.

(ii) For every locally convex space F, every T ∈ L(E, F) is Λ_{0}-quasinu-clear.

(iii) For every Banach space F, every T ∈ L(E, F) is Λ_{0}-nuclear.

(iv) For every p ∈ cs(E) there exists w ∈ Λ_{0}{E′} such that p ≤ p_{w}.

(v) The topology of E coincides with the topology of uniform convergence on the members of Λ_{0}{E′}.

Then (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) ⇒ (v).

If, in addition, P is stable, then properties (i) → (v) are equivalent.

It is well known (see, for example, [5], Proposition 5.4) that a locally convex space E is nuclear if and only if E is of countable type and every T ∈ L(E, c_{0}) is compact. Now, using Propositions 3.3 and 3.4 we get the following descriptions of Λ_{0}-nuclear spaces.

Theorem 3.5. For a locally convex space E, consider the following properties:

(i) E is Λ_{0}-nuclear.

(ii) For every Banach space F and every T ∈ L(E, F), there exists T_{1} ∈ L(E, Λ_{0}) Λ_{0}-nuclear and T_{2} ∈ L(Λ_{0}, F) such that T = T_{2} ∘ T_{1}.

(iii) E has the Λ_{0}-property and it is linearly homeomorphic to a subspace of Λ_{0}^{I} for some set I.

(iv) E is of countable type and has the Λ_{0}-property.

(v) E is linearly homeomorphic to a subspace of some product Λ_{0}^{I} and every T ∈ L(E, Λ_{0}) is Λ_{0}-quasinuclear.

Then (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) ⇒ (v).

If, in addition, P is stable, then properties (i) → (v) are equivalent.

Proof. By using Proposition 3.4, the implication (i) ⇒ (ii) can be proved as in Theorem 4.6 of [7].

(ii) ⇒ (iii): It follows from Proposition 4.5 of [7] and Proposition 3.3. (iii) ⇒ (iv): It is obvious (recall that Λ_{0} is of countable type). (iv) ⇒ (i): Let F be a Banach space and let T ∈ L(E, F). Since E

is of countable type, we have that the closure of R(T) is a Banach space of countable type, and so it is linearly homeomorphic to a subspace of c_{0}. By (iv) and Theorem 4.11 of [7] we derive that T is Λ_{0}-nuclear. Now, the conclusion follows by Proposition 3.4.(i) ⇔ (iii).

(iii) ⇒ (v): It is a direct consequence of Theorem 2.5.(i) ⇒(iii).

Finally, if P is stable, the implication (v) ⇒ (iii) follows from Proposition 4.5 of [7] and our Proposition 3.3.

Putting together Proposition 2.2, Theorems 2.4, 3.5 and the stability properties of spaces of countable type ([16], Proposition 4.12), we obtain the following extension of 5.7 of [5] and Proposition 3.5 of [19].

Corollary 3.6.

(a) Every subspace of a Λ_{0}-nuclear space is again Λ_{0}-nuclear.

(b) A locally convex space E is Λ_{0}-nuclear if and only if its completion Ê is Λ_{0}-nuclear.

(c) A quotient of a Λ_{0}-nuclear space is also Λ_{0}-nuclear.

(d) If P is stable, then the product of a family of Λ_{0}-nuclear spaces is also Λ_{0}-nuclear.

(e) If P is countable and stable, then the locally convex direct sum and the inductive limit of a sequence of Λ_{0}-nuclear spaces are also Λ_{0}-nuclear.

§ 4. Some Remarks and Examples

It is well known that if E is a nuclear space, then every bounded subset of E is compactoid. The corresponding counterpart is also true for Λ_{0}-nuclear spaces.

Proposition 4.1. Each bounded subset of a Λ_{0}-nuclear space E, is Λ_{0}- compactoid.

Proof. Let B be a bounded set of E and let p ∈ cs(E). Since π_{p} : E ⟶ E_{p} is Λ_{0}-compactoid (Proposition 3.5), we have that π_{p}(B) is Λ_{0}-compactoid in E_{p}. By [7], Proposition 3.10, we derive that B is Λ_{0}-compactoid in E.

Remark. The converse of Proposition 4.1 is not true in general. For an example see [20].

Now, we will give some examples of spaces which are, or are not, Λ_{0}- nuclear.

By Proposition 3.4 and with an argument analogous to the one used in the proof of Theorem 5.2 in [18], we can obtain the following result which will be crucial for our purpose.

Theorem 4.2. Let Q be a Köthe set (not necessarily of infinite type). Then the following properties are equivalent:

(i) Λ(Q) is Λ_{0}(P)-nuclear.

(ii) Λ_{0}(Q) is Λ_{0}(P)-nuclear.

(iii) For each α ∈ Q there exist β ∈ Q with α ≪ β, a permutation σ on N, and (λ_{n}) ∈ Λ_{0}(P) such that α_{σ(n)} ≤ |λ_{n}|β_{σ(n)} for all n ∈ N.

As a direct consequence we derive the following assertion (cf. [4], Proposition 3.5).

Corollary 4.3. Let K(B) be the Köthe space associated to an infinite matrix B = (b_{n}^{k}). Then K(B) is Λ_{0}-nuclear if and only if for every k there exist k_{1} > k, a permutation σ on N, and (λ_{n}) ∈ Λ_{0} such that b_{σ(n)}^{k}/b_{σ(n)}^{k_{1}} ≤ |λ_{n}| for all n.

Remark. The criterion in 4.3 can be used to decide easily whether a non-archimedean countably normed Fréchet space with a Schauder basis is Λ_{0}-nuclear (recall that a such space can be identified with some K(B)).

Observe that since Λ_{0} ⊂ c_{0}, every Λ_{0}-nuclear space is nuclear. But the converse is not true in general. Indeed, we know (see [7], Lemma 2.3) that Λ (or Λ_{0}) is nuclear if and only if there exists α ∈ P with α_{n} → ∞ . However, we have the following

Proposition 4.4. None of the spaces Λ and Λ_{0} is Λ_{0}-nuclear.

Proof. Suppose that one of the spaces Λ or Λ_{0} is Λ_{0}-nuclear. By Theorem 4.2, given α ∈ P, there exist β ∈ P with α ≪ β, a permutation σ on N, and (λ_{n}) ∈ Λ_{0} such that α_{σ(n)} ≤ |λ_{n}|β_{σ(n)} for all n. It is easy to see that the set N_{1} = {n ∈ N : n ≥ σ(n)} is infinite. For n ∈ N_{1} we have

α_{1} ≤ α_{σ(n)} ≤ |λ_{n}|β_{σ(n)} ≤ |λ_{n}|β_{n}.

This contradicts the fact that (λ_{n}) ∈ Λ_{0}.

Observe that every Λ_{0}-nuclear space has the Λ_{0}-property (Theorem 3.5). But the converse is not true in general. Indeed, if P is countable and K is not spherically complete, then Λ̄ has the Λ_{0}-property (see the examples in Section 2). However, with regard to the Λ_{0}-nuclearity of Λ̄, we have

Proposition 4.5. Λ̄ is Λ_{0}-nuclear if and only if Λ = Λ_{0}.

Proof. Assume that Λ = Λ_{0}. Let ξ = (ξ_{n}) ∈ Λ and let λ ∈ K with |λ| > 1. For each n ∈ N, choose λ_{n} ∈ K with |λ_{n}| ≤ \sqrt{|ξ_{n}|} ≤ |λλ_{n}|. Then (λ_{n}) ∈ Λ_{0} and |ξ_{n}| ≤ |λ_{n}|.|λ^{2}λ_{n}| for all n. By Theorem 4.2 we conclude that Λ̄ is Λ_{0}- nuclear.

Conversely, assume that Λ̄ is Λ_{0}-nuclear and let ξ ∈ Λ. By Theorem 4.2, there exist y ∈ Λ, a permutation σ on N, and (λ_{n}) ∈ Λ_{0} such that |ξ_{σ(n)}| ≤ |λ_{n}y_{σ(n)}| for all n. Since λ_{n} → 0, given ϵ > 0 and α ∈ P, there exits m ∈ N such that |λ_{n}|p_{α}(y) < ϵ if n ≥ m. Then, for n ≥ m we have

α_{σ(n)}|ξ_{σ(n)}| ≤ |λ_{n}|α_{σ(n)}|y_{σ(n)}| ≤ |λ_{n}|p_{α}(y) < ϵ.

Hence, ξ ∈ Λ_{0}.

Acknowledgement

This research was partially supported by Comision mixta Caja Cantabria-Universidad de Cantabria

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(Received 11.04.1995)

Authors’ addresses:

A. K. Katsaras Department of Mathematics University of Ioannina Ioannina Greece

C. Perez-Garcia Departamento de Matematicas Facultad de Ciencias Universidad de Cantabria 39071 Santander Spain