Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 223–231

WEAKLY COMPACT WEDGE OPERATORS ON KÖTHE ECHELON SPACES

José Bonet and Miguel Friz

Universidad Politécnica de Valencia, E.T.S.I. Arquitectura Departamento de Matemática Aplicada, E-46022 Valencia, Spain; jbonet@mat.upv.es Universidad Politécnica de Valencia, E.T.S.I. Telecomunicación Departamento de Matemática Aplicada, E-46022 Valencia, Spain; mfriz@mat.upv.es

Abstract. We study wedge operators defined on spaces of operators between Köthe echelon or co-echelon spaces of order 1 < p < ∞. In this case the wedge operator, defined by T → LT R for non-zero operators L and R, maps bounded sets into relatively weakly compact sets if and only if the operator L or the operator R maps bounded sets into relatively compact sets. This is an extension of a result of Saksman and Tylli for the sequence space l_{p}. The corresponding result for operators mapping a neighbourhood into a relatively weakly compact set does not hold, as an example shows.

1. Introduction and notation

A continuous, linear operator T ∈ L(X, Y) between Banach spaces is weakly compact if it maps the closed unit ball of X into a weakly relatively compact subset of Y . There are two possible extensions of this concept in case the continuous linear operator T ∈ L(E, F) is defined between locally convex spaces E and F. As in [5], we say that T is reflexive if it maps bounded sets into weakly relatively compact sets, and it is called weakly compact (as in [10, 42.2]) if there is a 0neighborhood U in E such that T (U) is weakly relatively compact in F. Here we complete our study [4] of reflexive and weakly compact wedge operators defined on spaces of operators between locally convex spaces, analyzing the case of echelon or co-echelon Köthe spaces and extending results due to Saksman and Tylli [15].

Let E_{1} , E_{2} , E_{3} , E_{4} be complete locally convex spaces. The wedge operator of R ∈ L(E_{1}, E_{2}) and L ∈ L(E_{3}, E_{4}) is defined by R∧L: L(E_{2}, E_{3}) → L(E_{1}, E_{4}), T → LT R. The compactness of the wedge operator in case each E_{i} is a Banach space was studied by Vala [17]. He proved that, if R and L are non-zero, R ∧ L is compact if and only if R and L are compact. Apiola [1] and Geue [7] presented

2000 Mathematics Subject Classification: Primary 46B07; Secondary: 46A45, 46A04, 46A32, 47B37.

The present research was partially supported FEDER and the MCYT project no. BFM2001-2670 and by AVCIT Grupo 03/050. We thank P. Domański for his suggestions concerning this article.

a complete generalization of Vala’s result to the locally convex setting for the nowadays so-called Montel operators which are those that map bounded sets into relatively compact sets. The case of compact operators between locally convex spaces was treated by Wrobel [19]. An operator is called compact if it maps a 0-neighborhood into a relatively compact set. Weakly compact wedge operators in the Banach space case were studied by Saksman and Tylli [15], [16], Racher [14] and Lindström, Schlüchtermann [11]. Saksman and Tylli proved in [15, 2.9] that if R and L are weakly compact operators between Banach spaces and if R or L is compact, then the operator R ∧ L is weakly compact. The extensions of this result to reflexive and weakly compact operators between locally convex spaces were obtained by the authors in [4, 2.8, 2.9, 2.14, 2.15]. Consequences about reflexive and weakly compact composition operators on weighted spaces of vector valued holomorphic functions on the disc were presented in [4, Theorem 4.3]. In the case the spaces E_{i} are all Fréchet, or all complete barrelled (DF)-spaces, if L: E_{3} → E_{4} and R^{t}: (E_{2})′_{b} → (E_{1})′_{b} are reflexive and if L or R^{t} is Montel, then R ∧ L is reflexive. The work of Saksman and Tylli [15] already ensures that this result is not optimal: by [15, 2.11], if E is a Banach space which is an ℒ_{1} -space or a ℒ_{∞} -space, E_{i} = E for each i and R and L are non-zero, then R ∧ L is weakly compact on L(E) if and only if L and R are weakly compact. On the other hand, they show in [15, 3.2, 3.3] (see also [16]) that R ∧L is weakly compact on L(l_{p}), 1 < p < ∞, if and only if L or R is compact. We extend this result below for reflexive operators in the case of Köthe echelon and co-echelon spaces of order 1 < p < ∞ in Theorems 7 and 8 respectively. The cases of Köthe echelon spaces of order one or zero is also treated in Propositions 1 and 6. Example 9 shows that the corresponding results for weakly compact operators does not hold. The sequence space representations of function spaces due to Valdivia and Vogt, see e.g. [18, Chapter 3], yield immediate consequences of our results.

We use standard notation for functional analysis and locally convex spaces [9], [10], [12]. The family of all closed absolutely convex 0-neighborhoods of a locally convex space E is denoted by 𝒰_{0}(E), the family of all closed absolutely convex bounded subsets of E by ℬ(E), and the family of all continuous seminorms on E by cs(E). The transpose of an operator T is denoted by T^{t}. The space of all (continuous, linear) operators between the locally convex spaces E and F is denoted by L(E, F), and we write L_{b}(E, F) when this space is endowed with the topology of uniform convergence on the bounded subsets of E. A basis of 0-neighborhoods in L_{b}(E, F) is given by W (B, V) := {f ∈ L(E, F) | f(B) ⊂ V }, as B runs in ℬ(E) and V in 𝒰_{0}(F). We briefly recall the notation and essential facts about Köthe echelon and co-echelon spaces, see [2], [12].

A Köthe matrix A = (a_{n})_{n∈N} is an increasing sequence of strictly positive functions on N. Corresponding to each Köthe matrix A = (a_{n})_{n} and 1 ≤ p < ∞, we associate the spaces

λ_{p}(A) = {x = (x(i))_{i∈N} ∈ C^{N} (or R^{N});

for all n ∈ N : q_{n}^{p}(x) = (∑_{i}(a_{n}(i)|x(i)|)^{p})^{1/p} < ∞},

λ_{∞}(A) = {x = (x(i))_{i∈N} ∈ C^{N} (or R^{N});

for all n ∈ N : q_{n}^{∞}(x) = sup_{i} a_{n}(i)|x(i)| < ∞},

λ_{0}(A) = {x = (x(i))_{i∈IN} ∈ C^{N} (or R^{N});

for all n ∈ N : (a_{n}(i)x(i))_{i} converges to 0},

the last space endowed with the topology induced by λ_{∞}(A). The spaces λ_{p}(A) are called (Köthe) echelon spaces of order p, 1 ≤ p ≤ ∞ or p = 0; they are Fréchet spaces with the sequence of norms p_{n} = q_{n}^{p}, n = 1, 2, …. If A consists of a single function a = (a(i))_{i}, we sometimes write l_{p}(a) instead of λ_{p}(A), 1 ≤ p ≤ ∞, and c_{0}(a) instead of λ_{0}(A). The Banach space l_{p}(a) is a diagonal transform (via a) of the space l_{p} = l_{p}(1), 1 ≤ p ≤ ∞. With this notation, λ_{p}(A) = proj_{n} l_{p}(a_{n}), i.e. λ_{p}(A) is the countable projective limit of the Banach spaces (l_{p}(a_{n}))_{n∈N} .

For a Köthe matrix A = (a_{n})_{n}, denote by V = (v_{n})_{n} the associated decreasing sequence of functions v_{n} = 1/a_{n}, and put

k_{p}(V) = ind_{n} l_{p}(v_{n}), 1 ≤ p ≤ ∞,  and  k_{0}(V) = ind_{n} c_{0}(v_{n}).

That is, k_{p}(V) is the increasing union of the Banach spaces l_{p}(v_{n}), respectively c_{0}(v_{n}), endowed with the strongest locally convex topology under which the injection of each of these Banach spaces is continuous. The spaces k_{p}(V) are called co-echelon spaces of order p. They are (LB)-spaces, and they are complete if p ≠ 0. For a given decreasing sequence V = (v_{n})_{n} of strictly positive functions on N or for the corresponding Köthe matrix A = (a_{n})_{n}, we associate as in [2] the system

λ_{∞}(A)_{+} = \{v̄ = (v̄(i))_{i} ∈ RN_{+}; for all n ∈ N : sup_{i} \frac{v̄(i)}{v_{n}(i)} = sup_{i} a_{n}(i)v̄(i) < ∞\},

which will be denoted by \overline{V} = \overline{V} (V). The system \overline{V} can be used to characterize the bounded subsets of λ_{p}(A), as follows, [2]: Let A be a Köthe matrix on N. A subset B of λ_{p}(A), 1 ≤ p ≤ ∞, is bounded if and only if there exists a strictly positive weight v̄ ∈ \overline{V} so that

B ⊂ v̄B(l_{p}) =

{y ∈ C^{N} (or R^{N}); y(i) = v̄(i)z(i), i ∈ N, for some z ∈ B(l_{p})}, where B(l_{p}) denotes the closed unit ball of the Banach space l_{p}. This result will be used several times in the proofs below.

The duality of the echelon and co-echelon spaces is as follows: For 1 < p < ∞ or p = 0, if (1/p) + (1/q) = 1 (where we take q = 1 for p = 0), then (λ_{p}(A))′_{b} = k_{q}(V) and (k_{p}(V))′_{b} = λ_{q}(A). See [2] for the remaining cases.

Every bounded subset of the co-echelon space k_{p}(V), 1 < p < ∞, is contained in the unit ball of one of the steps l_{p}(v_{n}). Moreover, the inductive limit topology of k_{p}(V), 1 < p < ∞, can be described using the system \overline{V} of associated weights as follows, [2]:

k_{p}(V) = proj_{v̄∈ \overline{V}} l_{p}(v̄).

2. Main results

We first treat the case of Köthe echelon spaces of order one which is easy.

Proposition 1. Let L: λ_{1}(A^{3}) → λ_{1}(A^{4}) and R: λ_{1}(A^{1}) → λ_{1}(A^{2}) be nonzero operators on Köthe echelon spaces of order 1. The wedge operator R ∧ L: L_{b}(λ_{1}(A^{2}), λ_{1}(A^{3})) → L_{b}(λ_{1}(A^{1}), λ_{1}(A^{4})) is reflexive if and only if L and R are Montel.

Proof. First suppose that R ∧ L is reflexive. By [4, 2.1], L and R^{t} are reflexive. By [5, 1.2], R is also reflexive. Since every weakly compact subset of a Köthe echelon space of order one is compact [18, Chapter 2, Section 2.2(2), p. 214], L and R are Montel operators. Now, if L and R are Montel, then L and R^{t} are Montel, see [6, 2.3]. The extension due to Apiola [1] of Vala’s classical result implies that R ∧ L is a Montel operator, hence reflexive.

The following well-known result will be used several times. It can be seen e.g. in [8, Proposition 3, p. 123]: Let E and F be locally convex spaces. Suppose that f ∈ L(E, F) and g ∈ L(F, E) satisfy that g ∘ f coincides with the identity on E. Then f is a topological isomorphism from E onto f(E) ⊂ F and f(E) is a complemented subspace of F. The following result about compact operators on classical Banach sequence spaces is proved for example in the proofs of [15, Propositions 3.2 and 3.3]. We state it in the form which will be needed later.

Lemma 2. (a) Let 1 < p < ∞. If f ∈ L(l_{p}, l_{p}) is not compact, then there exist a complemented subspace X_{0} of l_{p} isomorphic to l_{p} and a complemented subspace Y_{0} of l_{p} such that f | X_{0} is an isomorphism from X_{0} onto Y_{0} .

(b) If f ∈ L(c_{0}, c_{0}) is not compact, then there exist a subspace X_{0} of c_{0} isomorphic to c_{0} and a subspace Y_{0} of c_{0} such that f | X_{0} is an isomorphism from X_{0} onto Y_{0} . In particular, f is not weakly compact.

Proposition 3. Let 1 < p < ∞ and let A = (a_{n})_{n} and B = (b_{n})_{n} be Köthe matrices. If f ∈ L(λ_{p}(A), λ_{p}(B)) is not Montel, there is a complemented subspace X_{0} ⊂ λ_{p}(A) isomorphic to l_{p} and there is a complemented subspace Y_{0} of λ_{p}(B) such that f | X_{0} is a isomorphism from X_{0} onto Y_{0} .

Proof. Since f: λ_{p}(A) → λ_{p}(B) is not a Montel operator, there is a bounded subset M of λ_{p}(A) whose image f(M) is not relatively compact in λ_{p}(B). By [2, Proposition 2.5], there is v̄ ∈ \overline{V} (A) such that v̄(i) > 0 for each i ∈ N and

M ⊂ Mv̄ := \{x = (x(i)) ∈ λ_{p}(A) | ∑_{i=1}^{∞}(\frac{|x(i)|}{v̄(i)})^{p} ≤ 1\}.

Therefore M ⊂ Mv̄ and Mv̄ can be seen as the unit ball of the Banach sequence space l_{p}(1/v̄). We denote by j: l_{p}(1/v̄) → λ_{p}(A) the canonical inclusion. The set fj(Mv̄) is not relatively compact in λ_{p}(B). As λ_{p}(B) = proj_{n} l_{p}(b_{n}), there is n ∈ N such that π_{n}fj: l_{p}(1/v̄) → l_{p}(b_{n}) is not compact. Here π_{n}: λ_{p}(B) → l_{p}(b_{n}) is the canonical injection with dense range.

We apply Lemma 2(a) to obtain a subspace X_{0} of l_{p}(1/v̄) which is complemented, isomorphic to l_{p}, and a subspace Y_{0} of l_{p}(b_{n}) which is complemented and such that the operator (π_{n}fj) | X_{0} is an isomorphism from X_{0} onto Y_{0} . The proof is now completed in several steps:

(1): j: X_{0} → λ_{p}(A) is a topological isomorphism into.

Clearly j is injective and continuous. Suppose that (x_{k})_{k} ⊂ X_{0} satisfies that j(x_{k}) → 0 in λ_{p}(A). Then π_{n}fj(x_{k}) → 0 in l_{p}(b_{n}). Since (π_{n}fj) | X_{0} is an isomorphism, x_{k} → 0 in X_{0} .

We denote by P_{2}: l_{p}(b_{n}) → l_{p}(b_{n}) the continuous projection onto Y_{0} . We have P_{2}(y) = y for each y ∈ Y_{0} . The operator (π_{n}fj)^{−1}P_{2}(π_{n}f): λ_{p}(A) → X_{0} is well-defined, linear and continuous. Moreover, if x ∈ X_{0} , we have

(π_{n}fj)^{−1}P_{2}(π_{n}f)j(x) = (π_{n}fj)^{−1}(π_{n}fj)(x) = x,

since (π_{n}fj)(x_{0}) ∈ Y_{0} . We apply [8, Proposition 3, p. 123] mentioned above to obtain that

(2): X_{0} is isomorphic to j(X_{0}), hence to l_{p}, and j(X_{0}) is a complemented subspace of λ_{p}(A).

(3): If we consider fj(X_{0}) as a subspace of λ_{p}(B), then π_{n}: fj(X_{0}) → Y_{0} is a topological isomorphism.

Indeed, π_{n} is linear and continuous, and it is injective on fj(X_{0}) because π_{n}fj is an isomorphism on X_{0} . Moreover, π_{n} is surjective: given y ∈ Y_{0} , x = (π_{n}fj)^{−1}y ∈ X_{0} and π_{n}(fj(x)) = y. Suppose now that (x_{k})_{k} ⊂ X_{0} satisfies π_{n}(fj(x_{k})) → 0 in Y_{0} . Then x_{k} → 0 in X_{0} , thus fj(x_{k}) → 0 in λ_{p}(B).

Part (3) implies that π_{n}^{−1}(Y_{0}) is a Banach subspace of λ_{p}(B) which is isomorphic to X_{0} , hence to l_{p}.

(4): π_{n}^{−1}(Y_{0}) is a complemented subspace of λ_{p}(B).

Consider π_{n}^{−1}: Y_{0} → λ_{p}(B), which is linear and continuous by (3), and P_{2}π_{n}: λ_{p}(B) → Y_{0} which is also linear and continuous. As (P_{2}π_{n})π_{n}^{−1}(y) = P_{2}y = y for each y ∈ Y_{0} , the conclusion follows again from [8, Proposition 3, p. 123].

Proposition 4. Let 1 < p < ∞ and let V = (v_{n})_{n} and W = (w_{n})_{n} be decreasing sequences of strictly positive weights on N. If f ∈ L(k_{p}(V), k_{p}(W)) is not Montel, there is a complemented subspace X_{0} ⊂ k_{p}(A) isomorphic to l_{p} and there is a complemented subspace Y_{0} of k_{p}(B) such that f | X_{0} is a topological isomorphism from X_{0} onto Y_{0} .

Proof. This can be proved along the lines of the proof of Proposition 3. Since f: k_{p}(V) → k_{p}(W) is not a Montel operator, there is a bounded subset M of k_{p}(V) whose image f(M) is not relatively compact in k_{p}(W). There is n such that, if we denote by j_{n}: l_{p}(v_{n}) → k_{p}(V) the canonical inclusion, the image by fj_{n} of the closed unit ball of l_{p}(v_{n}) is not relatively compact in k_{p}(W). By the description of the topology of k_{p}(W) mentioned in the introduction, there is a strictly positive element \overline{w} ∈ \overline{W} such that πfj_{n}: l_{p}(v_{n}) → l_{p}( \overline{w}) is not compact, where π: k_{p}(W) → l_{p}( \overline{w}) is the canonical inclusion with dense range. The proof now continues as before, applying Lemma 2(a) to πfj_{n}: l_{p}(v_{n}) → l_{p}( \overline{w}).

Proposition 5. An operator f: λ_{0}(A) → λ_{0}(B) is reflexive if and only if it is Montel.

Proof. We proceed again as in the proof of Proposition 3. If the operator f: λ_{0}(A) → λ_{0}(B) is not Montel, we find v̄ ∈ \overline{V} (A) and n ∈ N such that π_{n}fj: c_{0}(1/v̄) → c_{0}(b_{n}) is not compact. Here j: c_{0}(1/v̄) → λ_{0}(a) and π_{n}: λ_{0}(B) → c_{0}(b_{n}) are the canonical inclusions. By Lemma 2(b), there are a closed subspace X_{0} of c_{0}(1/v̄) isomorphic to c_{0} and a closed subspace Y_{0} of c_{0}(b_{n}) such that (π_{n}fj) | X_{0} is an isomorphism from X_{0} onto Y_{0} . As in the part (1) of the proof of Proposition 3 we conclude that j: X_{0} → λ_{0}(A) is a topological isomorphism into, and as in the proof of part (3), π_{n}: fj(X_{0}) ⊂ λ_{0}(B) → Y_{0} is also a topological isomorphism into. Therefore, the restriction of f to j(X_{0}) ⊂ λ_{0}(A) is an isomorphism into λ_{0}(B) and j(X_{0}) is isomorphic to c_{0} . This implies that f is not reflexive.

If A is a Köthe matrix and we apply Proposition 5 to the identity id: λ_{0}(A) → λ_{0}(A), we conclude the following result of Valdivia [18, Chapter 2, Section 4, 2(1)]: a Köthe echelon space λ_{0}(A) is reflexive if and only if it is a Montel space.

Proposition 6. Let L: λ_{0}(A^{3}) → λ_{0}(A^{4}) and R: λ_{0}(A^{1}) → λ_{0}(A^{2}) be nonzero operators on Köthe echelon spaces of order 0. The wedge operator R ∧ L: L_{b}(λ_{0}(A^{2}), λ_{0}(A^{3})) → L_{b}(λ_{0}(A^{1}), λ_{0}(A^{4})) is reflexive if and only if L and R are Montel.

Proof. First suppose that R ∧ L is reflexive. By [4, 2.1], L and R^{t} are reflexive. By [5, 1.2], R is also reflexive. Proposition 5 implies now that L and R are Montel operators. Conversely, if L and R are Montel, then L and R^{t} are Montel, see [6, 2.3]. The extension due to Apiola [1] of Vala’s classical result implies that R ∧ L is a Montel operator, hence reflexive.

Theorem 7. Let 1 < p < ∞. Let L: λ_{p}(A^{3}) → λ_{p}(A^{4}) and R: λ_{p}(A^{1}) → λ_{p}(A^{2}) be non-zero operators between Köthe echelon spaces of order p. The wedge operator R ∧ L: L_{b}(λ_{p}(A^{2}), λ_{p}(A^{3})) → L_{b}(λ_{p}(A^{1}), λ_{p}(A^{4})) is reflexive if and only if L or R is a Montel operator.

Proof. As all the Köthe echelon spaces here are reflexive, the operators L and R are reflexive. Therefore, it is enough to show that the condition is necessary by [4, 2.11] and [6, 2.3]. Suppose now that neither R nor L is a Montel operator. We apply Proposition 3 to find a complemented subspace X_{0} of λ_{p}(A^{1}) isomorphic to l_{p} such that the restriction R_{1}: X_{0} → R(X_{0}) is a topological isomorphism and R(X_{0}) is complemented in λ_{p}(A^{2}). We denote by j_{1}: X_{0} → λ_{p}(A^{1}) the inclusion and by P_{1}: λ_{p}(A^{2}) → R(X_{0}) the continuous projection. Analogously, Proposition 3 provides us with a complemented subspace Y_{0} of λ_{p}(A^{3}) isomorphic to l_{p} such that the restriction L_{2}: Y_{0} → L(Y_{0}) is a topological isomorphism and L(Y_{0}) is complemented in λ_{p}(A^{4}). We denote by j_{2}: Y_{0} → λ_{p}(A^{3}) the injection and by P_{2}: λ_{p}(A^{4}) → L(Y_{0}) the continuous projection.

Since R_{1} and L_{2} are isomorphisms, it is easy to see that the wedge operator

R_{1} ∧ L_{2}: L_{b}(R(X_{0}), Y_{0}) → L_{b}(X_{0}, L(Y_{0}))

is an isomorphism, too. Moreover,

L_{b}(R(X_{0}), Y_{0}) ≃ L_{b}(X_{0}, L(Y_{0})) ≃ L_{b}(l_{p}, l_{p})

is not a reflexive space, because it contains a copy of l_{∞}.

Suppose that the wedge operator

R ∧ L: L_{b}(λ_{p}(A^{2}), λ_{p}(A^{3})) → L_{b}(λ_{p}(A^{1}), λ_{p}(A^{4}))

is reflexive. Then the operator

(j_{1} ∧ P_{2}) ∘ (R ∧ L) ∘ (P_{1} ∘ j_{2}): L_{b}(R(X_{0}), Y_{0}) → L_{b}(X_{0}, L(Y_{0}))

is also reflexive. It is easy to see that

(j_{1} ∧ P_{2}) ∘ (R ∧ L) ∘ (P_{1} ∘ j_{2}) = R_{1} ∘ L_{2},

which is an isomorphism between non-reflexive spaces; a contradiction.

If we use Proposition 4 instead of Proposition 3, the proof above shows the following characterization in the case of co-echelon spaces.

Theorem 8. Let 1 < p < ∞. Let L: k_{p}(V^{3}) → k_{p}(V^{4}) and R: k_{p}(V^{1}) → k_{p}(V^{2}) be non-zero operators between Köthe co-echelon spaces of order p. The wedge operator R ∧ L: L_{b}(k_{p}(V^{2}), k_{p}(V^{3})) → L_{b}(k_{p}(V^{1}), k_{p}(V^{4})) is reflexive if and only if L or R is a Montel operator.

As a consequence of [4, 2.14, 2.15], if 1 < p < ∞, L: λ_{p}(A^{3}) → λ_{p}(A^{4}) and R: λ_{p}(A^{1}) → λ_{p}(A^{2}) satisfy that L and R^{t} are weakly compact and L or R^{t} is compact, the wedge operator R ∧ L: L_{b}(λ_{p}(A^{2}), λ_{p}(A^{3})) → L_{b}(λ_{p}(A^{1}), λ_{p}(A^{4})) is weakly compact. It is a natural question whether a result similar to Theorem 7 would hold for weakly compact operators on Köthe echelon spaces of order 1 < p < ∞. The answer is negative as the following example shows. The authors thank P. Domański for his suggestions concerning the example.

Example 9. Let 1 < p < ∞. Grothendieck gave an example of a Köthe echelon space λ_{p}(A) which is a Fréchet Montel space with a quotient isomorphic to the Banach space l_{p}. See e.g. [10, 33.6(2)]. We denote by Q: λ_{p}(A) → l_{p} the quotient map. The operator Q is weakly compact but not compact.

We set λ_{p}(A^{1}) = λ_{p}(A), λ_{p}(A^{2}) = l_{p}, λ_{p}(A^{3}) = λ_{p}(A), λ_{p}(A^{4}) = l_{p}, R: λ_{p}(A^{1}) → λ_{p}(A^{2}), R = Q and L: λ_{p}(A^{3}) → λ_{p}(A^{4}), L = Q.

Since R and L are bounded (i.e. they map a 0-neighbourhood into a bounded set), it is easy to see that the wedge operator

R ∧ L: L_{b}(l_{p}, λ_{p}(A)) → L_{b}(λ_{p}(A), l_{p})

is also bounded. As λ_{p}(A) is a Montel Köthe echelon space, the space L_{b}(λ_{p}(A), l_{p}) is a reflexive (DF)-space by [3]. Therefore the wedge operator R ∧ L is weakly compact. However, neither R nor L is compact.

An example in which all the spaces E_{1} , E_{2} , E_{3} , E_{4} coincide can be easily obtained as follows: define the spaces E_{1} := λ_{p}(A) × l_{p}, i = 1, 2, 3, 4, and the operators

\begin{aligned}R & : λ_{p}(A) × l_{p} → λ_{p}(A) × l_{p}, R(x, y) = (0, Q(x)), \\ L & : λ_{p}(A) × l_{p} → λ_{p}(A) × l_{p}, L(x, y) = (0, Q(x)). \end{aligned}

In this case T: λ_{p}(A) × l_{p} → λ_{p}(A) × l_{p} has the form
( \begin{matrix}T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix} )

and

(R ∧ L)(T)(x, y) = (0, QT_{12}Q(x))

for each (x, y).

Weakly compact wedge operators on Köthe echelon spaces 231

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Received 25 August 2003