CONTINUOUS HOMOMORPHISMS BETWEEN ... - Project Euclid

Oct 3, 2003 - The natural topology on spaces of holomorphic germs is the Nachbin ported topology Ïw. It is defined on H(U) by the family of all semi-.

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 36, Number 5, 2006

CONTINUOUS HOMOMORPHISMS BETWEEN TOPOLOGICAL ALGEBRAS OF HOLOMORPHIC GERMS LUCIANO O. CONDORI AND M. LILIAN LOURENC ¸O ABSTRACT. We study τw -continuous homomorphisms on algebras of holomorphic germs. In this setting we give conditions for these homomorphisms to be composition operators. We also present equivalent conditions for the above homomorphisms to be Montel or reflexive.

1. Introduction. Let E be a Banach space, U an open subset of E and H(U ) the space of all holomorphic functions on U . A holomorphic germ on a compact subset K of E is an equivalence class determined in the set of all holomorphic functions on open neighborhoods of K by the relation, f ∼ = g if f and g coincide on an open neighborhood of K. We will denote by H(K) the algebra of all holomorphic germs on K. The natural topology on spaces of holomorphic germs is the Nachbin ported topology τw . It is deﬁned on H(U ) by the family of all seminorms ported by the compact subsets of U . A semi-norm p on H(U ) is ported by the compact subset L of U if for every open subset V , L ⊂ V ⊂ U , there is a c > 0 such that p(f ) ≤ cf V = c supx∈V |f (x)| for every f ∈ H(U ). Now, the topology τw on H(K) is the locally convex topology deﬁned by the inductive limit of the spaces (H(U ), τw ), where U varies over all open neighborhoods of K. We remark that the space (H(K), τw ) can be represented as an inductive limit of Banach spaces, namely the inductive limit of the spaces H ∞ (U ), where H ∞ (U ) denotes the Banach space of all bounded holomorphic functions on U . We are interested in the study of continuous homomorphisms between locally m-convex algebras of holomorphic germs. The continuous homomorphisms between topological algebras of holomophic functions have been extensively studied lately. For example see [4 6]. In [16] Nicodemi has 2000 AMS Mathematics Subject Classiﬁcation. Primary 46G20, 32A38, 47B33. Key words and phrases. Holomorphic germs, continuous homomorphisms. Research of the first author supported by a doctoral fellowship from CNPq, Brazil. Received by the editors on October 3, 2003, and in revised form on Feb. 18, 2004. c Copyright 2006 Rocky Mountain Mathematics Consortium

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studied the continuous homomorphisms between algebras of holomorphic germs with the compact open topology. She proved that, under some conditions, the continuous homomorphisms A : H(K) → H(L), where L is a compact subset of a Banach space F , are exactly those induced by a holomorphic mapping Φ : V → E such that Φ(L) ⊂ K and V is an open set in F containing L, in the sense that A(f ) = f ◦ Φ, for every f ∈ H(K). It is natural to study the continuous homomorphisms between algebras of holomorphic germs endowed with the τw -topology. We show that every τw -continuous homomorphism between locally m-convex algebras of holomorphic germs factors through a continuous homomorphism between Fr´echet algebras of holomorphic functions of bounded type. Using this factorization we establish when the continuous homomorphisms between locally m-convex algebras of holomorphic germs are composition operators. As a consequence we get that if E is a Tsirelson space each continuous homomorphism between locally m-convex algebras of holomorphic germs is a composition operator. Furthermore, this factorization also allows us to characterize when continuous homomorphisms between locally m-convex algebras of holomorphic germs are Montel or reﬂexive. We refer to the books of Dineen [2] or Mujica [13] for background information from inﬁnite dimensional complex analysis. 2. The space of holomorphic germs. Let (H, τ ) = lim −→ (Hi , τi ) i∈I denote the locally convex inductive limit of locally convex spaces (Hi , τi ). The inductive limit (H, τ ) is called regular if, for each bounded set B in (H, τ ), there exists i = i(B) ∈ I such that B ⊂ Hi and B is τi -bounded. The inductive limit (H, τ ) is called strongly boundedly retractive if (H, τ ) is regular and, for each i ∈ I, there exists j > i such that (H, τ ) and (Hj , τj ) induce the same topology on each bounded subset B of (Hi , τi ). Let τw denote the compact-ported topology introduced by Nachbin [15] on the space H(U ) of all holomorphic functions on an open subset U of a complex Banach space E. Let H(K) denote the space of all germs of holomorphic functions on a compact subset K of E and let us also denote by τw the locally convex inductive limit topology on H(K) which is deﬁned by (H(K), τw ) = lim −→ (H(U ), τw ). U⊃K

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Let B1/n (0) = {x ∈ E : ||x|| < 1/n}. If we ﬁx a decreasing fundamental sequence of open neighborhoods (Un )n∈N in E with Un = K+B1/n (0) ⊂ E, we remark that the space (H(K), τw ) can be represented as an inductive limit of Banach spaces, namely, (H(K), τw ) = lim −→ H ∞ (Un ) Un ⊃K

where H ∞ (Un ) denotes the Banach space of all bounded holomorphic functions on Un , with the supremum norm. Mujica showed in [10] that the inductive limit (H(K), τw ) = lim −→ H ∞ (Un ) is strongly boundedly n∈N

retractive, since all the spaces H ∞ (Un ), n ∈ N, are normed spaces. Before stating our results, let us ﬁx some additional notation and K denote the inclusion mapping from terminology. For m ∈ N we let Jm ∞ K H (Um ) into H(K) given by Jm (f ) = [f ]. If U is an open subset of E, then a set A ⊂ U is said to be U -bounded if A is bounded and is bounded away from the boundary of U . As usual the space of all holomorphic mappings from an open set U of E into C that are bounded on U -bounded subsets of U, endowed with the topology τb of uniform convergence on U -bounded sets is denoted by Hb (U ). Since H ∞ (Un ) → Hb (Un ) → (H(K), τw ) for each n ∈ N, we have (H(K), τw ) = lim −→ Hb (Un ) and we deﬁne the natural inclusion from n∈N

Hb (Un ) into (H(K), τw ) by InK (f ) = [f ], where [f ] is the equivalence class determined by f . Now, we are going to give some results about inductive limits, which will be useful in Section 4. Proposition 2.1. Let (Hn , τn )n∈N denote a sequence of locally convex spaces and (H, τ ) = lim −→ (Hn , τi ) denote a strongly boundedly n∈N retractive inductive limit. Then for each m ∈ N there exists n > m such that σ(H, H  ) and σ(Hn , Hn ) induce the same topology on each bounded subset of (Hm , τm ). Proof. Given B ⊂ (Hm , τm ) a bounded subset, the closed absolutely convex hull of B will be denoted by Γ(B). Since (H, τ ) is a strongly boundedly retractive limit, then there exists n > m such that (H, τ ) and (Hn , τn ) induce the same topology on Γ(B). Now, by Grothendieck’s lemma [8, Proposition 3.11.1] we have that for each f ∈ Hn and ε > 0

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there exists g ∈ H  with |f (x) − g(x)| < ε, for all x ∈ Γ(B). Hence given f1 , . . . , fp ∈ Hn , there exist g1 , . . . , gp ∈ H  such that {x ∈ Γ(B) : |fj (x)| < 2ε for j = 1, . . . , p} ⊃ {x ∈ Γ(B) : |gj (x)| < ε for j = 1, . . . , p} and therefore σ(Hn , Hn )|B ≤ σ(H, H  )|B . Since the opposite inequality is always true, we conclude that σ(Hn , Hn )|B = σ(H, H  )|B . Proposition 2.2. Let E be a Banach space, and let K ⊂ E be a compact subset. Then the inductive limit (H(K), τw ) = lim −→ Hb (Un ) n∈N is strongly boundedly retractive. Proof. (H(K), τw ) is the strongly boundedly retractive inductive limit of an increasing sequence of Banach spaces H ∞ (Un ). (H(K), τw ) = lim −→

(1)

n∈N

Therefore, given X ⊂ H(K) bounded set there exists m0 ∈ N such that X is a bounded subset of H ∞ (Um0 ). Since the inclusion mapping im0

H ∞ (Um0 ) → Hb (Um0 ) is continuous, we have that X is a bounded subset of Hb (Um0 ) and consequently (H(K), τw ) = lim −→ Hb (Un ) n∈N

is a regular inductive limit. Now we can prove that (H(K), τw ) = lim −→ Hb (Un ) is strongly boundedly retractive. Let m ∈ N and let n∈N

K (X ) is a bounded subset of H(K). By X ⊂ Hb (Um ) be bounded. So Im (1), we can ﬁnd n > m such that (H(K), τw ) and H ∞ (Un ) induce the same topology on the bounded set X of H ∞ (Um ).

Now, it follows at once from the commutative diagram: Hb (Um )

in,m Ý

Hb (Un )

im

H ∞ (Um )

in jn,m Ý

H ∞ (Un )

K In

  

Ý

JnK

H(K)

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that (H(K), τw ) and (Hb (Un ), τb ) induce the same topology on the bounded subset X of Hb (Um ). An immediate consequence of Proposition 2.1 and Proposition 2.2 is the following: Corollary 2.3. Let E be a Banach space and let K ⊂ E be a compact subset. Let (H(K), τw ) = lim −→ Hb (Un ) denote a strongly boundedly n∈N retractive inductive limit. Then for each n ∈ N there exists m > n such that σ(H(K), H(K) ) and σ(Hb (Um ), Hb (Um ) ) induce the same topology on each bounded set X of (Hb (Un ), τb ). 3. Homomorphisms between algebras of holomorphic germs. A topological algebra is locally m-convex if its topology is deﬁned by the family of the convex 0-neighborhoods W with W 2 ⊂ W . For any compact subset K of a Banach space E, Mujica in [12] showed that the inductive limit (H(K), τw ) = lim −→ H ∞ (Un ) is a n∈N

locally m-convex algebra. From now on (H(K), τw ) stands for a locally m-convex algebra and (Hb (Un ), τb ) stands for a Fr´echet algebra. By a homomorphism between topological algebras we mean an algebra homomorphism, which is not identically zero. Next we show that each continuous homomorphism between locally m-convex algebras of holomorphic germs factors through some continuous homomorphism between Fr´echet algebras of holomorphic functions of bounded type. Theorem 3.1. Let E and F be Banach spaces. Let K ⊂ E and L ⊂ F be compact sets, and let A : (H(K), τw ) → (H(L), τw ) be a continuous homomorphism. Then for each open subset Un ⊃ K there exists an open subset Vmn ⊃ L and a continuous homomorphism L n : Hb (Un ) → Hb (Vm ) such that Im n = A ◦ InK . A ◦A n n Proof. Given Un ⊃ K an open subset of E, we consider the continuous homomorphism A ◦ InK : Hb (Un ) → H(L). By Grothendieck’s theorem, see [7, p. 16], there exist an open subset Vmn ⊃ L and a continuous

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linear mapping A ◦ InK : Hb (Un ) → Hb (Vmn ) such that the diagram H(K)

A

H(L)

L Im n

K In

Hb (Un )

K A◦In

Hb (Vmn )

L is commutative. Since A, InK and Im are continuous homomorphisms, n we have that

A ◦ InK (f · g) = A ◦ (InK (f ) · InK (g)) and therefore A ◦ InK (f · g) ∼ A ◦ InK (f ) · A ◦ InK (g), for all f, g ∈ Hb (Un ). So, there exists an open set Vmln ⊂ Vmn such that A ◦ InK (f · g)|Vm = (A ◦ InK (f ) · A ◦ InK (g))|Vm . Since each connected compoln ln nent of Vmn contained an open set of Vmln , by the Identity Principle for holomorphic functions we have A ◦ InK (f · g) = (A ◦ InK (f ) · A ◦ InK (g)) on Vmn for all f, g ∈ Hb (Un ). n (f ) = n : Hb (Un ) → Hb (Vm ) deﬁned by A Now, the mapping A n

A ◦ InK (f ) for each f ∈ Hb (Un ) is clearly a continuous homomorphism L n . Thus, for every open set Un ⊃ K, ◦A such that A ◦ InK = Im n there exist an open set Vmn ⊃ L and a continuous homomorphism n : Hb (Un ) → Hb (Vm ) such that the diagram A n H(K)

A

H(L)

L Im n

K In

Hb (Un )

n A

Hb (Vmn )

is commutative. Before stating and proving Theorem 3.2 we need some preparation. Let U and V denote open subsets of complex Banach spaces E and F , respectively. Given a holomorphic mapping of bounded type Φ : V → E such that Φ(V ) ⊂ U where Φ takes a V -bounded set in a U -bounded set, the mapping AΦ : Hb (U ) → Hb (V ) deﬁned by AΦ (f ) = f ◦ Φ is called a composition operator.

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Let K and L be compact subsets of E and F respectively. A homomorphism A : H(K) → H(L) is a composition operator if there exist an open subset V ⊃ L and a holomorphic mapping Φ : V → E such that Φ(L) ⊂ K and A([f ]) = [f ◦ Φ] for each holomorphic function f deﬁned on a neighborhood of K. We denote a composition operator A by AΦ . Now, we give some conditions for τw -continuous homomorphisms between algebras of holomorphic germs to be composition operators. Theorem 3.2. Let E and F be Banach spaces. Let K ⊂ E and L ⊂ F be compact subsets, and let A : (H(K), τw ) → (H(L), τw ) be a continuous homomorphism. If, for each open subset Un ⊃ K there exist an open subset Vmn ⊃ L and a composition operator L n : Hb (Un ) → Hb (Vm ) such that Im n = A ◦ InK , then A is A ◦A n n a composition operator. Proof. We suppose that for each open subset Un ⊃ K there exist n : Hb (Un ) → Hb (Vm ) a composition an open subset Vmn ⊃ L and A n L n = A ◦ InK . We shall prove that A is a operator such that Imn ◦ A composition operator. Without loss of generality we may assume that Vm1 ⊃ Vm2 ⊃ · · · . n : Hb (Un ) → Hb (Vm ) is a composition operator, then there Since A n n (f ) = f ◦ Φn , for every exists a mapping Φn ∈ Hb (Vmn , E) such that A f ∈ Hb (Un ) and for each n ∈ N. So, L n )(f ) = [f ◦ Φn ], ◦A (2) A([f ]) = (A ◦ InK )(f ) = (Im n

∀ f ∈ Hb (Un ),

for each n ∈ N. Now, we consider the open subset Vm1 as above and deﬁne a mapping Φ : Vm1 → E by Φ(y) = Φ1 (y), for all y ∈ Vm1 . We claim that Φ(y) = Φn (y) for all y ∈ Vmn and A = AΦ . Since (E  ,  ) → Hb (Un ), for all n ∈ N, we have that by equality (2)  A([x ]) = [x ◦ Φ1 ], for every x ∈ E  and (3) A([x ]) = [x ◦ Φn ], for every x ∈ E  , for every n ∈ N; consequently, [x ◦ Φ1 ] = [x ◦ Φn ], for every x ∈ E  and for every n ∈ N. Then there exists an open subset W such that L ⊂ W ⊂ Vmn

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and (x ◦ Φ1 )|W = (x ◦ Φn )|W for all x ∈ E  . Now, for each y ∈ Vmn there exists z ∈ L such that y ∈ B1/mn (z). Since L ⊂ W we have that W ∩ B1/mn (z) = ∅ and consequently (x ◦ Φ1 )|W ∩B = 1/mn (z)      for all x ∈ E and (x ◦ Φ1 )(y) = (x ◦ Φn )(y) (x ◦ Φn )|W ∩B 1/mn (z) for all y ∈ Vmn and for all x ∈ E  . By the Hahn-Banach theorem we have that (4)

Φ(y) = Φ1 (y) = Φn (y),

for every

y ∈ Vmn .

Thus to show our claim it suﬃces to prove that Φ(L) ⊂ K. By (4) we have that Φ(L) ⊂ Un for all n ∈ N. So, for each y ∈ L, there is a ξn ∈ K such that Φ(y) − ξn  < 1/n, for each n ∈ N. Then d(Φ(y), K) = 0 for each y ∈ L. Now the proof of the theorem is complete. Our next corollary shows that in the case of the Banach space E to be the Tsirelson space, deﬁned by Tsirelson in [17], every continuous homomorphism between algebras of holomorphic germs is a composition operator. Corollary 3.3. Let E be a Tsirelson space and F a Banach space. Let K ⊂ E be an absolutely convex and compact subset and L ⊂ F a compact subset. Then every continuous homomorphism A : (H(K), τw ) → (H(L), τw ) is a composition operator. Proof. Let K ⊂ E be an absolutely convex and compact subset. Since Un = K + B1/n (0) for each n ∈ N, Un is also absolutely convex. Now, by Theorem 3.1 there exist an open subset Vmn ⊃ L and a continuous L n = A ◦ InK . n : Hb (Un ) → Hb (Vm ) such that Im ◦A homomorphism A n n n is a composition operator. Now By [4, Proposition 3] we have that A it suﬃces to apply Theorem 3.2. Before giving our next corollary we need additional notation. Let E be a complex Banach space. We denote Pf (n E) the space generated by all polynomials of the form P (x) = ψ(x)m , with ψ ∈ E  . Corollary 3.4. Let E be a reflexive Banach space such that Pf (n E) is dense in P(n E) for each n ∈ N. Let K ⊂ E be an absolutely

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convex and compact subset of E. Let F be a Banach space and L ⊂ F be a compact subset. Then every continuous homomorphism A : (H(K), τw ) → (H(L), τw ) is a composition operator. Proof. The result follows arguing as in Corollary 3.3 and using a result of Mujica [14, Theorem 1.6]. 4. Montel and reflexive homomorphisms. Our purpose in this section is to study the Montel and reﬂexive homomorphisms between the algebras of holomorphic germs and their relationship. Let U be an open subset of a complex Banach space E. We say that a mapping g : U → F is compact, respectively weakly compact, if it takes U -bounded sets into relatively compact, respectively weakly compact, sets. A linear operator between locally convex spaces is Montel, respectively reflexive, if it takes bounded subsets into relatively compact, respectively weakly compact, subset. All these deﬁnition can be found in the paper [1]. Let us recall that a linear mappings T : X → Y , where X and Y are locally convex spaces, is called weakly compact, respectively compact, if it maps some 0-neighborhood into relatively weakly compact, respectively relatively compact sets. If X and Y are normed space T is weakly compact, respectively compact, if and only if T is reﬂexive, respectively Montel. Theorem 4.1. Let E and F be Banach spaces. Let K ⊂ E and L ⊂ F be compact subsets, and let A : (H(K), τw ) → (H(L), τw ) be a continuous homomorphism. Then A is a Montel homomorphism if and only if for each open subset Un ⊃ K there exist an open subset Vmn ⊃ L and a Montel continuous homomorphism Ψn : Hb (Un ) → Hb (Vmn ) such L ◦ Ψn = A ◦ InK . that Im n Proof. Suppose A is a Montel homomorphism. Given Un ⊂ E an open subset, by Theorem 3.1 there exist an open subset Vmrn ⊃ L n : Hb (Un ) → Hb (Vm ) such that and a continuous homomorphism A rn L K  Imrn ◦ An = A ◦ In .

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By Proposition 2.2 we can ﬁnd mn > mrn such that the spaces (H(L), τw ) and Hb (Vmn ) induce the same topology on each bounded subset B of Hb (Vmrn ). iL mr

n

L L ◦ iL Since Hb (Vmrn ) → Hb (Vmn ) and Im mrn = Imrn , we can n deﬁne a continuous homomorphism Ψn : Hb (Un ) → Hb (Vmn ) by K L  Ψn (f ) = iL mrn ◦ An (f ) for each f ∈ Hb (Un ). So A ◦ In = Imn ◦ Ψn and this means the following diagram

H(K)

A

H(L)

L Im rn

K In

Hb (Un )

n A

 

Hb (Vmrn )



L Im n

iL mr Ý

n

Hb (Vmn )

commutes. We claim that the homomorphism Ψn is Montel. Indeed, n (X ) ⊂ Hb (Vm ) is let X ⊂ Hb (Un ) be a bounded subset. Then A rn a bounded subset. Since A is a Montel homomorphism, Ψn (X ) =  iL mrn (An (X )) ⊂ Hb (Vmn ) and the topology induced by (H(L), τw ) and Hb (Vmn ) on Ψn (X ) is the same. Then Ψn (X ) is a relatively compact subset of Hb (Vmn ) and therefore Ψn is Montel. Conversely, suppose that for each open subset Un ⊃ K there exist an open subset Vmn ⊃ L and a Montel homomorphism Ψn : Hb (Un ) → L ◦ Ψn = A ◦ InK . We will prove that A is a Hb (Vmn ) such that Im n Montel homomorphism. Let X ⊂ H(K) be a bounded subset. By Proposition 2.2 there exists an r ∈ N such that X is a bounded subset of Hb (Ur ). Then using the hypothesis, there exist an open subset Vmr ⊃ L and a Montel L homomorphism Ψr : Hb (Ur ) → Hb (Vmr ) such that Im ◦ Ψr = A ◦ IrK . r L As Ψr (X ) is relatively compact subset and A(X ) = Imr (Ψr (X )), we have that A is a Montel operator and the proof is complete. We now discuss under which conditions a homomorphism between algebras of holomorphic germs is reﬂexive. Theorem 4.2. Let E and F be Banach spaces. Let K ⊂ E and L ⊂ F be compact subsets, and let A : (H(K), τw ) → (H(L), τw ) be a continuous homomorphism. Then A is reflexive if and only if for

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each open subset Un ⊃ K there exist an open subset Vmn ⊃ L and a reflexive continuous homomorphism Ψn : Hb (Un ) → Hb (Vmn ) such that L Im ◦ Ψn = A ◦ InK . n Proof. Suppose A is reﬂexive. Given Un ⊂ E an open subset, by Theorem 3.1, there exist an open subset Vmrn ⊃ L and n : Hb (Un ) → Hb (Vm ) such that a continuous homomorphism A rn L n = A ◦ InK . By Corollary 2.3 there exists mn > mr such Im ◦A n rn that σ(H(L), H(L) )|B = σ(Hb (Vmn ), Hb (Vmn ) )|B , for each bounded subset B of Hb (Vmrn ). Now, a slight modiﬁcation of arguments from Theorem 4.1 gives the proof. Before proving Proposition 4.3 we need some preparation. Let P(E) denote the algebra of all continuous polynomials on E. We denote by (E, σ(E, P(E))) the space E endowed with the coarsest topology making all P ∈ P(E) continuous. The topology σ(E, P(E)) is a regular Hausdorﬀ topology such that (E,  · ) ≥ (E, σ(E, P(E))) ≥ (E, σ(E, E )). Thus it follows that σ(E, P(E)) is angelic, see [3, Corollary 1], and consequently the concepts (relatively) countably compact, (relatively) sequentially compact and (relatively) compact all agree with respect to this toplogy. A Banach space E is called a Λ-space, if all null sequences in σ(E, P(E)) are norm convergent, and hence, convergent sequences in σ(E, P(E)) are also norm convergent. All superreﬂexive spaces and 1 are Λ-spaces [9]. Proposition 4.3. Let E be a Λ-space with the approximation property and F be a Banach space. Let K ⊂ E be an absolutely convex compact set, L ⊂ F a compact set and A : (H(K), τw ) → (H(L), τw ) a continuous homomorphism. If for each open subset Un ⊃ K there n : Hb (Un ) → exist an open subset Vmn ⊃ L and a reflexive operator A L K  Hb (Vmn ) such that Imn ◦ An = A ◦ In , then A is a Montel operator. Proof. Since Un = K + B1/n (0) is an absolutely convex subset and E has the approximation property, by [4, Proposition 9] we have that Φn maps Vmn -bounded into σ(E, P(E))-relatively compact subsets in Un ⊂ E. Since the topology σ(E, P(E)) is angelic, see [3, Corollary 1], we have that Φn maps Vmn -bounded subset into relatively compact

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n : Hb (Un ) → subset in Un ⊂ E. Consequently Φn is Montel and A Φn Hb (Vmn ) is a Montel homomorphism by [4, Proposition 13]. Now, by Theorem 4.1 the proof is complete. The next example gives a non Montel composition operator which is a reﬂexive composition operator. Example 4.1. Let E be a Tsirelson space, and let Φ ∈ H(E, E) be deﬁned by Φ(z) = z/2 for all z ∈ E. If we take U and V as open sets such that Φ(V ) ⊂ U , then it is not diﬃcult to see that Φ takes V bounded sets in U -bounded sets. If K ⊂ E is a balanced compact set, then the composition operator AΦ : (H(K), τw ) → (H(K), τw ) given by AΦ (f ) = f ◦ Φ is reﬂexive and however AΦ is not Montel. Indeed, for each open subset Un ⊃ K by Theorem 3.1 there exist an open subset Umn ⊃ K with mn ≥ n and a continuous homomorphism K n : Hb (Un ) → Hb (Um ) such that Im n = AΦ ◦ InK . For each A ◦A n n n (f ) = f ◦ Φn , n ∈ N, let Φn = Φ|Umn . Then it is possible to prove A for each f ∈ Hb (Un ). As E is a Tsirelson space, we have that Hb (Un ) n is reﬂexive. Therefore, according to is reﬂexive and consequently A Theorem 4.2, AΦ is a reﬂexive operator. Now suppose that AΦ is a Montel operator. By Theorem 4.1 we have n is a Montel operator. From Proposition 13 of [4] we have that that A Φn is a compact operator. As mn > n, then Φ(Umn ) is Un -bounded and so (1/4)BE ⊂ mn · Φn (Umn ) is a compact subset. Thus E has a ﬁnite dimension. This is a contradiction; consequently, AΦ is not a Montel operator. Acknowledgment. We would like to thank Prof. Jorge Mujica for helpful comments on the subject of this paper. REFERENCES 1. J. Bonet, P. Domanski, M. Lindstr¨ om and M. S. Ramanujan, Operator spaces containing c0 or ∞ , Results Math. 28 (1995), 250 269. 2. S. Dineen, Complex analysis on inﬁnite dimensional spaces, Springer-Verlag, London, 1999. 3. K. Floret, Weakly compact sets, Lecture Notes in Math., vol. 801, Springer, New York, 1978.

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