Condori, Luciano O.

Lourenço, M. Lilian

English

ESE - Salento University Publishing

Note di Matematica 26, n. 1, 2006, 131–137.Composition operators between Fre´chetspaces of holomorphic functionsLuciano O. CondoriiICET, Universidade Paulista,CEP: 06500-000 Sa˜o Paulo, Brazillch@ime.usp.brM. Lilian Lourenc¸oDepartamento de Matema´tica, Universidade de Sa˜o Paulo,CP 66281 - CEP: 05311-970 Sa˜o Paulo, Brazilmllouren@ime.usp.brReceived: 9/8/2004; accepted: 19/5/2005.Abstract. Let E, F and G be Banach spaces. Let V a balanced open subset of F . Thereflexive and Montel composition operator TΦ(f) := f ◦ Φ acting between the Fre´chet spacesof all G-valued holomorphic functions of bounded type on E is studied in terms of Φ, whereΦ is a G-valued holomorphic functions of bounded type on V .Keywords: holomorphic functions, composition operatorsMSC 2000 classification: primary 46G20, 46J05, secondary 47B331 IntroductionLet E andG be complex Banach spaces. For an open subset U of E,Hb(U,G)denotes the space of all holomorphic functions from U into G which are boundedon U -bounded subsets of U . It is endowed with the topology τb of uniformconvergence on U -bounded sets. It is known that Hb(U,G) is a Fre´chet space.As usual, we will always omit G in the notation in case G = C. So, for instance,we will write Hb(U) for Hb(U,C).If F is a complex Banach space and V an open subset of F , given a holo-morphic mapping of bounded type Φ : V → E with Φ(V ) ⊂ U we will considerthe composition operator TΦ : Hb(U,G) → Hb(V,G) defined by TΦ(f) = f ◦ Φ.Recently the study of composition operators has deserved some attention. Sev-eral results of composition operators between the Fre´chet spaces Hb(U,G) ofholomorphic functions of bounded type have appeared when G = C (see forexample [2], [4]). For instance, in [3] and [4] M. Gonza´lez and J. Gutie´rrezhave found results relating the (weak) compactness of holomorphic functioniResearch supported by a doctoral fellowship from CNPq, Brazil.132 L. O. Condori, M. L. Lourenc¸oof bounded type Φ ∈ Hb(F,E) with the (weak) compactness of compositionoperator TΦ from Hb(E) into Hb(V ).It seems natural to study Montel and reflexive composition operators be-tween the Fre´chet spaces Hb(U,G) of all vector valued holomorphic functions ofbounded type.In this note we study reflexive and Montel composition operators TΦ betweenthe Fre´chet spaces of all G-valued holomorphic functions of bounded type interms of Φ.Let us mention that if G is a complex Banach algebra the space Hb(U,G) en-dowed the τb topology of the uniform convergence on U -bounded set is a Fre´chetalgebra and the composition operator TΦ is a continuous homomorphism.Preliminaries. Our notation is standard and we refer to the books ofDineen [1] and Mujica [7] for background information on holomorphic functionson infinite dimensional Banach spaces, to Jarchow [6] and Horvath [5] regardinglocally convex spaces theory.Let E and G be complex Banach spaces. If U is an open subset of E, thena set A ⊂ U is said to be U -bounded if A is bounded and is bounded away fromthe boundary of U .If X and Y are complex Hausdorff locally convex spaces and T : X → Y isa linear map, then T ∗ : Y ∗ → X∗ defined by T ∗(y∗) = y∗ ◦ T is a well definedlinear map, it is called the algebraic adjoint of T . Under some conditions T ∗induces a map T ′ : Y ′ → X ′ and we call T ′ the adjoint or transposed map of T .A continuous linear map from X into Y is called Reflexive (resp. Montel),if it transforms bounded sets into relatively weakly compact (resp. relativelycompact) sets.Let us recall that a continuous linear mappings T from X into Y is calledweakly compact (resp. compact), if it maps some 0-neighborhood into relativelyweakly compact (resp. relatively compact) sets. If X and Y are normed space Tis weakly compact (resp. compact) if and only if T is Reflexive (resp. Montel).Let now X be a complex Hausdorff locally convex space and let X ′ be itstopological dual. As usual σ(X,X ′) is the weak topology on X and σ(X ′,X)is the weak-star topology on X ′. τβ denotes the strong topology on X, τµ de-notes the Mackey topology on X ′ and τc denotes the topology of the uniformconvergence on compact subsets of X on X ′.We will denote BX(0) a fundamental system of neighborhoods of 0 of theHausdorff locally convex space X.Let A a bounded set in Hb(U,G), we associate the following neighborhoodComposition operator 133of 0 in (Hb(U,G)′, τβ):U0,A,ǫ = { f ′ ∈ Hb(U,G)′/ supf∈A|f ′(f)| < ǫ } ∈ B(Hb(U,G)′,τβ)(0).For each finite subset { g1, . . . , gr } ofHb(U,G) and for each ǫ > 0 we considerthe following neighborhood of 0V0,g1,...,gr,ǫ = { f ′ ∈ Hb(U,G)′/|f ′(g1)| < ǫ, . . . , |f ′(gr)| < ǫ }in B(Hb(U,G)′,σ(Hb(U,G))′,(Hb(U,G)))(0). To simplify the notation from now we usew∗ instead of σ(Hb(U,G)′, (Hb(U,G))).2 Composition operatorsIn this section we study Montel (resp. reflexive) composition operators.1 Lemma. Let E and G be Banach spaces. Let a ∈ G and a′ ∈ G′ such that‖a‖ = 1, ‖a′‖ = 1 and a′(a) = 1. Then:(i) Ja : (E′, ‖‖) → Hb(E,G) given by Ja(x′)(x) = x′(x)a, for all x′ ∈ E′and for x ∈ E is a continuous linear mapping. Moreover, the transposedmapping J ′a : (Hb(E,G)′, τβ)→ (E′′, ‖‖) is continuous.(ii) Let V be an open set of E and let δa′ : V → (Hb(V,G)′, w∗) defined byδa′(y)(g) = a′(g(y)), for all y ∈ V and for g ∈ Hb(V,G). Then δa′ is acontinuous mapping and δa′ maps V -bounded sets into bounded sets forthe topology τβ on Hb(V,G)′.Proof. (i) It is clear that Ja is linear. Now, let B be a bounded set of Eand ǫ > 0. So, there is λB > 0 such that ‖x‖ ≤ λB for all x ∈ B. If x′ ∈ E′with ‖x′‖ < ǫλB we have ‖Ja(x′)‖B ≤ ǫ and consequently Ja is continuousat the origin. For the continuity of J ′a we consider B′′ǫ (0) ∈ BE′′(0) and theunit ball BE′ of E′. Then Ja(BE′) is a bounded set inHb(E,G) and we have|f ′(Ja(x′))| < ǫ for all x′ ∈ BE′ and f ′ ∈ W0,Ja(BE′ ),ǫ ∈ B(Hb(E,G)′,τβ)(0).So J ′a(f ′) ∈ B′′ǫ (0) for all f′ ∈ W0,Ja(BE′ ),ǫ.(ii) Let y0 ∈ V and Vδa′ (y0),g1,...,gr,ǫ ∈ B(Hb(V,G)′,w∗)(δa′(y0)) with gi ∈ Hb(V,G).Since each gi, 1 ≤ i ≤ n is continuous in y0 ∈ V , there exists Bλi(y0) ⊂ Vsuch that for each y ∈ Bλi(y0) we have ‖gi(y) − gi(y0)‖ ≤ ǫ for each1 ≤ i ≤ n. Let λ = min1≤i≤r{λi } and y ∈ Bλ(y0). So y ∈ Bλi(y0) and|δa′(y)(gi) − δa′(y0)(gi)| < ǫ for each i = 1, . . . , r. Consequently δa′(y) ∈Vδa′(y0),g1,...,gr,ǫ and δa′ is continuous.134 L. O. Condori, M. L. Lourenc¸oTo complete the proof it suffices to show that δa′ maps V -bounded sets ofV into bounded sets of (Hb(V,G)′, τβ). Let B ⊂ V be a V -bounded andV0,g1,...,gr,ǫ ∈ B(Hb(V,G)′,w∗)(0) with gi ∈ Hb(V,G). So there exist λi > 0such that supy∈B‖gi(y)‖ < λi for 1 ≤ i ≤ r. Let λ = max1≤i≤r{λi } and g′ ∈ǫ2λδa′(B). Then |g′(gi)| < ǫ, i = 1, . . . , r and g′ ∈ V0,g1,...,gr,ǫ. Consequentlyǫ2λδa′(B) ⊂ V0,g1,...,gr,ǫ and it is w∗-bounded. Since Hb(V,G) is a barrelledspace we have that δa′(B) is bounded in the space (Hb(V,G)′, τβ), thuscompleting the proof.QEDIn the next theorem we study the Montel composition operator.2 Theorem. Let E, F and G be Banach spaces. Let V ⊂ F an open subset,Φ ∈ Hb(V,E) and TΦ : Hb(E,G) → Hb(V,G) a composition operator. Considerthe following conditions:(a) TΦ is a Montel operator(b) The adjoint operator T ′Φ : (Hb(V,G)′, τβ) → (Hb(E,G)′, τβ) is a Monteloperator(c) Φ maps V -bounded sets into relatively compact sets in E.Then (a)⇒ (b)⇒ (c).Proof. (a)⇒ (b) First we show that T ′Φ : (Hb(V,G)′, τc)→ (Hb(E,G)′, τβ)is continuous. Let X ⊂ Hb(E,G) be a bounded set andV0,X ,ǫ = { f ′ ∈ Hb(E,G)′ / ‖f ′‖X < ǫ } ∈ B(Hb(E,G)′,τβ)(0).Since TΦ is a Montel operator we have that TΦ(X ) is a relatively compact setof Hb(V,G). Then the closed absorbing convex hull Γ(TΦ(X )) is compact, sinceHb(V,G) is a Fre´chet space.Now, if we considerW0,Γ(TΦ(X )),ǫ = {h′ ∈ Hb(V,G)′ / ‖h′‖Γ(TΦ(X )) < ǫ } ∈ B(Hb(V,G)′,τc)(0),we have that T ′Φ(W0,Γ(TΦ(X )),ǫ) ⊂ V0,X ,ǫ, since for each h′ ∈ W0,Γ(TΦ(X )),ǫ weget |h′(h)| < ǫ for all h ∈ Γ(TΦ(X )). So |h′(TΦ(f))| < ǫ for all f ∈ X andT ′Φ(h′) ∈ V0,X ,ǫ. Therefore T ′Φ : (Hb(V,G)′, τc)→ (Hb(E,G)′, τβ) is continuous.Now, let A ⊂ Hb(V,G)′ a τβ-bounded set. Since Hb(V,G) is barrelled wehave that A is an equicontinuous set and consequently σ(Hb(V,G)′,Hb(V,G))-relatively compact. By Banach-Dieudonne´ theorem we have that A is τc-rela-tively compact. As T ′Φ is τc − τβ continuous we have that T ′Φ(A) ⊂ Hb(E,G)′ isτβ-relatively compact on Hb(E,G)′.Composition operator 135(b) ⇒ (c) Let a ∈ G with ‖a‖ = 1. By Hahn-Banach theorem there existsa′ ∈ G′ with ‖a′‖ = 1 and a′(a) = 1. Now, let ψ : V → (E′′, ‖‖) defined byψ := J ′a ◦ T ′Φ ◦ δa′ . By Lemma 1 we have that ψ maps V -bounded sets intorelatively compact set of E′′.As ψ(y)(x′) = x′(Φ(y))a′(a) = C(Φ(y))(x′), for all y ∈ V , for all x′ ∈ E′where C : E → E′′ is the natural inclusion, we have that ψ(y) = C(Φ(y)) for ally ∈ V . So C ◦Φ = ψ and Φ maps V -bounded sets of V into relatively compactsets in E′′. QEDIn [3] Gonza´lez-Gutie´rrez proved these conditions of Theorem 2 are equiv-alent when G = C. However, in the general case, the following example showsthat the assertions of Theorem 1 are not equivalent.3 Example. Let Φ be a identity on C, which is a trivially Montel mapping.Give an infinite dimensional Banach space G, we consider the composition oper-ator TΦ : H(C, G) −→ H(C, G) given by TΦ(f) = f ◦Φ = f for all f ∈ H(C, G).Let (yn) be a sequence of norm one vectors in G such that ||yn − ym|| ≥ δ > 0for all n 6= m. Define fn : C −→ G by fn(λ) = λyn. Then the sequence (fn) isbounded in H(C, G) but is not relatively compact, since (fn(1)) = (yn) is notrelatively compact.4 Corollary. Let E,F and G be Banach spaces. Let U ⊂ E, V ⊂ F anopen set, Φ ∈ Hb(V,E) and let TΦ : Hb(U,G) → Hb(V,G) be a compositionoperator. Then Φ maps V bounded sets into relatively compact sets of U if TΦis compact.Proof. It suffices to observe that the composition operator A : Hb(E,G)→Hb(V,G) given by A(f) = f ◦ Φ for all f ∈ Hb(E,G) is compact if TΦ :Hb(U,G)→ Hb(V,G) is compact. QEDThe next theorem studies, in the same cases, when the composition operatorTΦ is reflexive.5 Theorem. Let E, F and G be Banach spaces. Let V be a balanced open setof F , Φ ∈ Hb(V,E) and TΦ : Hb(E,G) → Hb(V,G) be a composition operator.Consider the following conditions:(a) TΦ is a reflexive operator(b) The adjoint operator T ′Φ : Hb(V,G)′ → Hb(E,G)′ maps τβ-bounded sets intorelatively σ(Hb(E,G)′, (Hb(E,G)′, τβ)′)-compact sets(c) Φ maps V -bounded sets into relatively weakly compact sets in E.Then (a)⇒ (b)⇒ (c).136 L. O. Condori, M. L. Lourenc¸oProof. (a)⇒ (b) First we show that T ′Φ : (Hb(V,G)′, τµ)→ (Hb(E,G)′, τβ)is continuous. Let X ⊂ Hb(E,G) be a bounded subset and letV0,X ,ǫ = { f ′ ∈ Hb(E,G)′ / ‖f ′‖X < ǫ }be a τβ-neighborhood of zero in Hb(E,G)′. Since TΦ is a reflexive operator itfollows that TΦ(X ) is a relatively weakly compact set of Hb(V,G). As Hb(V,G)is a Fre´chet space we have that the closed convex absolutely hull of TΦ(X ),Γ(TΦ(X )), is a weakly compact set of Hb(V,G).Now, we consider W0,Γ(TΦ(X )),ǫ = {h′ ∈ Hb(V,G)′ / ‖h′‖Γ(TΦ(X )) < ǫ } ∈B(Hb(V,G)′,τµ)(0). Then it is clear that T ′Φ(W0,Γ(TΦ(X )),ǫ) ⊂ V0,X ,ǫ. We claim thatT ′Φ : (Hb(V,G)′, w∗)→ (Hb(E,G)′, σ(Hb(E,G)′, (Hb(E,G)′, τβ)′))is a continuous mapping. Indeed letV0,f ′′1 ,...,f ′′k ,ǫ ∈ B(Hb(E,G)′,σ(Hb(E,G)′,(Hb(E,G)′,τβ)′))(0)with f ′′i ∈ (Hb(E,G)′, τβ)′ for each i = 1, 2, . . . , k. Then we have that T ′′Φ(f ′′i ) ∈(Hb(V,G)′, τµ)′. Thus there exist fi ∈ Hb(V,G) such that T ′′Φ(f ′′i )(f ′) = f ′(fi)for all f ′ ∈ Hb(V,G)′ and i = 1, 2, . . . , k. If we considerW0,f1,...,fk,ǫ ∈ B(Hb(V,G)′,σ(Hb(V,G)′,Hb(V,G)))(0),it is easy to see that T ′Φ(W0,f1,...,fk,ǫ) ⊂ V0,f ′′1 ,...,f ′′k ,ǫ and consequently T ′Φ isσ(Hb(V,G)′,Hb(V,G)) − σ(Hb(E,G)′, (Hb(E,G)′, τβ)′) continuous.Now, let X ⊂ Hb(V,G)′ be a τβ-bounded set. As Hb(V,G) is barrelled wehave that X is a w∗-relatively compact.So T ′Φ(X ) is σ(Hb(E,G)′, (Hb(E,G)′, τβ)′)-relatively compact, and the im-plication (a)⇒ (b) follows.(b)⇒ (c) Let a ∈ G with ‖a‖ = 1. Then there is a′ ∈ G′ such that ‖a′‖ = 1and a′(a) = 1.First we show thatJ ′a : (Hb(E,G)′, σ(Hb(E,G)′, (Hb(E,G)′, τβ)′))→ (E′′, σ(E′′, E′))is a continuous mapping at origin, where J ′a is the adjoint operator of Ja definedin the Lemma 1.Let V0,x′1,...,x′r,ǫ ∈ B(E′′,σ(E′′,E′))(0). For each i = 1, 2, . . . , r, we considerx′i ⊗ a ∈ Hb(E,G) defined by x′i ⊗ a(x) = x′i(x)a for all x ∈ E and f ′′i :(Hb(E,G)′, τβ) → C given by f ′′i (f ′) = f ′(x′i ⊗ a) for all f ′ ∈ Hb(E,G)′. Now,it holds that W0,f ′′1 ,...,f ′′r ,ǫ ∈ B(Hb(E,G)′,σ(Hb(E,G)′,(Hb(E,G)′,τβ)′))(0) and we haveJ ′a(W0,f ′′1 ,...,f ′′r ,ǫ) ∈ V0,x′1,...,x′r,ǫ. So by Lemma 1 (ii) the mapping ψ : V →Composition operator 137(E′′, σ(E′′, E′)) given by ψ = J ′a ◦ T ′Φ ◦ δa′ maps V -bounded set into σ(E′′, E′)-relatively compact set in E′′.Let (yn)n∈N ⊂ V be a bounded sequence. Since ψ is σ(E′′, E′)-relativelycompact and ψ(y)(x′) = C(Φ(y))(x′) for all y ∈ V and x′ ∈ E′ where C is thenatural inclusion from E into E′′, there exist a subsequence (ynk)k of (yn) andx′′ ∈ E′′ such that x′(Φ(ynk)) → x′′(x′) for all x′ ∈ E′. Consequently, Φ mapsbounded sets of V into σ(E,E′)-relatively compact sets in E. QED6 Remark. Slight modifications of example 1 give an example that shows ingeneral the assertions of Theorem 2 are not equivalent. Indeed, let Φ be a reflex-ive mapping and let G be a non-reflexive Banach space, and choose a sequence(yn) of norm one vectors in G without any weakly convergent subsequence.Define (fn) as the example 1. Then (fn) is not relatively weakly compact.In [4] M. Gonza´lez and J. Gutie´rrez showed that the conditions of Theorem2 are equivalent if G = C and E has Dunford-Pettis property.References[1] S. Dineen: Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, LondonBerlin Heidelberg 1999.[2] P. Galindo, M. L. Lourenc¸o, L. A. Moraes: Compact and weakly compact homomor-phisms on Fre´chet algebras of holomorphic functions, Math. Nachr., 236, (2002), 109–118.[3] M. Gonza´lez, J. Gutie´rrez: Schauder type theorems for differentiable and holomorphicmappings, Monatsh. Math., 122, (1996), 325–343.[4] M. Gonza´lez, J. Gutie´rrez: Gantmacher type theorems for holomorphic mappings,Math. Nachr., 186, (1997), 131–145.[5] J. Horvath: Topological Vector Spaces and Distributions, Vol. 1. Addison-Wesley, Mas-sachusetts 1966.[6] H. Jarchow: Locally Convex Spaces, Mathematische Leitfa¨den, B. G. Teubner, Stuttgart1981.[7] J. Mujica: Complex Analysis in Banach Spaces, Math. Stud., 120, North-Holland 1986.

Composition operators between Frèchet spaces of holomorphic functions

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Composition operators between Frèchet spaces of holomorphic functions

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