gantmacher type theorems for holomorphic mappings
TRANSCRIPT
Math. Nachr. 186 (1997), 131 - 145
Gantmacher Type Theorems for Holomorphic Mappings
By MANUEL GONZALEZ of Santander and JOAQU~N M. C ; U T I h , E Z of Madrid
(Received September 28, 1994)
(Revised Version May 24, 199G)
Abstract. Givena holomorphicmapping of bounded typeg E Hb(u, F), where U E is a balanced open subset, and E , F are complex B a n d spaces, let A : Hb(F) 4 H b ( U ) be the homomorphism defined by A ( f ) = f o g for all f E Hb(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W C U. To obtain these results, we prove that the class of Dunford-Pettis sets is stable under projective tensor products. Moreover, we diaracterize the reflexivity of the space Hb(U,F) and prove that E' and F have the Schur property if and only if H b ( u , F ) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.
1. Introduction
If T : E + F is a (linear bounded) operator between Banach spaces, Schauder's theorem states that T is compact if and only if its adjoint T' : F' -+ E' is compact [J, Theorem 17.1.31. Similarly, by GANTMACHER'S theorem [J, Theorem 17.2.51, T is weakly compact if and only if so is T".
In [CXhch] the authors gave Schauder type theorems characterizing the compact homomorphisms between certain algebras of differentiable functions on Banach spaces, and the compact composition operators between algebras of holomorphic functions of bounded type on Banach spaces. Namely, given a mapping g : U - F, where U E E is a balanced open subset, and E, F are complex Banach spaces, then g induces a continuous homomorphism A : Hb( F ) - Hb(U) between the corresponding algebras of holomorphic functions of bounded type, given by A( f) = f o g for all f E Ha(F) , if and only if g E Ha(U, F ) . Moreover, given g E Hb(U, F ) and the induced A as above, then g is a compact mapping if and only if A is compact, and this is equivalent to the
1991 Maihemaiics Subjeci Classificaiion. Primary 46 E 25; Secondary 46 G 05, 46 G 20,47 B 38. Keyword8 and phrases. Algebras of analytic fullctions of bounded type, holomorphic mappings,
:omposition operators, weakly compact homomorphisms, linearization, Dunford- Pettis set.
132 Math. Nachr. 186 (1997)
adjoint A* being compact for the strong topologies. A similar result was obtained in [GGM].
Our purpose in the present paper is to establish similar results for the weakly com- pact and the completely continuous composition operators. Thus in Section 4 we prove (the definitions will be recalled later) that:
(a) A is completely continuous (i.e., A takes weakly null sequences into null se- quences) if and only if g( w) is a Dunford - Pettis set for every U -bounded W c U , and
(b) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact.
As a consequence of our techniques, we also prove that: (c) Hb(U,F) is reflexive if and only if, for every k E IN, the space P( 'E ,F) of
(d) Hb(U, F ) has the Schur property if and only if so do E' and F. As a preparation to these results, we study in Section 2 tensor products of sets and
sequences, showing for instance that the tensor product of Dunford-Pettis sets is a Dunford - Pettis set. In Section 3 we obtain properties of certain composition operators between spaces of polynomials, refining previous results on reflexivity of these spaces. Finally, in Section 5, we show the relationship between our composition operators and MUJICA'S linearization of holomorphic mappings of bounded type, constructed in [Mu - Val], proving that the composition operator A has certain properties if and only if the linearization of its inducing mapping g has the same properties.
Many of our results and proofs are inspired by the work of ARON et al. ([AAD], [AAF], [AHV], [AS]) and RYAN ([RyT], [RyDPP], (Ry]). Some of the results in this paper were announced in [GGCR].
Throughout, E and F denote Banach spaces, E' the dual of E , BE its closed unit ball, and U E E a balanced open subset; the weak-star topology on E' is denoted by w'; IK is the scalar field (real, or complex C ) , and IN the natural numbers. Without explicit mention, we assume that E is embedded in E".
For j E IN, P (iE, F) represents the space of all j -homogeneous (continuous) poly- nomials from E into F . Its subspace P,b(jE, F) consists of those polynomials which are weakly (uniformly) continuous on bounded sets (see [AHV]). We identify P(OE, F ) with F . For P E P(JE , F ) , stands for the symmetric j-linear continuous mapping associated to P. As usual,
k -homogeneous polynomials is reflexive,
P ( E , F ) := P(jE, F ) . j = O
By L(jE, F ) we denote the space of all j-linear continuous mappings from (1 1 E x . . . x E into F . Whenever the range space is omitted, it is understood to be
the scalar field; e. g., P(jE) = P(jE, IK). For complex E and F, H ( U , F) is the space of all holomorphic mappings from U
into F . We denote by Hb(U, F) the space of holomorphic mappings of bounded type (i.e., bounded on U-bounded sets) from U into F. We recall that a bounded set W C U is U -bounded if the distance from W to the boundary of U is positive. The
Gonz&lez/GutiCrrez, Gantmacher Type Theorems 133
notation H,,(f l ,F) stands for the space of all g E H ( U , F ) so that for every U - bounded set W C U , the restriction g(w : W -+ F is uniformly continuous when W is given the restricted weak topology (see [LL, $111). An operator T : E -+ F belongs to H,,(E, F) if and only i f T is compact. The spaces H,,,,(U, F) and Ha(U, F) are F'rkchet spaces when endowed with the topology of uniform convergence on U - bounded sets. The algebras H,,(E) and Ha(E) are relevant to the Michael problem on automatic continuity of complex valued homomorphisms on Frkchet algebras. Indeed, given an infinite dimensional complex Banach space E, if every complex homomorphism on H,,(E) is continuous, then every complex homomorphism on a commutative Frkchet algebra is continuous [Mu, p. 2441. The same is true if we replace H,,(E) by Ha(E). We refer to [Din], [Mu] for the general theory of holomorphic mappings.
We say that a mapping g : U * F is (weakly) compact if it takes U-bounded sets into relatively (weakly) compact sets. An operator between locally convex spaces is (weakly) compact if it takes bounded subsets into relatively (weakly) compact subsets. Although this is not the standard definition, we prefer to keep it in this way for the sake of symmetry of the results.
2. Tensor products of sets and sequences
In this Section, we investigate weak convergence of tensor products of sequences,
We write @: E for the linear subspace of @'E := E@ . . . @ El introduced in
[RyT], generated by the elements of the form d k ) := z @ . . . Wz, for z E E. For A c El we denote A(') = { d k ) : z E A} c @fE. The space @,"E is normed as a subspace of BkE with the projective norm, and the convex hull of B!$) is a bounded neighbourhood of zero. The dual space of &E is isomorphic to P('E) by means of the duality (dk) , P) := P ( z ) for each P E P('"E) and t E E.
For the projective and injective tensor products we use the symbols @r and BC, respectively. The closure of 8," E in & E is denoted by @:,, E.
Some conventions will be needed for the next Lemma. We say that a set A C E is relatively pwk-compact if given any sequence (tn) c A, we can find a subsequence (zn,) and z E E such that P (zn,) -+ P ( z ) for every P E P()E). If q3 E E*, we denote by dk the polynomial given by d k ( z ) := (r$(z))' for each z f E.
and. show that Dunford- Pettis sets are stable under tensor products. (k)
( k )
Lemma 2.1. Given a sequence (zn) c E and k E IN $zed, we have: (a) if (a)) is weakly convergent to some z E @t,,E, then (zn) has a weakly
(b) suppose E has the approzimation property (AP), (2,) is weakly convergent t o z,
(c) if a set A C E is relatively PWk - compact, then A ( k ) is relatively weakly compact.
n convergent subsequence ;
and k i s weakly convergent to z E Br,# E ; then z = x ( ~ ) ;
Assuming E has the AP, the converse i s also t rue .
134 Math. Nachr. 186 (1997)
Proof. For k: = 1, there is nothing to prove. Suppose k: > 1. (a) For every qi E E*, the sequence ((z!ik),qik)) = ((qi(tn))k) is convergent. If
its limit is 0 for all qi E E', then (zn) is weakly null and the r e h t is proved. If, for some qi E E', limn (qi(sn))k # 0, then we can find a subsequence (En,) such that (qi (zn,)k-l)i converges to some cr # 0.
Themapping [email protected]@yk I+ #(yl).. .:qi(yk-~)yk definesanoperator T:@:E-+E. Hence weak- limi (4 (z,,,))~-' xn, = T ( z ) . Therefore (zn,) is weakly convergent to cr-'T(z) and the proof is finished..
yk E E, defines an operator 4 : @:E -+ @:-'E. Write z = Cui 8 vi, with ui E E, V i E @:-'E, and C ~ ~ u , ~ ~ ~ ~ ~ v , ~ ~ < 00. Consider the natural inclusion
n
(b) For every # E E', the mapping y~ €3 . + . LB yk ++ #(y~)yz @ - 3 . @ yk, for y1, . . .,
which is injective since E has the AP [Gr, Ch. I, Prop. 351. Suppose first x = 0. Then, for every qi E E',
since qi(xn) -t 0. Therefore J ( r ) = 0 and so z = 0. Suppose now x # 0, and choose cp E E' with p(x) = 1. Assume the result is true for k - 1. We have
@(z) = weak- 1im p(~,,)x&~-') = weak -1im tik-') = x('-'), n n
using the induction hypothesis. Moreover,
For every qi E E', we have Ci #(ui)vi = weak - limn qi(tn)tn (k-1) = qi(z)dk-'). There- fore ~ ( z - t ~a ~ ( ~ - 1 ) ) = 0, and so z = dk).
c A ( k ) be a sequence. WI, can find an increasing sequence (? t i ) c IN and t E E such that P (z,,) -+ P(x) for all P E P ( k E ) . Therefore
Then there are an increasing sequence (ni) c IN and z E @:,,E such that
(c) Suppose A is relatively pwk-compact. Let zn ( (k)) is weakly convergent to %('I.
Conversely, let A ( k ) be relatively weakly compact and take a sequence ( t n ) C A .
tit),^) - ( z , ~ ) for all P E P ( ~ E ) .
( ;onzilez/Guti&rez, Gantmacher Type Theorems 135
I3y (a), (zn,) has a subsequence weakly convergent to some E E E . Since E has the D
A subset A c E is a Dunford - Pettis (DP) set [An] if for every weakly null sequence (&) C E * , and every sequence (zn) C A, we have lim(z,, &) = 0. Equivalently, A is it DP set if and only if every weakly compact operator on E takes A into a relatively compact set [An]. A Banach space E has the Dunford-Pettis property (DPP, for short) if, for every Banach space F, every weakly compact operator from E into F is completely continuous.
We need a result whose proof follows the. ideas of [GGdw, Theorem 3.21. Given Banach spaces E l , . . . , Ek, we denote by L ( E 1 , . . . , Ek; F) the space of all k- linear (continuous) mappings from El x . . . x Ek into F. It is well known that the dual of El @ w . . . @* Ek may be identified with L(E1 , . . . , Ek; IK).
AP, we get from (b) that z = dk) , and so A is relatively pWk -compact.
Theorem 2.2. Lei Banach spaces Ej and the Dunford- Pettis sets Dj c E j , 1 5 j 5 k , be given. Then D1 €3 . . . C3 Dk as a Dunford - Pettis set in El @* . . . @= En:.
Proof . By associativity of the tensor product, it is enough to prove the result for k = 2. Suppose it fails to be true. Then we can find a weakly null sequence (An) C L ( E l , E z ; I K ) , sequences (zy),”=, c D j , j = 1,2, and 6 > 0 so that for all 91 E IN, we have IA, (z: ,z!j) I > 6. For each I E E,”, define the operator
T, : L(E1 ,Ez ; IK) - E:
by ( t l , T , ( A ) ) := j(z1, z) for every A E L(E1, E2;IK) and 21 E E l , where A is the :xtension of A ( z 1 , . ) to E2f* by w* -continuity. Then, since D1 is a DP set,
An (x:,~) = T z ( A n ) ( x Y ) - 0 as n -+ 00 .
Iefine &, E E2f by ( 2 , &) := A,, (xy, z) for all 3: E E2. Since the sequence (&) C E,’ s weakly null, and D2 is a DP set, we have 0 = lim,(z?, q?~,,) = limn An(zy , E ! ) , a :ontradiction. 0
A Banach space has the Schur property if weakly null sequences are norm null. i straightforward consequence of Theorem 2.2 is the following result, contained in RyDPP, Theorem 3.31:
Corollary 2.3. Given Banach spaces E l , . . . , Ekr ihe space L ( E 1 , . . . , Ek, K) has
;. Operators and spaces of polynomials
Given an operator T : E + F, we construct its “adjoint” 2’; : P(’F) -+ P ( k E ) )r each k E IN, and give some properties, refining previous results on reflexivity of ‘ ( ‘E) . We also show that the weak polynomial topology is angelic. We say that E has the compact approximation property (CAP) if for each com-
act subset D c E and > 0 , there is a compact operator K : E + E so that
136 Math. Nachr. 186 (1997)
sup {(JIC(z) - zll : x E D} < c . We say that E has the A-CAP if I< may be chosen so that 1111' - 111 5 A, where I(x) := 2 [AT]. If E has the A-CAP for some A 2 1, then it is said to have the bounded compact approximation property (BCAP). It is well known that the AP and the bounded approximation property (BAP) are defined in the same way, replacing compact operator by finite rank operator. Then the following implications hold:
BAP BCAP
U U AP CAP
None of these can be reversed: it is shown in [Wl that the BCAP does not imply the AP, and in [BP, 6.51 that the A P does not imply the BCAP.
Lemma 3.1. Suppose E has the A -CAP and let k E IN. Then the (A + l ) k -bal l o ~ P , ~ ( ~ E , F ) is dense in the unit ball of P('E, F ) for the topology T~ of pointwise convergence.
Proof . Take c > 0, P E P('E, F ) with llPll 5 I , and afinite set {xl, . . . , zn} C BE. Since P is uniformly continuous on bounded sets, there is 6 > 0 such that whenever x, x' E (A + BE satisfy the condition IIx - 2/11 < 6, then llP(x) - P(x')ll < c . Take a compact operator Ii' on E such that 11K11 5 A + l and IlK(z,)-zill < 6 for i = 1, , . . , n. Let Q := P o K E ?wb(kE, F). Then
IlQl l I (A + I l k and
IJQ(xi) - P(~i)ll = IIP(K(zi)) - P(xi)ll < c for i = 1, . . .,a,
and the proof is finished. 0
Given an operator T : E 4 F and k E IN, let Tk : @:,8E - @i,,F be the operator taking y ( k ) into ( T v ) ( ~ ) . Its adjoint Ti : P ( k F ) - P('E) is given by Ti(P) := POT for all P E P ( k F ) . We denote by T+k : 7)wb(kF) - ?wb('E) the restriction of Ti to Pw6 ( k F ) .
Theorem 3.2. Lei T : E + F be an operator and k E IN. Consider the following
(a) T(BE) is relatively pwk -compact ; (b) T; is weakly compact; ( c ) is weakly compact.
assertions :
Then (a) implies (b) which implies (c). If F has fhe AP, then also (b) implies (a). If F has the BCAP, then (c) implies (b).
P roof . (a) e (b). By Lemma 2.1 (c), if T(BE) is relatively pwk-compact, then T( B E ) ( k ) = Tk( Bg') is relatively weakly compact. This implies that 'Ik, and there - by Ti, is weakly compact. Assuming F has the AP, the converse is also true by Lemma 2.1 (c).
Gonzilez/GutiQrez, Gantmacher Type Theorems 137
(b) j (c) is obvious. (c) a (b). Let P E P('FF) with IlPll 5 1. By the BCAP of F and Lemma 3.1,
we can find a net (Pa) C Pwb(,F), with llPall 5 ( A + I), for some X 2 1, which is r,, -convergent to P. Since Tl is continuous for the rp -topologies, we have Fl(Pa) = T;(Pa) 2 T'(P). As is weakly compact, (Ti(Pa)) is weakly convergent to T'(P). Therefore T l ( P ) is in the weak closure of a relatively weakly compact set. 0
Using this result and Lemma 2.1 (b), we easily obtain:
Corollary 3.3. Given k E IN, consider the following assertions: (a) E is reflezive and P('EE) = P,,,b(,E); (b) P(,E) is refieciue; (c) ?wb(,E) is reFezive.
Then (a) implies (b) which implies (c). If E has the AP, then (b) implies (a). More- over, if E has the BCAP, then (c) implies (b).
The relationship between (a) and (c) may be seen in [RyT]. I t is shown in [AAD] that Tsirelson's original space T" satisfies the conditions of this Corollary.
We say that a net (za) C E converges to E in the weak polynomial (pw) topology [CCG] if P(E,) tends to P ( E ) for every P E P ( E ) . We denote by ( E , p w ) the space E endowed with the pw topology. Recall that in an angelic topological space a subset is relatively compact if and only if it is relatively sequentially compact [Fl, Theorem 3.31.
Proposition 3.4. The p w iopology is angelic.
Proof . Supposing P(E) endowed with the norm topology, let i : E -+ P(E)* be the mapping taking t into the linear form i given by i ( P ) = P ( E ) for each P E P ( E ) . Clearly, i is well defined, and the preimage by i of the w+ topology on P(E)' is the pw topology on E. 'Since the range space is regular (i. e., given a closed subset C and z 4 C, there are disjoint open sets U 3 E and V 2 C), ( E , p w ) is regular too. Now,
0
We say that an operator is pw-compact if it takes bounded sets into relatively pw - compact sets, equivalently (by Proposition 3.4), if it takes any bounded sequence into a sequence having a pw - convergent subsequence. We now give an improvement of Theorem 3.2 in the presence of the DPP.
as pw is finer than the weak topology, ( E , p w ) is angelic [Fl, Theorem 3.31.
Theorem 3.5. Suppose F has the DPP, and let T : E -, F be an operator. The
(a) T is weakly cornpact ; (b) T is pw - compact ; (c) for every k E IN, Tl i s weakly compact; (d) f o r some k E IN, T , as weakly compact.
Proof . (a) =$ (b). Since F has the DPP, it is well-known that every P E P ( F ) takes weakly convergent sequences into convergent ones (see e. g. [GGdw, Corollary 3.71).
following assertions are equivalent :
* .
138 Math. Nachr. 186 (1997)
From Proposition 3.4 we get the result. (b) =$ (c) by Theorem 3.2. (c) 3 (d) is obvious. (d) (a) is as in [GGsch, Proposition I l l . U
4. Composition operators
In this Section we charac.terize the weakly compact and the completely continuous composition operators between algebras of holomorphic functions of bounded type. As a consequence, we prove that E* and F have the Schur property if and only if the weakly convergent sequences in Hb(U, F) (or in H,,(U, F)) are convergent. The following Lemma will be needed in several places. It is easily proved using standard techniques (see for instance [I]).
Lemma 4.1. Lei (f,) c Hb(U, F ) be a sequence with Taylor’s expansions a t the origin f, = Ckq)==o P:, where P,” E ?(‘“El F ) . Then ( f,) is (weakly) convergent if and only i f ( f , ) as bounded and (P,”), is (weakly) convergent i n P(’”E, F ) for e v e y k E I N .
Given an operator T : E 4 F, and balanced open sets U C E and V E F such that T ( U ) C V , let A : H b ( V ) 4 Hb(U) be the homomorphism given by A ( f ) = f o T for all f E f f b (V) , and R : H,,(V) + H,,(U) the restriction of A to H,,(V). With this notation, we can prove:
Theorem 4.2. Consider the following asserlaons : (a) T is “Ha - compact” in the sense that for any U -bounded sequence (x,) C U ,
there exist a subsequence ( 2 n k ) and z E E with T x E V svch that f(T ( x n k ) ) -+ f ( T ( z ) ) f o r all f E H b ( V ) ;
(b) T is pw - compact; (c) for every k E IN, Ti is weakly compact; (d) A is weakly compact; (e) the range of ihe second adjoin2 A** is contained an Hb(U) ; (f) ihe adjoint A* i s weakly compaci for the strong topologies; (g) B is weakly compact; (h) for every k E IN, Ti i s weakly compact.
Then (a) implies ( b ) which implies (c), and (c), (d), (e) and (f) are equivalent. More- over, (f) implies (g), and (g) and (h) are equivalent If F has the AP, then also (c) implies (b). If F has the BCAP, then (h) implies (b). If V is polynomially convez; then (b) implies (a).
Proof . (a) + (b). Let (z,) C B E be a sequence. We can find a U-bounded set W C U and X > 0 such that (Xz,) c W. Then there are a subsequence (z,,) and x such that f o T (Axc,,) + f o T(Xr), for every f E H b ( v ) . Therefore
P ( T (Znj)) 4. P(T(z)) for all P E P ( F ) ,
and T is pw-compact.
( ;onzQez/GutiCrrez, Gantmacher Type Theorems 139
(b) 3 (a). Given (2,) c W c U , where W is U-bounded, there are a subsequence ( J , , ~ ~ ) and 2 E E such that P (T(z ,&>)) ---f P ( T ( 2 ) ) for all P E P(F). Since V is polynomially convex, T ( z ) E V . Given f E H b ( V ) and t > 0, since
I V [I, Proposition 11 we can find Q E P ( F ) such that ( f ( T ( z ) ) - Q ( T ( z ) ) ( < ~ / 3 whenever : E W U {z}. Then, for j large enough, we have
I f (T (ZnJ)) - f(T(z))l L I f (T (znj)) - Q (T (zn,))l + 14 (T (znj)) - Q(T(z))( + IQ(T(z)) - f(T(c))l
< € .
(c) (d). Let (fn) C H b ( V ) be a bounded sequence with Taylor's expansions at 1.he origin fn = CF=o P,", where P," E P ( k F ) . Since T' is weakly compact, passing to it subsequence by a diagonal procedure, we can assume that (T' (P,")), = (P," o T)n is weakly convergent in P ( k E ) for every L E IN. Since A(fn) = fn o T = Cp=o P," o T, ;i.pplying Lemma 4.1, we get that (A(fn))n is weakly convergent in H b ( u ) .
(d) 3 (e). If A is weakly compact, then the adjoint
A" ( H b ( U ) * , q J - ( H b ( v ) * , r p )
s continuous, where r,, is the Mackey topology p (Hb(U)* , Hb(U) ) and 7-0 the strong ,apology. Therefore the range of A'" is contained in H b ( u ) .
(e) 3 (f). If the range of A** is contained in H b ( U ) , we easily conclude that A" is u* -weak continuous. Since H , ( U ) is barrelled, if K c Hb(U)* is strongly bounded, .hen K is relatively W" -compact. Hence A * ( K ) is relatively weakly compact.
(f) j (c). For each 6 E IN, let J k : P('F) + H b ( V ) be the natural inclusion. If 1" is weakly compact, so is J; o A" = (A o Jk)". Since A o J k ( P ) = T'(P) for all ' I E P(kF'), we deduce that Ti is weakly compact.
(d) (h) e= (g) can be proved using the argument of (c) 3 (d) and the fact that H,, (U)
0
(g) s- (h) and (d) 3 (c) are obvious.
3 a closed subspace of Hb(U) . (b) e (c) and (h) j (b) are contained in Theorem 3.2.
An immediate consequence of Theorem 3.3 is the following:
Corollary 4.3. Cons ider the following assertions : (a) for every L E IN, P ( ~ E , F ) is refleziue ; (b) Nb(U, F) is ref lez ive; (c) H,,,,(U, F) i s ref lez ive; (d) for every k E IN, Pwa(kE, F ) is ref le t ive .
'hen (a) and (b) are equivalent, (b) implies (c), and (c) and (d) are equivalent. If E as the BCAP, then (d) implies (a).
140 Math. Nachr. 186 (1997)
Remark 4.4. (1) The proof of (d) 3 (a), which is similar to that of Theorem 3.2, part (c) (b), shows that for E having the BCAP, if the space H,,(U, F) is reflexive, then Hb(U, F) = H,,(u, F) and P ( ~ E , F ) = Pwb('E, F ) .
(2) It is proved in [AAF, Corollary 81 that P(&E, F) is reflexive for all k E IN i f E = T* and F = lP (1 < p < m).
(3) The equivalence (a) ($ (b) was obtained in [AnP, Corollary 31 for U an open ball centred at the origin, and (c) ($ (d) may be found in [P, Corollary 81 for U = E and F = CC.
Theorem 4.2 may be improved considerably if F has the DPP. To this end we need a result similar to [GGsch, Lemma 101. We recall that given a polynomial P E P(kE, F ) , its adjoint is the operator P* : F* + P('E) defined by P*($J) = $J o P for all II, E F * . Then, P is weakly compact if and only if P* is weakly compact [Ry, Proposition 2.11.
Lemma 4.5. Given polynomials Qi E P('*E, F ) with 1 5 i 5 k, for k E IN fixed, define the operator
by Q ( * ) ( P ) ( z ) = F ( Q 1 2 , . . . , Q k z ) , for P E P(kF) and z E E . W e have:
compact ;
E ) 9'') : P ( k F ) - p(rl+ "'+rk
(a) if Qj is weakly compact for 1 6 i 5 k and F has the DPP, ihen 9'') is weakly
(b) i f Q i ( B E ) is a D P set in F for 1 5 i 5 k, then Q(*) is completely continuous.
Proof. (a) Let Q : E + @:E' be the polynomial given by
Q(c) = (QIz) @ * . . @ ( Q k z ) .
Since the Qi's are weakly compact and F has the DPP, the set Q(B,y) is relatively weakly compact [Di, Theorem 163. The adjoint of Q is the operator
Q* : L('F) - P(rr+".+rkE) given by Q*(A) = A o Q .
Then Q* is weakly compact. To finish the proof, it is enough to observe that 9'') is the restriction of Q' to P()F) .
(b) As in (a), using Theorem 2.2, and the easy fact that a polynomial takes the unit 0 ball into a DP set if and only if its adjoint is completely continuous.
Theorem 4.6. ,Suppose F has the D P P . Given g E H a ( [ / , F ) , let
A : H b ( F ) -----) H b ( u )
be the induced homomorphism. The following assertions are equivalent : (a) g is weakly compact; (b) A is weakly compact; ( c ) the range of the second adjoint A** is contained in H,(U) ; (d) the adjoint A* i s weakly compact for the strong topologies.
If $J o g E H,,(U) for all $ E F', then ihey are also equivaleni 2 0 : (e) the induced hoinoinorphism B : H,,(F) --.$ H,,(U) is weakly compoct.
( hnzaez/Gu tikrrez, Gantmacher Type Theorems 141
Proof . (a) + (b). Since g is weakly compact, the derivatives d'g(0) are weakly (.ompact polynomials [Ry, Theorem 3.21 for 1 E IN. Let (fn) C Ha(F) be a bounded sequence. For each k E N, by the chain rule, d k ( f n o g)(O) may be written as a finite sum of terms of the form
where I 1 +. . .+ l j = k, and Bjfn(g(0)) denotes the symmetric k-linear mapping asso- c,iated to the polynomial djf,(g(O)). By Lemma 4.5 (a), passing to a subsequence of (m), we can assume that (dk(fn o g)(0))n is weakly convergent. A diagonal procedure enables us to suppose that this is true for every k E IN. Therefore, by Lemma 4.1, (Afn)n = (fn o g)" is weakly convergent.
(b) + (c) =+ (d). The proofs are as in Theorem 4.2. (d) + (a). Let J : F' + Hb(F) be the natural inclusion, and 6 : U - Hb(u)' the
mapping given by 6(z) = i for z E U, where k(f) = f(x) for f E H , ( U ) . If W c U is U -bounded, then a( W) is w' -bounded and, therefore, strongly bounded. Since y = J" o A* o 6, we get that g(W) is relatively weakly compact.
(b) + (e). This statement is obvious. (e) + (a). Given (&) C BF*, we can assume that ($n o g)n is weakly convergent.
Then (& o dkg(0))n is weakly convergent in P(kF) for every k E IN. Therefore the adjoint dkg(0)' is weakly compact, and so is d'g(0) for every k. Now, adapting the argument of [AS, Proposition 3.41 it is not difficult to show that g is weakly com- pact. 0
Remark 4.7. We do not know if a nonlinear version of Theorem 4.2 holds. The proof of such a result would need an affirmative answer to the following questions.
Problem 4.8. (1) Given two pw-null sequences (zn), (yn) c E, is the sequence
(2) Given two pw-null sequences (zn), (yn) C El is the sequence (xn+yn) pw-null? (3) Given a pw-compact set A C E, is the absolutely closed convex hull of A
(z, @ yn) weakly null in E 8% E ?
pw - compact?
We remark lhat the answer to these questions is affirmative if E has the DPP, and also if E is a A-space [CCG, is], i .e. , every pw-null sequence is norm null. While 1.he final version of this paper was in preparation, a partial positive answer to (1) and
WawwWicribe a class of holomorphic mappings, which will be shown to generate Ithe completely continuous homomorphisms. The proof of the following result may be c 4 l y adapted from [AS, Proposition 3.41, using well-known facts about DP sets.
h piYj.
Proposition 4.9. Let f E H ( U , F ) . The following assertions are equivalent: (a) every x E U has a neighbourhood W C U such that f ( W ) i s a DP set in F ; (b) f o r all k E IN and z E U , dkf (z ) ( i3~ ) is a DP sei in F; ( c ) f o r all k E IN, d k f ( o ) ( B E ) i s a DP set in F;
Math. Nachr. 186 (1997) 142
(d) there is a 0 -neighbourhood W C U such that f ( W ) is a D P se t i n F
(e) f takes U -bounded s e i s inio DP seis . I f , tnoreover, f E Hb(U, F ) then these s f d e m e n t s are equivalent t o :
Given a locally convex space X, we say that B C X' is a n (L)-set [Em] if for every weakly null sequence (r,) C X , and every sequence (4,) C B, we havr lim (zn, 4,) = 0.
Theorem 4.10. For g E Hb(U, F ) , lei A : Ha(F) ---t Hb(U) be the homomorphism. given by A(f) = f o g for all f E Hb(F), and A' the adjoint of A . T h e following assert ions art. equivaleni :
(a) g(W) i s a DP set f o r every IJ -bounded W c I J ; (b) A as coniplelely cont inuous; (c) A* lakes bounded se t s of (Hb(U)*,Tp) into (L) - se t s in Hb(F)*.
Proof . (a) =+ (b). By Proposition 4.9, d ' g ( 0 ) f B ~ ) is a DP set, for all 1 E IN. Choose a weakly null sequence (fn) C Hb(F). Clearly, the sequence (djf,(g(O))), is weakly null in P(JF) for all j . For each k E N, applying the chain rule as in the proof of Theorem 4.6 and using Lemma 4.5(b), we get, that the sequence (dk(fn og)(O)), is norm null. By Lemma 4.1 f, o g + 0 and A is completely continuous.
(c). Let B C Hb(U)* be strongly bounded, and (fn) C Hb(F) a weakly null sequence. Since Hb(U) is a Frichet space, the inclusion of Hb(U) into its bidual, equipped with the strong topology, is an injective isomorphism. Therefore, since A is completely continuous, I(f,,A*(u))J = I (A( f , ) ,u )J + 0 uniformly for u E B.
(c) + (a). Defining the mappings J and 6 as in Theorem 4.6, part (d) + (a), lei, W c U be U-bounded and (+,) c F' a weakly null sequence. Then (J(+n)) is weakly null in H b ( F ) . Therefore
(b)
uniformly for 2 E W since 6( W ) is bounded in ( H b ( l J ) ' , T O ) . 0
Corollary 4.11. T h e following assertions are equivalent : (a) E* and F have the Schur proper t y ; (b) every weakly convergent sequence in Hb(U, F ) is convergent; (c) every weakly convergent sequence in H,,(U, F ) as convergent.
Proof . (a) + (b). If E* and F have the Schur property, then the space L ( E , F ) of linear operators from E into F has the Schur property [RyDPP, Theorem 3.3) and so does, by induction on k, the space L(kE, F ) and therefore its subspace P ( k E , F). Since this is true for all k, to get the result we only need to apply Lemma 4.1.
(b) 3 (c). This assertion is obvious. (c) =$ (a). The proof of this implication is clear since E' and F may be embedded
in H,,(U, F ) . 0
( :onzaez/Gutibrrez, Gantmacher Type Theorems 143
6 . Linearization of holomorphic mappings
The idea of linearization of holomorphic mappings is due to MAZET [Ma]. MUJICA obtained in [Mu- Val] the following result, slightly stronger than a previous one in I ( X M b t] :
Proposition 5.1. Let U be a balanced open subset of E . Then there are a complete (LB) -space Gb(U) and a mapping 6EHb(U, Gb(U)) with the following universal prop- r r t y : For each Banach space F and each mapping g f Ha(U, F ) , there is an operator Tg : Gb(U) -+ F such that Tg o 6 = g. This property characterizes Gb(U) up t o a lopologrcal isomorphism.
The space Gb(U) is constructed as a subset of Ht,(U)*. The mapping 6 : Gb(U) is defined by 6(z) = x, for c E U , where x(f) = f(z) for all f E Hb(U).
In this Section, we show that MUJICA'S linearization may be obtained from our composition operators, and we prove that a mapping g E Hb(U, F ) is (weakly) compact (resp. takes IJ -bounded sets into DP sets) if and only if the linearization Tg is (weakly) compact (resy . has completely continuous adjoint).
Given g E Hb(U, F ) , let A : Hb(F) + Hb(U) be the induced homomorphism, A* : Hb(U)* -+ Hb(F)* its adjoint, and J : F* -+ Hb(F) the natural inclusion. It is easy to check that Tg = J* o A*IG~(u). Therefore Tg may be considered as a restriction of A*.
Theorem 5.2. Given a mapping g E Hb(U, F ) , let Tg be its linearization. Then g is (weakly) cornpact if and only if Tg i s (weakly) compact. Moreover, the following assertions are equivalent :
(a) g fakes U -bounded sets into DP sets; (b) Tg fakes bounded sefs o f G b ( U ) into DP s e f s ; (c) the adjoint Ti is cornpletely continuous.
Proof . Suppose Tg is (weakly) compact, and W C IJ is U-bounded. Since 6(W) is bounded in Gb(U), the set g(W) = Tg6(W) is relatively (weakly) compact in F .
Let g be (weakly) compact. Given a fundamental sequence (U,,) of balanced open U - bounded subsets, Gb(U) is a boundedly retractive inductive limit of Banach spaces Gm(Un). The unit ball of Gm(Un) is the absolutely convex closed hullFb(U,,) of 6(Un) [Mu-lin, Theorem 2.11. If B c Glb(U) is a bounded set, then B is contained and bounded in Gm(Un) for some n E IN. For some A > 0, we have B AF6(Un). Hence Tg(B) 2 AfCT,6(Un) = Ar'g(Uh). Therefore Tg(B) is relatively (weakly) compact. This also shows that (a) 3 (b).
To prove (b) =+ (c) and (c) (a), it is enough to consider that for every II E F*, 0 we have (Ti(II))(z) = ( 6 ( ~ ) , Ti($)) = (Tg6(c), $) = (g(z), 16).
Acknowledgements
The first named author was supported in part by DGICYT Grunt PB 94 - 1052 (Spain). The second named author was supported in part by DGICYT Grant PB 93 - 0452 (Spain).
144 Math. Nachr. 186 (1997)
The authors wish to thank the referees for their many suggestions which allowed improvement of the paper.
References m C 4 R , R, AWN, R M, and m, s.: A Reflexive Space of Holomorphic h c t i o n s in Infinitely Many Variables, Proc. Amer. Math. SOC. 90 (1984), 407-411
JkI?NCm, R, h N , RM, and FhcxE, G.: Tensor Products of Tsirelson’s Space, Jllinois J. Math. 31 (1987), 17-23 hDREW3, KT.: Dullford-Pettis Sets in Spaces of Bochner Integrable Functions, Math. Aim. 241 (1979), 35-41 hSJLWL, J.M, and Fbm, S.: An Example of a Quasi-Normable FkCchet Function Space Which is not a Schwarts Space, in: S. Machado (Ed.), Functional Analysis, Hole morphy, mid Approximation Theory, Ftio de Janeiro 1978, Lecture Notes Math. 843 1-8, Springer- Verlag, Berlin, 1981 &N, R K, f k R V b , c:., and V O W , h.z: Weakly Continuous Mappings on B a n d Spaces, J. h c t . Anal. 52 (1983), 189-204 &N, R M and ! h l m H E R , M:, Compact Holomorphic Mappings on Banach Spaces and the Approximation Property, J. Funct. Anal. 21 (1976), 7-30 /!STALA, K, and ’hU, H . 0 . : On the Bounded Compact Approximation Property and Measures of Noncompactness, J. h i c t . Anal. 70 (1987), 388-401 %ESAGA, c:., and KI, A: Some Aspects of the Present Theory of B a n d Spaces, in: S. Banadi, (Euvres, 11, 221 -302, P. W. N., Wmaawa, 1979
BrslRiini, P., JARAMULO, J. A, and W , M: Polynomial Compactness in Banach Spaces, to appear in Rocky Mountain J. Math. CARNE, T. K, am, B., and w, T. w.: A Uniform AlUgebra of Analytic Functions on a Banadi Space, ’bans. Amer. Math. SOC. 314 (1989), 639-659
DIESTEL, J.: A Survey of Results Related to the Dunford-Pettis Property, in: W.H. Graves (Ed.), Proc. Cod. on Integration, Topology and Geometry in Linear Spaces, Chapel Hill 1979, Contemp. Math. 2, 15-60, American Mathematical Society, Provi- dence, 1980
m, s.: Complex Analysis in Locally Convex Spaces, Math. Studies 57, North- Holland, Amsterdam, 1981
EhMNUEW, G.: A Dual Characterization of B a n d Spaces not Containing t’, Bull. Polisli Acad. Sci. Math. 34 (1986), 155-160
b R E T , K: Weakly Compact Sets, Lecture Notes Math. 801, Springer-Verlag, Berlin, 1980
C;ALnvw , P., Ce’A, D., and hhESTRE, M.: Holomorphic Mappings of Bounded Type, J. Math. Anal. Appl. 166 (1992), 236-246
c;ALINDo, P., w, D., and MAESIRE, M.: Entire Functions of Bounded Type on RCdiet Spaces, Math. Nadir. 161 (1993), 185-198 a&, M., and GhJdWG?,, J. h.2: Weakly Continuous Mappings on Bruiach Spaces with the Dunford-Pettis Property, J. Math. Anal. Appl. 173 (1993), 470-482
GoNZhZ,, M., and -, J.M: RCultats de DualitC pour Applications non LinCaires, C. R. Acad. Sci. Paris SCr. I 316 (1993), 901 -904 CbNZhE?,, M, and CkdRRE?, , J. M: Sdisuder Type Theorems for Differentiable and Holoniorpluc Mappings, Monatsh. Math. 122 (1996), 325 - 343 m E C X , k: Produits Tensoriels Topologiques et Espaces NuclCaires, Mem. Amer. Math. SOC. 16, American Mathematical Society, Providence, 1955
( lonzilez/GutiCrrez, Gantmacher Type Theorems 145
hIMu), J.M: Topological Duality on the F’unctional Space ( H ~ ( U , F ) , T ~ ) , Proc. R. Irish Acad. 79A (1979), 115-130
JARCHOW, H.: Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981 U V O N A , J. G.: Approximation of Continuously Differentiable Functions, Math. Studies 130, North- Holland, Anisterdatn 1986
MAZFT, P.: Analytic Sets in Locally Convex Spaces, Math. Studies 89, North-Holland, Amsterdam, 1984
J. MUJICA, Complex Analysis in Banach Spaces, Math. Studies 120, North-Holland, Amsterdam 1986
[Mu - lin] MUJICA, J.: Linearization of Bounded Holomorphic Mappings on Banach Spaces, ’lkans. Amer. Math, SOC. 324 (1991), 867-887
[Mu - Val] MUJICA, J.: Linearization of Holomorphic Mappings of Bounded Type, in: K. D. Bierstedt et al. (Ed.), Progress in h c t i o n a l Analysis, Peiiiscola 1990, Math. Studies 170, 149- 162, North-Holland, Amsterdam, 1992
mEn>, A.: The Bidual of Spaces of Holomorpllic Functions in Infinitely Many Variables, Proc. R. Irish Acad. 92A (1992), 1-8
RYAN, R A.: Applications of Topological Tensor Products to Iihli te Dimensional Holo- morphy, Ph. D. Thesis, Trinity College, Dublin, 1980
RYAN, RA.: The Dunford- Pettis Property and Projective Tensor Products, Bull. Polish Acad. Sci. Math. 35 (1987), 785-792
RYAN, R A.: Weakly Compact Holoinorpllic Mappings on Banach Spaces, Pacific J. Math.
WLLIS, G.: The Compact Approximation Property does not Imply the Approximation Property, Studia Math. 103 (1992), 99-108
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