The property (Hu) and (Ω) with the exponential representation of holomorphic functions

Publicacions Matemàtiques, Vol 38 (1994), 37-49 . THE PROPERTY (H a ) AND (S2) WITH TH E EXPONENTIAL REPRESENTATION O F HOLOMORPHIC FUNCTION S NGUYEN MINH HA AND NGUYEN VAN KHU E Abstract  The main aim of this paper is to prove that a nuclear Freche t space E has the property (Hu) (resp . (S~)) if and only if ever y holomorphic function on E (resp . on some dense subspace of E ) can be written in the exponencial form . Let E be a locally convex space . We say that E has the propert y (Hu) and write E E (Hu) if every holomorphic function f on E is o f uniform type . This means that there exists a continuo .ussemi -norm p oi l E such thatf can be factorized holomorphically through the canonica l map w p : E --~ E p , where E p denotes theBanach space associated to p . On the other hand, we recall that E is called a space having the propert y (í) if for every neighbourhood U of C E E there exists a neighbourhoo d V of 4 E E and d ~ o such that for every neighbourhood W of C E E there exists C ~ o such tha t II u IIV +d ~ c II u IIWII u II U for u E E*, the dual space of E, wher e II u II- = sup{ l u(x) 1 : x E K } for every subset K of E . The properties (H a) and (ñ) were introduced and investigated b y Meise and Vogt in [5] . In the present paper we investigate the property (H a ) and (9) by the relation with the exponential representation o f entire functions . 38 N . MINH HA, N . VAN KHU E 1 . The property (H,,) and the exponential representatio n of entire function s In this section we shall prove the followin g Theorem . Let E be a Frechet space . Then E is nuclear and has th e property (H u ) if and only if every entire function on E with values in a Banach space B can be written in the fore n (Exp) B f (x) _ ~_ k expu k( x ) k7 l where the series is absolutely convergent in the space H(E, B) of holomorphic functions on E with values in B equipped with the compact-ope n topology . Proof : First prove sufficiency of the theorem . Given f E H( E, B) wit h B is a Banach space . Since E is a Frechet space we can find a continuou s semi -norm p on E such tha t E ~kII exp II u kIIP < oo , k ~ l with Ilull* p = sup{lu(x)I : p(x) < 1} . Indeed, in the converse case let {II • ii p } is a fundamental system of seminorms on E . Then for every p we hav e E  exp II u kIlp = 00 . k> l Hence for every p there exists k p such tha t E IKkII exp II u kIIP > p . k<k P This inequality implies that for each k Ç kp there exists x~ with II xP k IIp ç 1 such that E Ik)i > p • k<k y Put K= THE PROPERTY (Hu) AND THE EXPONENTIAL REPRESENTATION 39 Then K is compact in E an d E IIekH exp ll u k ii K > p for every p > 1 . k> 1 This is impossible, becaus e E IIkII exp I G 00 • k ~ 1 Thus the form El - 1c expu k (x) for x E E P ,  iixll < 1 k> 1 defines a holomorphic function on U P , the open unit ball in E P which i s Gateaux holomorphic on E/ Ker p . Let x E E P . Pu t W={(1—t)y+tx :tE(C\{0}, yEU P } . Then W is a non -empty open set in E P . Hence there exists z E W n E/ Ker p . Let z = (1 — t o)yo +tax o with yo E U P , to E C\{0} . Then x = z/to + ((1 — to) /to)y o and hence E lil k ll exp Iu k (x) l k> 1 5 E IkII ex v[I(1/ t o)I lur :(=)I + I(1— to)/to)I luk(u0)I1 5 k> 1 ~ E I+exP 2 1( 1 — t o)/ t o~ u k(YO)I] < k> 1 Coo . 9 = E exp u k k7 l Thus 40  N . MINH HA, N . VAN KHU E is a Gateaux holomorphic function on E p . Since g is holomorphic on U p by the Zorn Theorem [6], g is holomorphic on E p . Obviously f = gw P and hence f is of uniform type . To prove the nuclearity of E for every continuous semi -norm p oil E write the canonical map wp : E —> Ep in the for m Wa(x) = E eX P u k( x ) k> 1 in which E IIklI eX P 1I'ukIIc < 0 0 k ~ l for every compact set K in E . Then w P (X) = E 1k u k (x) for x E E k> l E IIII Ic oo for every compact set K c E . k> 1 As aboye there exists a continuous semi -norm ) 3>p on E such tha t E Il~kll I u kIIá < cc . k> 1 This means that the canonical map w p , p from  to E p is nuclear . Henc e E is nuclear . Assume that E is nuclear and has the property (H a ) . Given f E H(E, B), with B is a Banach space . By hypothesis there exists a continuous semi -norm p on E and a holomorphic function g on E p such tha t f = gw p . Take a continuous semi -norm p on E such that T = wp , p is nuclear . Write T(x) =  t~u i (x)e i j ~ 1 a = E It i l < oo and ilu j il + lle i ll < 1 for j  1 . j> 1 Consider the Taylor expansion of g at 0 E E , 9( x ) _ E P n9( x ) and with THE PROPERTY (Hu) AND THE EXPONENTIAL REPRESENTATION 4 1 with Pn,9( x ) = (1/2i) f  (9(tx)/t 1 ) dt . l= r Choose the two sequences {e k } and {a k } in C such tha t z = YTk expc kz for z E C k> l Cr =  IlkI expria k ~ < oo for all r > O . k> 1 Such sequence exist by [2] . Formally we hav e (gT)(x) = g(Tx) = E P n9 ( Tx ) = E Pn9 E t i u J ( x ) e i n>0  n>0  j> 1 andd  = E  E t,%1 . . tj n u 3 x (x) . . . u jn ( x ) P ngl  . . .,ejn) ~  = E  E  tj1 . . . t jn .P n g ( e j l , . .  n70  7 1 E exp aku il (x) . . . ~ k eX P a k u i n ( x ) k>1  k> 1 = E E tj 1 . . . tj n . n7 0 P n g ( e j1 , . . . , e jn } exp[a h u jx (x) + . . . + a kn u jn (x)] . It remains to check that the right hand side is absolutely convergent i n H(E, B) . For each r 7 o take s ~C T a .e . Sinc e I . . , ejn  (n n /n!s n ) II g II ~ where Il g llS = sup {Il g ( x )II ~ Il x ll < s} , and without loss of generality by the nuclearity of E, we may assum e that g is bounded on every bounded set in B P , we hav e E  E  I t iiII t jIIkjIftI - n7o 1~ 1 , . , k,, , 7 1 • 1I  1 + . . . +l a kn T annn / n!sn  < oo for ~~ x I C r . e jn } C n ~ o 42  N . MINH HA, N . VAN KHU E The theorem is completely proved . ■ 2 . The property ~SZ} an d the exponential representation of entire function s The relation between the property (II) and the exponential representation of entire functions is given b y Theorem 2 .1 . Let E be a nuclear Frechet space having the approximation property . Then E has the property (S2) if and only if there exist s a balanced convex compact set B in E such tha t (i) E(B) is dense in E, where E(B) denotes the Banach space spanned by B , (ii) every holomorphic function on (E(B),T E ), where T E is the topology of E(B) induced by the topology of E, can be written in th e form (Exp) : E k eX P u k k> 1 in which the series is absolutely convergent in H(E(B), T E ) . Proof : Since every nuclear Frechet space having the property (ñ) ha s also the property (H a ) [5], and since every holomorphic function o n (E(B), TE) can be extended holomorphically to E [5], where B is a balanced compact set in E as in [5], the necessity of the theorem is a s in Theorem 1 .1 . Conversely, by [5] it suffices to show that every holomorphic functio n on (E(B), T E ) is holomorphic on E . As in Theorem 1 .1 there exists a continuous semi -norm p on E such tha t E IkI ex PM u kMJflE(B) C ° O • k ~ 1 Since E(B) is dense in E, it follows that U P n E(B) is dense in U P , an d hence E Ikt exp IukII c oo . k7 1 Given x E E . As in Theorem 1 .1 pu t W = {(1-t)y-tx :tEC\{O}, y E U P } . THE PROPERTY (Hu) AND THE EXPONENTIAL REPRESENTATION 4 3 Then W is an non-empty open set in E and hence there exists z E W f1 E(B) . Le t z = (1 — to)yo + tox with to E C\{0},  yo E U P . Henc e E kI eX A I  I ~ E lekl eX P[I u k( z )/ t ol + I( t o — l )l t ol l u k( y o)I] Ç k>1  k> 1 ~ E K+exp21to — 1 / t ol k'k(YO)I] < oo . k> 1 By the Zorn Theorem [6], it follows that f is holomorphic on E . Theore m 2 .1 is proved . ■ 3 . The property (H u ) and (S1 ) Proposition 3 .1 . Let E be a Frechet-Schwartz space with the propert y (H a ) . Then every holomorphic function on E with values in a Banac h space is of uniform type . Broa : Write E = limproj E n , where E n are Banach spaces such tha t E is dense in E n for every n ~ 1 and thecanonical maps cv n+l,n : E n + l — + E n are compact . By hypothesis the canonical ma p S : limind H b ( E n ) ---~ [H(E)]~,o r where [H(E) l bor denotes the bornological space associated to H(E) an d H(E) far each n ~ 1 is the Frechet space of holomorphic functions o n E n which are bounded on every bounded set in E n , is a continuous bi - jection . Since H(E) is complete, [H(E)] b O r is untrabornological . By th e open mapping theorem S is an isomorphism . Given f : E —> B a holo - morphic function, where B is aBanach space . Consider the continuou s linear map f : B* H(E) associated to f . Then f : B* --} [H(E)lbo r is continuous . SinceS is isomorphic, we can find no such that Im f Ç H b (En o ) and f : B* --> H b ( E no ) is continuous . This yield s sup{lu f(x} I : I l u ll ~ 1, IIxM ç r} _ ~ = su p {If (u)(x) : Iç 1, IIxM ç r} c o 0 for allr~O . Thus f induces a holomorphic function g : E n() -WB such that gwn p = f . ■ Remark . Proposition 3 .1 is a particular case of a recent result o f Galindo,Garcia and Maestre [3] . 44  N . MINH HA, N . VAN KHU E Theorem 3 .2 . Let E be a nuclear Frechet space with the property (Q ) and F a Schwartz space with F E (H u ) . Then E X F E (H u) . We need the followin g Lemma 3 .3 . Let E be a nuclear Frechet space with the property (SZ ) and F a Banach space . Then every holomorphic function on F x E whic h is bounded on every bounded set in F x E is of u,niform type . Proof .• Lemma 3 .3 will be proved as in [5] by use Lemmas 3 .1 an d 3 .2 in [5] . Indeed, choose p and b > 0 such that if f is bounded o n Bó x U P , where f is a holomorphic function on F x E as in the lemm a and Bó = {z E F : < S} . Since E E (S2), by Vogt [8] there exists a balanced convex compact set K in E such tha t II ,  5 II ' IIKII ' II p for some q > p and d > O . We can asume that E(K), E q and E p are Hilbert spaces . Write th e canonical map A from E(K) to E p in the for m A(x) = E Ai( X i e j)E(K)Y i , j 7 1 where {e j } is a complete orthonormal system in E and { y j } a orthonormal system in E p and a = (a i ) E s . Let (p i denote the continuous linea r functional on E q induced by v i . The n Iail for j ~ 1 . TakeoCeCSsuchthatfor ,u, =(e/j) wehav e xEE :x = I .j 1, j l i < µ j forj>1 C{xEE p ilxií <1} ; j> 1 Put M = {m = for each k > o and m E M pu t f ll-I= 1 Cbk .(z) =(1/27f2) n+1 ! p iI = i ¡ p nI = n 9~TZ7ply1 + . . ,r~+1pm1 +1 . .  + p n yn ) dTd . pñ n +1  p 1 =(1/A m )(1/21ri) n+1 . . dp n f T1 =1 f Il v il=7-1  l wn l =r n f (z,w1e1 + . . + wn en ) dTdw . . . d2 v ,rk+1w1--1 . . . w n mn+1  1  n 1 THE PROPERTY (H a ) AND THE EXPONENTIAL REPRESENTATION 4 5 where g is the holomorphic function on B 6 x {y E E p : llyil < 1} i s induced b y f and A m - - Ami • • • n • 1 For s, t > 0 pu t B(s,t)=B s x xEE :x = By hypothesis N(s,t) = sup{lf (w)l : w E B(s,t)} < o 0 and hence Sup{lak,m(z)l : IzII < S} < MS,t)p, m f .G m t l m i with 1mI=mi++m . Letr~= 1 / 1+d,v =-y= r7/2, fi =1--y . Givens~0 . Take o- > o,suc h that ory é 3 > s . Since A E s, the sequence (À/1i) = (j a . 7/E) E 1 1 and henc e R = sup{iAliµ k 1 : k > 1} < oo . Put t = (2Rr) l h . Then as in [5] we hav e E  rsup I G mEM k>0  zEBs  j> 1 ~ E E rI mi (( 5 l a ) kN ( o Al l u mcl mI r(1 A I m Y( M ( s /E) k /m m ) Q = mEM k> 0 = N(Q, tr'M Q E(3/Ury E a ) k H (1 — Iñk/2RF .Gk) —1 G 0 0 /c>0  k> 1 where N = sup{l f (w)l : w E B 6 x U p } . As in [5] this implies the serie s Z E a k,m( z ) n ( x ) m i mEM k ~ O  j?l . Iei i  tµ ifor j > 1 3»