BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC

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1 Iteat. J. Math. & Math. Sci. Vol. 9 No. (1986) BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC LEON M. HALL Depatet of Matheatics ad Statistics Uivesity of Missoui-Rolla Rolla, Missoui U.S.A. (Received Mach 8, 1985) ABSTRACT. This [ape is coceed lth fuctios of seveal coplex vaiables aalytic i the uit olydlsc. Cetai Baach spaces to hich these fuctios ight belog ae defied ad soe elatioships betee the ae developed. The space of liea fuctloals fo the Baach space of fuctios aalytic i the ope uit polydisc ad cotiuous o the uit tous is the descibed i tes of aalytic fuctios usig a extesio of the Hadaad l)oduct. KS/ WOiIDS AND P,RASES. Baach space, liea fuctioal, aalytic fuctio. lso MATHE/4AICS L UBJECT CLASCFICATION CODE ]. INTRODUCTION. I 1950 A. E. Taylo [I] studied Baach spaces of fuctios aalytic i the uit disc. Oe of his picipal esults as a epesetatio of liea fuctioals i tes of fuctios aalytic i the uit disc. I this pape, the esults of Taylo ae exteded to fuctios aalytic i the uit polydisc i -diesioal coplex space. The goal is the epesetatio theoe fo liea fuctios, Theoe 4.5, i hich the fuctloels ae expessed i tes of a Hadaad poduct. Taylo s esults have poved to be vey useful i ok ivolvig cetai sigula liea diffeetial opeatos. His epesetatio theoe fo liea fuctioals as a key pat of the ok of Giu ad Hall i [2], ad of subsequet ok of Hall i [3] ad [4]. The folloig otatio ill be used extesively: U ope uit disc i the coplex plae C, (ii) uit cicle i C, (i2) Iz e C: IZl I-6}, 0 e i, (1.3) oegatlve iteges, (1.4) U U U.. U, ( col)ies of U) the uit polydisc i C, (1.5) [... [ ( colies of ) the uit tous i C (1.6) I I I... I ( copies of I ), (i 7) Z 7 Z... ( copies o Z ). (i 8)

2 40 L.M. HALL If z (Zl,Z 2,,z ad (Wl,W2,, ae i C defie z by z (ZlW 1,z22,...,z ). Futhe, if z e C ad (c1,2,...,c) e Z defie z e by z el 2 a z z ad I 2 Z lal by II i 2 " A Deote by the class of fuctios o C hich ae aalytic i U ad defie the fuctio u e A fo each a e Z by u (z) z Also defie the opeatos U A A ad 7. A A by x IR Ux f g hee x (x I x2,.,x) e ad (1.9) ix ix I 2 g(z) f(ze TM) f(zle..,z e z2e ix) (l.10) 7. f g, hee e U ad g(z) f(z) (l.ll) The opeatos./ ad 7. ae easily see to be liea. x If f the f has the poe seies expasio f(z) Z f z (1.12) EZ Deote fa by y(f) hee ya is the fuctioal defied by -i II Ya (f) (al!e2! a 1) Zl Z z--(0,..,0). (1.13) If f ad g ae i A ith poe seies f(z) f z ad g(z) aezz defie the Hadaad poduct of f ad g by H(f,g;z) Z Z fgz aez gez (1.14) (1.15) H(f,g;z) is clealy i A, ad the folloig also ca be poved: y((7" f) Ya (f) (1.16) H(f,g;z) H(g,f;z), (1.17) H(f,g;z) (2#) 17. H(afbg,h;z) ah(f,h;z) bh(g,h;z), H(7. f,g;z) H(f,g;z), Zl Z.) dl d f(l )g( 1 i " -- E (1.18) (1.19) (i20) hee Izil lil fo l=l. Equatio (1.20) ca also take the fo H(f,g;z) (2) 0 f(le hee. 1 _. ad z e U. i@ z I -i8 1 z -i8 e g (e 1,----e d@ I..dO (1.21)

3 BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC 41 2 SPACES OF TYPE A k. Let B be a otivial coplex Baach space, each eleet of hich belogs to A Such a space ill be called a space of type A. B ay also have oe o oe of the folloig popeties. Thee exlsts a costat A such that PI: Iye(f) AI Ifl if f e B ad e E Z. The least such costat ill be deoted AI(B) Z P2: U B fo all e Thee exists a costat A such that lu < A fo all e. The least such costat ill be deoted A2(B). P3: U f B if f B ad x is a eal -tuple. Also X lluxfll Ifl P4: T f B if f e B ad (l,2,..,) ith 0. i. Thee exists a l costat A such that ll[fll Allfll. The least such costat ill be deoted A4(B). B ill be called a space of type A k if B is of type A ad also satisfies axios Pl,...,Pk. Let B* deote the space of cotiuous liea fuctioals o B. The by PI Y e B* ad t ca be sho that A 1 (B) supl IyeIl, ad (2.1) Othe elatios satisfied by the costats (B) ae A4(B) suplltll, (2.3) 1 <_ AI(B)A2(B), (2.4) 1 < A 4 (B) (2.5) Let B be a space of type A I. Fo ay fixed g e ad z defie A N(g;z) sup IH(f,g;z) I. (2.6) U The folloig ae clea fo the popeties of the Hadaad poduct. N(gh;z) < N(g;z) N(h;z) N(ag;z) N(T g;z) N(g;z), ll i. W (2.7) (28) (2.9) THEOREM 2.1. Let B be a space of type A 3. The the fuctio N(g;z) has the popeties: N(g,z) N(g,z), hee g e z e U ad z N(g;( I )) is a odeceasig fuctio of each., i=l,, hee 0. i, (2.10) (2.11) N(g;( I, )) 0 fo all (l, if ad oly if g 0. (2.12)

4 42 L.M. HALL PROOF. Let deote the set of all ozeo eleets of B ad defie, fo f A{ (2.13) hee g is fixed i A Also defie M(z) Obseve that M(z) N(g;z) ad also, fo x e R ad z e U Let z e U be fixed. The M(U f;z) x M(f;zeXX). f (2.14) (2.15) 1 I%1 Izl- Hece, fo fixed z e U, IM(f;z) is bouded as f vaies ove 4. No fo x e R U f vaies ove all of as f vaies ove all of sice f U (U f), ad so by (2.15) e get M(z) M(z) if x is chose caefully, ad (2.10) is x x. poved. Next, let R (RI Ri R) ad (l,...,i,...,) ith 0 _< k < i fo k=l,...,, k i ad 0. < R < 1 Also suppose f e The let z ad z be l i R espectively, at hich the poits o the toi {z: IZkl } ad {z: IZkl k}, IM(f;z) assues its axiu value. The axiu piciple fo polydisc fuctios o yields I(f;Z) But IM(f;zR) M(z R) M(ZR) M(R) ad so! I;zl. hich iplies M() M(R), ad (2.11) is poved. (2.17) IM(f;) <_ IM(f;Z) < M(R), (2.18) Fially, if g 0, N(g,) 0 fo all. If g 0, the 0 fo soe e. e a Let f= u fo this I e The H(f,g;) e ye(g)... hich is I ozeo if k M 0 B. k=l,...,, ad the poof is coplete. 3. THE SPACES B AND Suppose B is a space of type A 3. To elated spaces hich ill be used late i chaacteizig B* ill be defied as follos: B {FeA: N(F;) is uifoly bouded i, 0<_k<l k=l,...,}, (3.1)

5 BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC 43 B 0 {FEA: li H(f,F;) exists fo each f e B}. (l,i) Fo F e 1 B defie N(F) sup{n(f;) 0 k < i, k=l,...,} By (2.11) e have (3.2) N(F) li N (F;) (3.3) (i, I) THEOREM 3.1. If B is a space of type A 3 the B is a Baach space ith o N(F). PROOF. Fo the popeties of the Hadaad poduct ad N(F;), ad fo theoe 2,1, B is clealy a oed liea space ith o N(F). To pove copleteess poceed as follos. Let {F be a Cauchy sequece i B o The thee is a J > 0 such that if > J, N(F.) N(F i, ad N(F.) is bouded by, say, Eo Let 0 < 1 fo k=l,..., ad J k coslde, fo g Z, 17(F -F )el H(u F -F ;) (34) e Thus, {ya Z (F3)} is also a Cauchy sequece fo each No defie ae=li, y(fj) ad 3 ote that covegece is uifo i ad that the sciece {a }, e e Z, is bouded. Usig a the above aguet ith 0 e obtai F ((Fj) _< A2(B)E hich yields _< A2(B)E. Next defle F(z) a z (3.5) Z Clealy F e A, ad it ill be sho that F e B ad li F. F. Cosde fo fxed ad f e IH(f,F ;) -H(f,F;) < ZIy(f)esuply(F a (3.6) Sice li[7 (F0)-a 0 uifoly i s, j e e Futhe, fo f e B li H(f,F.;) H(f,F;). (3.7) j hich iplies IH(f,F;) < E Ifl o N(F;) < E. Hece, F e B. Fially, fo e > 0, choose jo(e) such that if j, > jo(e) the N(F.-F < e. Fo f e B (3.5) yields IH(f F -F ;) < N(F -F )I Ifll < El Ifll (3.9) As (3.4) ylelds IH(f,Fj--F;) _< ellfll if > j0(e), hich eas N(F.-F)3 < e ad the poof is coplete. THEOREM 3.2 If B is a Baach space of type A 3 the B 0 is a liea subset of B ad is a Baach space of type A 4 ith o N(F)t,

6 44 L.M. HALL PROOF: Clealy B 0 is a liea subset of A. H(f,F;) defle a fuctioal of f ove B. The B 0 Fo f e B ad F e let (f) sup l(f) N(F;), (3.10) ad sice F e B 0 the N(f;) is bouded as a fuctlo of by the uifo boudedess 0 piciple. Thus, B is a liea subset of B o I ode to sho that B 0 is closed i B, let {F0} B 0 ad F e B be such that l N(F -F) 0 Fo f 6 B j IH(f,F;)-H(f, F ;0)I<IH(f,F-F.;)IIH(f,F.;)-H(f,F.;0) <2N(F.-F) If IIH(f,F. ;)-H(f,F. ;0) (3.11) Fo f B flxed ad 0 choose lage eough so that 2N(F.-F) Ilfll < /2 ad choose ad Q close eough to (i,i,...,i) so that IH(f,F.;)-H(f,F.;0)<e/2 Theefoe li IH(f,F;)-H(f,F;p) 0 ad F B0., 0 (i,...,i) 4..REPRESENTATION OF LINEAR FUNCTIONALS. I thls sectio a seies of esults hich culiate i a epesetatio of the space B* ad ts elatioship to the spaces B ad B 0 ill be poved. THEOREM 4.1. Let B be a Baach space of type A 4. If y B* let The G B ad IGI N(G) <_ A 4(B) Iyl I. PROOF: G A sice IY(ue) IYl IA 2(B)- ad e U The [ f e B ad e G(z) Z y(u )z z e U (4.1) ez (- y(t f) E f y(z) Also y(u) y(g) G Let f e I f G H(f,G;). (4.2) A If f B the IH(f,G;) Iy(Tf) A4(B) IYI Ifl I. (4.3) Hece N(G;) sup [H(f,G;) A 4(B) [y[ ad G B. Also, llfll-- IGII N(G) SUp N(G;) <_ A4(B) II l- F(7) G. The passage fo to G give by (4.1) defies a opeato F: B* B hee THEOREM 4.2. Let B be a Baach space of type A 4. The F is a liea opeato ad IFI A4(B). F has a ivese if ad oly if the liea subspace of B spaed by {u }, e Z is dese B PROOF By Theoe 4.1 F is liea ad IFI A4(B) By (4.2) IY(If) <_ I[fll IIGtl < Ilfll IIFII I111" (4.4) NOW choose y so that I1tl i ad (</f) Ilfll so that ((f[( _< Ilfll Illl.

7 BANACH SPACES OF FUNCTIONS ANALYTIC IN A POLYDISC 45 The lit II Illl ad A4(B) Illl,. Hece Illl Suppose F(y) 0 The y(u 0 fo all a e Z ad hece sayig that F -I exists 2 if ad oly if (u 0 fo all e is equivalet to sayig y 0. Thus F -I exists if ad oly if {u is total i B hich is equivalet to the subspace spaed by {u beig dese i B. 0 Fo f e B, H(f,F;) defies a liea fuctioal ith o N(F;), so if F e B y(f) ll H(f,F;) (4.5) (l,...,l) defies a eleet of B* coespodig to F. The passage fo F to y give by (4.5) defies a opeato.% B 0 B* hee A(F) y THEOREM 4.3. Let B be a Baach space of type A 4. The A is a liea opeato, FA(F) F, fo F e B0, ad (4.6) IA()I <_ (F) <_ N () 1^() I, fo F e BO. (4.7) Thus A defles a ijective appig, ith bouded ivese, of B 0 oto a subspace of B*., y(f), PROOF: Let F s B0, A(F) y ad F(y) G. The y(g) y(u li H(u F;) li y (F) >(i i) (l, i) ad so F(A(F)) F. Also, sice IH(f,F;) < lfll IFII it follos that Iy(f) < lfll IFII hee A(F). Thus, fo (4.6) (4.8) If the space B satisfies a additioal axio, viz., P5 If f e B the T f B ad li lltf-fll 0 (l,...,l) 0 the B*, B ad B tu out to be isoetically isoophic. THEOREM 4.4. Let B be a Baach space of type A 4 also satisfyig P5" The B 0 B A is bijective ad isoetic, ad -i (4.9) (4.10) (4.11) PROOF: By (1.19) H(T f,f;p) H(T f,f;) ad so p H(f,F;) -H(f,F;0) H(f-[ f,f;) H(] f-] f,f;0) H(T f-f,f;0) p p p Theefoe, if f e B ad F e B IH(f,Ft) H(f ;)1 [-I If- fll If-pfl (4.12) (4.13) ad by P 5 so F g B 0 li IH(f,F;) H(f,F;p) O,,p (i,...,i) 0 Sice B B aleady, this eas B B o (4.14)

8 46 L.M. HALL Fo li P5 IITll I1=11. if Il < 1 i=l..,, ad ]" f is i (l i) aalytic i the ITfll ltfll by Hece fo the axiu odulus theoe P3" fo fuctios ith values i a Baach space (see [3], p. 211) ITfll is a odeceasig cotiuous fuctio of. This yields A4(B) 1 ad by Theoe 4.3. is isoetic. No suppose f e B ad B*. The [A(F())] (f) lia H(f,F(y);) (l, i) li y(t f) (f). (l,,i) (4.15) As a coollay of Theoe 4.4 the folloig theoe povides a epesetatio of B* i tes of the Hadaad poduct_ THEOREM 4.5. Ude the hypotheses of Theoe 44 evey liea fuctioal y e B* is epesetable i the fo li (l,i) y(f) lia H(f,F;) (l i) ) f(l ()F --, T x...xt () i e I e (4.16) < < 1 i=l,, ad F B 0 F uiquely deteies ad hee, 0 6. I, 0 < i i 1 is uiquely deteied by y ad Y If N(F). Equatio (4.16) ca be expessed i tes of eal itegals usig (1.21). REFERENCES I. TAYLOR, A. E. Baach Spaces of Fuctios Aalytic i The Uit Cicle, I, II, Studia. Math. II (1950), pp ; Studia. Math. 12 (1951), pp GRIMM, L. J. ad HALL, L. M. A Alteative Theoe fo Sigula Diffeetial Systes, J. Diff. Equa. 18 (1975), HALL, L. M. A Chaacteizatio of The Cokeel of a Sigula Fedhol Diffeetial Opeato, J. Diff. Equa. 24 (1977), I HALL, L. M. Regula Sigula Diffeetial Equatios Whose Cojugate Equatio Has Polyoial Solutios, Sia J. Math. Aal. 8 (1977), TAYLOR, A. E. Itoductio To Fuctioal Aalysis, Joh Wiley ad Sos, Ic., Ne Yok, 1967.