AOKAAAbIAKAAEMVTTN HAYK CCCP

1974

r. 215, N! 5

I o x a a A u A a a A e M r r r r H a y x C C C Pl974.Tou 215. N 5

vAK 513.88 MATEMATNI{A

c. c. ItyTATEIIAASE

HEKOTOPbIE TEOPEMbI OTIEPATOPHOU TEOPIIN IIIOTTE

(Ilpe1craeneuo ana1enunon JI. B. Itanropoluqeil 13 VIII 1973)

Pa6orrr trocJreAlrefo BpeMerru (t-') uoHaarrBaror, qro ecrecrBeurrofi xarero-prrei,i, B paMnax Horopofi Aon)r(Ha crpor{rbcfl reopufi ITIoHe, fiBJrsercfl Hareropustrpocrparrcrn ltanroponlFra. OAHaKo rrMeroqr{ec.E B aToM Ha[paBJreHrrrr pe3ynb-'raTbr KaH rrpaBlrno ucnoJrbSyloT gna o6peueHrrTeJrbHbrx ycJroBtrfi - Tpe6oBaHtrel{onFirrqrranbHocTtr fiopo]r(Aaroqero ylopsAoqeHHocrb IIIone KoHyca rr rpe6oBa-nlre norsoft Jrl{Herluocrtr paccMarprrBaeMr,rx oneparopoB. I{elr Hacronrqefi pa-tjorrr - BarIoJIHrrrb IrMerorqlrec.s qeJrrr B paaltrrrHr,rx HaupaBJrenugx KaH AJIfl tro-JyqeHfifl TeopeMbl AeHoMnoSIIquII, TaK r{ AJrfl xapaKTepr{cTr{r{rr MaKcr{MaJrbrrbrxorrepaTopoB.

flycru X - apxuue4on K-nuueaJr, a )a - Henoropoe K-upocrpaucrBo. 9epeaL* (X, Y) o6oauauaerca. Muo?fiecrBo noJro?fitrreJrbubrx lnuefiuux oneparopoBr.ra X a Y. flycrr, AaJree, X" - coornercrByroqafl. crerreub X, a A: x+ (r, ... , r) -

ArrarouaJrbHrrii oueparop, A: X*X". rlepee A- o6oaHasuM orreparop aB X" B X,gei.icrnyroqufi uo Soprvryne A. (rr, . . . , x,):rrY . . .Y t,.

Onepa rop S : .X " * I uaa r r sae rc f l cBepxp aa6neHueM o t repa ropa ^S uL*(X,Y), ectu S:SA-. Oueparop S ua.L+(X", Y) uaarrnaercf f p ae6neHr4 eM.oleparopa S ua L+ (X, Y), euru r(oMMyrarr{Brra ArrarpaMMa

v !-x3 x" j, r.I lpegJroEreuue 1. a) J lunef i ,nar , f i oneparop A onopen n ceepupaa1ue-

u,uto S a roM u roJlbno row crlAvcre, ectu A - paa6uenue S.6) Iaa nactc)oao r ua X" cAuqecrsAer paa6uenue S ranoe, uro St:Sx.a) I lneer Mecro npe1craoaenue Sr:sup{Se: ,SA:S} , reX".

I Iq"4ootne^H:ce 2. I lycra_5, T=L*(X, Y) u H -K,or . tyc o X". Toa1a(V7tr ,S: SeSpr(?, H) \+YheV Sh>Th.

30ecu 1na oneparopa TeL+ (X, Y) u noH,Aca H e X no"r,ouceno

Spr(r , H) : {T 'eL* (X, Y) : T 'h2Th, heH}.

Hauounulr, qro Muoxtecrno Spr(?, 11) uaarrnaerc.s rr o Jr o rr(u r e Jrr'H brM. p o c r n o M o t r e p a r o p a ? u a n o u y c e f I .

3 aueqa H u e 1. Buepnr,re no4o6uoe [peAJrox(eHrre AJrfl cryqa.fl Bbrnyr(JrbrxtroBepxuocreft ycrauour.rr IO. l. PeurernnH (u).

B reopun IIIore cyqecrBeHuyro poJrb r4rpamr crrryaqtrr4, norAa rlpeAJroxe-uue 2 uontno o6parurr. B rrolr cryqae roBopflT, qro rrMeer Mecro reopeMa [e-HOMIIOsurIZU.

Teope:ora I . Eca,u H-no1npocrpaHcrso ur , r .u ec l l t t , X-or1e. tunut t iJr,otr,&rlbruo-eatnynnatrt K-auruea,tv, Y - K-npocrparLcrlo oaparLuveH,H,brr al,t,eil,eH,-roo u S, T ru,enpepot,erdbl, To u,t(,eer tt,ecro reoperta 0enounoauquu.

SaueqaHtre 2. f loHaaareJrr,crBo reopeMr,r t n neprolr cilyqae Momer6rrrr uponeAeHo tro cxeue ltaprr,e-@enta-Mefie (cu., naupuuep, (6)). BoBTOpOM CJIJ^rae, K COE(aneHtrrO, UptrXOAr{rCn UpU6erarr, H O6XOgUOMy MarreBpy -ucnoJlbaoBarb cB.sab cy6nunefinux o[eparopoB u ceJrenropoB (t) u EaBecrnyroreopeMy Xacyuu (').

IIepefi4eu rerrepb fi xapaxrepucrrxe MaxcuMaJrbrrbrx oneparopoB. OneparopTeL+ (X, Y)

' nasuaaercfi MaxcuMaJrbHbrM orrrocrrreJrbno HoRyca I/, ecJru

Spr(, H):{f l .Mancuuanr,Hbre oreparop6r BoBHtr(aror B paanoo6paaHr,rx Ba-Aaqax reopuu IIIone - B BbrlynJroM auaJrtrBe, reoprrr{ npu6luxeurfi, reolrerpuuc([op u r. ;1. Baxua.fi xapanrepucrur{a raxux oleparopoB cBfiBaEa c aagauefio upo6nrrx dtyuHqrnx (cu., uaupuMcp, (') ).

T eop eua 2. Crc1yrcu4ue yreeptc}err,urL )nru6arleH,rtl,bt;a) Oneparop T ttancuuarlen ornocurerr,bno nou,yca H;6) Ec,tu noroqeuHn oapaHurteHnatr cerb (7,) oneparopor ua I+(X, y)

rarcoea, aro l im T"h>Th, heH, ro (o) - l im T,r:Tt 1ta xeX.

IrIa reopeunr AefioMtroauqtrs cJreAyer, qro MancuMalrbr{bre oruocuTeJrr,rronepxneli peluerxa otreparopbr aatroJrn.aror riouyc, nnnnrorquftcfl Bepxuefi pertrer-nofi n upocrparrcrBe otreparopoB. Bonee roro, otrpaBeAJrtrBa

Teopeua 3. I lycra H nounur4ua,neu, X. Toe1a 1na ecanoeo oneparopa Tua L+(X, Y) cyu4ecreAer oneparop TeSpr(T, II) ranoti, uro Spr(T, H)::Spr(7, H-H).

tr{s erofi TeopeMbr (npegcranrnrcqeft arraJror JreMMbr o BbrMerautrn) cre-Ayer, qro Mar(cuMaJrbuble otreparopbr cyqecrByror Ha AocrarorrHo 6onrmux no-Eycax. To.ruee, cnpaBeAnuBo

f Ip"gJroxrenue 3. Hanortyce H cyugecreyrcr na.Hcumar lbnbLe oneparo-pbt, co BHaqeH,uai;u 6 Henoropow (a, artaaur, u e a,ro6on) peeA.nnpHo ynopn1o-ttetdH,oJ4, K-npocrparLcree I ToJvt u roJLbHo ron cJrVqae, ec,nu tfir,oJrcecr6o eeprtlurepaH,ut4 orle.rcenr o I H nt orn o e X e p e ey n apnofi, ronorLo euu.

Kax npanuro, rrgj^{euue MarrcrrMirJrr,r.ri,rx otreparopoB BeAercE c rroMorr{r'rorroAteMa Ea Eenoropoe (peryuapuo ynopflAorreuuoe) K-npocrpancrso Z, tr4uelr-EO, paccMarpuRaeTcfi rioMMyraTr{DHarr Alrarpar\IMa

v I xi, z'---v.Polr, rpanuqrr IIIoHe rrprr DToM, ecrecrBerrHo, AoJrlnrra r4r'parb rtounoneurrr n Z.Orpnnrvzlrt,flArrn npocrorr,r cJryqaeM, Hor4a [/ ecrr (uonoroHrroe) RnomeRue Xs Z. B aroii caryaqr{rr orreparop PeL+(2, Y) Eaabrrlaercrr 11-na4lranculranr,-Hr,rlt, ecJru / lraHcu,rrareu.

Teopeua 4. I3 6yaeeort a,nae6pe. npoenropos Z cyut ,ecreyer u,au6o, tuutu i rH -rua1 n an c twtanun ot li np o e tn o p.

l lpoenrop Pcr , , o lPe[enennr, r i i : reopel to i i 1r , sagr tsaror t rpoeKTopoMITToHe , a coo rBe rc rBy ro rqy rc r {oMt roge r r r y - r {o M I ro Hes ro f i I I I one .I4cuorrayrorcfl ral{}He o6osnaseHu fl P.o\t,, x , tt w Cb(II , X, Z) .

3al rcsanue 3. Oruerul r , qro ! i l t t t rpoexropa P ct rpaBei lJruRoSpr(P, H):Spr(P, P(H) ), rAe P(H) - rar{rrerrbrtrafi Bepxufif i perrrerna, tro-polr(AeuHafl .I1. Tanuu o6paaom, upuBeAeHrroc orpeAeJlerrue trpoel{ropa IIIor:eB clyqae iiordur4rlr{a,nrJrrocrr{ ff npocrparrcrtv X conna4aet c BBeAeIrHbrMB. H. , (nrro, : r ' r .v ( t )

T e o p e u a 5 . I l y u t , P ( i l ) - P ( I I ) : X u T e L + ( 2 , Y ) - n e n o r o p a u i ,enoJllde "out+efiu,utit H-u,a0uancu,rta"a,bLLbui oneparop. Toe1(r noil,noHeH,ra e?o cU-u+ecrseHHoti norlolrcurerlbu,ocru coAepzrcurcfl, 6 nownoHetrre IIIone Ch(11, X, Z),

Saueqagtre 4. CyrqecrBeuHo, r r ro reoper{br 4 ra 5 ornocf l rcs n o5qemycilyqarc u pa6oraror, B qacr[octv, n Lo. O4ua uo"neaHafl B sToM cil] rae cBfl3r)rpanurl oilr{cblBaercfl coorrrouresuem Ch (II , X, Z)-Ch(H, 7, Z)t ecrru X:Z(r4e eaurrnaHue, Hafi o6rruno, 6epetca B peryJlflplrofi touonoruu).

Saueqague 5. O.renuAHo, qro o6parqenue reopeMbl 5 noo6rqe ronopf l ,Ee ctrpaBeAJrnno. [nn HouuuqtraJrbHoro Horryca Eytnfioe AorIoJIHureJIbHoe ycJto-BEe Baxrroqaercfl B rpe6onanuu frunr,rpaqnn BtrpaBo otroplrbrx Mno)*tecrB.

B aaxnrosentre npuBe4erI Herioropble xapanreptrcrtrHu He o6seatentnoErroJrHe ;rurrefr nr,rx MaHctrMaJrbHbIx otreparopoB.

I (onyc H s X r raal rBaercf l Z-rora.n inrru (e4ccr X=Z), ecJ lu . { ro6of totreparop T us L+(2, Z) flBJrflercfl (H-H)-uagMaKcuMaJIbHbrM.

flpoenrop P p Z HaabrBaerc.s H-rpaHtrqrrbrM (n crrarrcue Illunona),ecnuYheH Pla<O+D(0.

IIpe4ao'fterru.e 4. Ilycru noH,Ac II aenaerca Z-rora.nanout u nouvut+u-a,rnrdbt Jtt Z. T o a0 a en sul alrerrrt+w, cae1y rcu4ue y r o e pcr7 en ua:

a) Ilpoenrop P aeaaerca H-aparuuvnttu;6) il,na aro6oao H-nancunalrbtr,oeo oneparopa, TeL+ (X, Y) uneer Mecro

ebtcnaabr,ga,tLue Y teX Pr{0+7c(0.IIycffi rerrepb I/-Korlr{trquanrnr,rfi fiorryc s Z u. Pt (f/) - uatrMenbmas yc-

JIoBHo troJIHafi Bepxrrfi.E pemerr(a n Z, nopoxtgeuHag 11 (:nouyc .I/-rsrnynnrrxoJreMeHToB n Z). Ionopflr, qro BbrtroJrHeuo ycJroBr.re noAbeMa (n upocrpaucrnoY), ecrn

Spr (2 , 11) :Spr (? , Pu(H) ) , TeL+(2 , Y) .

Teopervra 6. Ilycru HcXeZ u ebtnorlHer"r,o Acrrolue no7zena e npocr-paHcreo Y. Oneparop T aenaerca H-na0nancuu,a.nbH,btM 6 ToJtt u rortbno rowc.nyqae, ec,nu TP"h (a-CO"r):0, xeX.

3gecr Pd - trpoenTop, AotroJrrrrrrenrnrrfi nI/-nnrnynlafi o6oilo.rna r.

Cne,qcrBu e l . Ecau TPlnr :O, ceX,t{,blfit.

CneAcrBrr e 2. IIycra U -nnowecreo H-ua0ttancuJ,t&lrbtrbtx oneparopor.Toe1a noil.noH,eHrd (Udd1, nopottclerunaa U, ecra i ltrroncecreo {f : l f leUT,

Korrn6nnaqufl npeAnorilenus. 4 u reopeMa 6 uognorser, HarrpuMep, rroJr)FrurbcJleAylou{yrc DHBraBaJreHTEocrb reopeusr IIIoHe tr trpilHr{rrna Mafi orrMyMa.

Teop ev 'a T. I lycra H - rcounuq,uaa,aH,at t i , Z-roranau,ut f i nonyc, npuqewH : P r, (H ) . C n e 0 y toutue V r 6 e p ?r 0 enua a r{ 6 u I &lr er!,r rdbl :

a) II p o enr o p III on. e fl, e lLrr er c fl, H - e p an uvnr'rw;6) Oneparop T ua L* (2, Y) aenaercn Mancuwa,rlbtrbtw orrdocure,nbrdo tto-

nyca H e rort a rolrbno rotr. crr,Aqae, ec.nu TP(,:O.

P n.COna:sup{h: h{x, heH}-

ro T ae,taerca H-rua0ttaqcuwa.rlb-

Ilocrynlrro13 VI I I 1973

?Ilacrwryr MareMarrrnfiCu6rpcroro orAeJrenrff Ana4euura nayr CCCPHoaocr.r6npcn

III{TI4POBAHHAB JI'ITEPATYPA

I G. F. Vincent-Smitb, J. London Math. Soc., v.44,3,553 (1969). 2 C. C. Ityrare-4qAf, 4. M.-P_y6unoa, VMH, r. 27, 3, L28 (L572). 3 C. C. Ityrareaa|ae,,{AH, r. 208,N 4,771 (1973). L Z. Semaileni,.Banach Spaces of Continuous Functions, Warszawa,7571.. 5 n. l. Peuternan, fiuccepraqnr, JL, 1954, 6 E. Allsen, Compact Convex Setsan4 Bounda,ry I_ntegrals, Berlin, N. Y., 1971. 7 IO. 0. Jlunne, AAH, r. 207, N! 3, b3tt!972). --8 M. Hasumi, Math. Ann., v. 179,2,83 (1969). s B. H. f i,artoa, [,AH, r.2L2,J\! 5 (1973).