luật mạnh số lớn trong đại số von neumann
TRANSCRIPT
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Mc lcLi m u 1
1 Kin thc chun b 41.1 i s von Neumann v vt . . . . . . . . . . . . . . . . 4
1.1.1 i s Banach . . . . . . . . . . . . . . . . . . . . 41.1.2 Php tnh lin hp . . . . . . . . . . . . . . . . . 51.1.3 i s von Neumann . . . . . . . . . . . . . . . . 61.1.4 Phim hm tuyn tnh dng v biu din . . . . 6
1.2 Ton t o c theo mt vt . . . . . . . . . . . . . . . 71.2.1 Cc tnh cht c bn ca ph . . . . . . . . . . . 71.2.2 Khi nim v ton t khng b chn . . . . . . . 91.2.3 M u v php chiu . . . . . . . . . . . . . . . 111.2.4 L thuyt v ton t o c . . . . . . . . . 12
1.3 Khng gian Lp theo mt vt . . . . . . . . . . . . . . . . 211.3.1 Hi t hu u trong i s von Neumann . . . . 241.3.2 Cc kiu hi t hu chc chn trong i s von
Neumann . . . . . . . . . . . . . . . . . . . . . . 261.3.3 Dng khng giao hon ca nh l Egoroff . . . . 281.3.4 Khi nim v lut s ln . . . . . . . . . . . . . . 29
2 Lut mnh s ln trong i s von Neumann 312.1 Tnh c lp . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Hi t hu y trong i s von Neumann . . . . . . 322.3 nh l gii hn mnh cho dy trc giao . . . . . . . . . 342.4 M rng khng giao hon ca nh l Glivenko-Cantelli . 412.5 Bt ng thc Kolmogorov i vi vt v mt s h qu 442.6 Lut mnh s ln i vi vt . . . . . . . . . . . . . . . . 47
i
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1
2.7 Tc hi t trong lut mnh s ln . . . . . . . . . . . 622.8 Ch v ch thch . . . . . . . . . . . . . . . . . . . . . 68
Kt lun 70
Ti liu tham kho 72
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Li ni uCc ti liu khoa hc hin ti a ra nhiu bng chng rng cc
phng php i s vn cch mng ha ton hc thun thy th nayli ang gy nh hng tng t i vi khoa hc vt l. Tip cn is i vi c hc thng k v l thuyt trng lng t l mt v dcho nh hng mi ny.
Gn y nhiu tc gi m rng cc nh l hi t im c bntrong l thuyt xc sut v l thuyt ergodic sang (ng cnh) i svon Neumann.H cung cp mt s cng c mi cho vt l ton vng thi to ra nhiu k thut hp dn cho l thuyt i s ton t.
Mc ch chnh ca ti l trnh by bn cht ca mt s tngv kt qu t lnh vc ni trn,chuyn cc kt qu c in bit trongl thuyt xc sut n cc phin bn khng giao hon ca chng, avo cc ng dng trong l thuyt trng lng t.
i s von Neumann l mt s tng qut ha khng giao hon rt
t nhin ca i s L v nhng cu trc tt ca n em li kh nngthu c cc phin bn hu chc chn ca cc nh l gii hn.Trong i s von Neumann, ta c th a ra khi nim hi t huu tng ng vi khi nim hi t hu chc chn trong i sL.Kiu hi t ny s l nn tng cho ton b ti.
Ni dung ca ti gm hai chng :Chng 1 Trnh by mt s kin thc chun b cho vic nghin
cu chng 2 bao gm cc kin thc nn tng v gii tch v xc
sut.Mt s tnh cht ca hi t
hu u
trong i s von Neumann.Chng 2 Ni dung chnh ca ti: Lut mnh s ln trongi s von Neumann
Trnh by cc kt qu ca Batty ,cng mt s kt qu khc. Nu nhtrong xc sut c in cc dng hi t theo xc sut ,hi t hu chcchn v hi t trung bnh ca dy bin nhin ng vai tr then cht
2
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Li ni u 3
trong Lut s ln th y chng ta c tip cn vi 1 kiu hi thon ton mi hi t hu u .Cc nh l c chng minh i vitrng thi , i vi vt. V mt trng thi th khng cng tnh di trndn cc php chiu nn cc qui tc v cc k thut i vi trng thikhng vt khc nhiu v kh khn hn nhiu so vi cc qui tc i vivt. ng ch y l cc lp lun cn dng i vi vt thng rt
ging trng hp c in. Nhng trong mt s thng hp th chng tacn hng tim cn mi .Hu ht cc kt qu c trnh by trong ti l kt qu mi.
Mt s ni dung ca ti l mt trong s cc bi ging ti iHc Tenessee Knoxville v ti Trung Tm Qu Trnh Ngu Nhin(Centerfor Stochastic Processes) ti i hc North Carolina ChapelHill (bi R.Jajte)
Ni dung ca ti vn ang l hng quan tm ca nhiu nhton hc v vt l hc trong cc lnh vc i s ton t v cc ngdng ca chng nh Lance 1976-1978 ; Goldstein 1981; Watanabe 1979;Yeadon 1975-1980; Kiinrinerre 1978;.......
Trong qu trnh tm hiu , nghin cu ni dung cc cng trnh cangi khc chng ti h thng , gii thiu nhm phc ha trin vngng dng cc kt qu nghin cu ca mnh trong tng lai.
Tc gi xin c gi li cm n su sc n ngi thy, ngi hngdn khoa hc ca mnh l PGS. TS. Phan Vit Th, ngi a ra
ti v hng dn tn tnh trong sut qu trnh nghin cu bn lunny. ng thi tc gi cng chn thnh cm n cc thy c trong khoaTon - C - Tin hc trng i hc Khoa hc T nhin, i hc Qucgia H Ni, v cc thy c gio Vin Ton hc -Vin KHCNVN tn tnh ging dy ,to iu kin thun li trong sut thi gian Tc gihc tp v nghin cu.
H Ni, nm 2009Hc vin
V Th Hng
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CH NG 1. KI N TH C CHU N B 5
A c php nhn:A A A
(x, y) xy
tha mn cc tnh cht sau :
1. x(yz) = (xy)z;
2. (x + y)z = xz + yz; x(y + z) = xy + xz;
3. (xy) = (x)y = x(y), x,y,z A, CKhi , A c gi l mti s phc. Hn na , nuA l mtkhng gian Banach (vi chun ||.||) tha mn cc tnh cht sau :
4. ||xy|| ||x||.||y||; (x A, y B)
5. A cha phn t n v e sao cho xe = ex = x (x A);
6. ||e|| = 1;thA c gi l mti s Banach.
Nu php nhn giao hon th gi l i s (i s Banach) giaohon .
1.1.2 Php tnh lin hp
nh ngha 1.1.2. Gi sA l mt i s phc , nh x x A x
gi lphp lin hp nu tha mn cc iu kin sau :
(i) (x + y) = x + y
(ii) (x) = x
(iii) (xy) = yx
(iv) (x) = x
i s phc A ng i vi php lin hp gi l i s . Nu Al i s Banach ng i vi php lin hp th gi l mt i sBanach .NuA l mt i s Banach tha mn iu kin ||x|| = ||x|| thAgi li s Banach lin hp. Nu i s Banach tha mn iukin ||xx|| = ||x||2, x A th gi l mtC i s
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CH NG 1. KI N TH C CHU N B 6
Phn t x A (A l i s ) gi l chun tc nu xx = xx.Gi l Hermit nu x = x, unitar nu xx = xx = e, (e l n v caA). Nu A l C i s th ||x|| = ||x||, x A. Vy mi C i su l i s Banach lin hp.
1.1.3 i s von Neumannnh ngha 1.1.3. Gi sH l khng gian Hilbert, B(H) l i s ccton t b chn. A B(H) l mt i s con.i sA B(H) gi l i s von Neumann nu:
(i) A l kn i vi php ly lin hp;
(ii) I A
(iii) A ng i vi topo hi t yu, tc lAnw
A nu :< Anx, y >< Ax, y > vi mix, y H
Nh vy i s von Neumann l mt C i s.
1.1.4 Phim hm tuyn tnh dng v biu din
nh ngha 1.1.4. (*) Gi sA l i s von Neumann. Phim hmtuyn tnh :
: A C
gi ldng nu:(xx) 0, x A
(*) gi lphim hm dng chnh xc (ng) nu:
(xx) 0
v t(xx) = 0 suy rax = 0
c bit : |||| = (I)(*) Nu(I) = 1 th gi ltrng thi
nh ngha 1.1.5. K hiu : A+ = {x A : x 0} . nh x : A+ [0, ] tha mn tnh cht :
(i) (x + y) = (x) + (y), x, y A+
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CH NG 1. KI N TH C CHU N B 7
(ii) (x) = (x), 0; x A+ (quy c 0. = 0)
(iii) NuU l unitar th : (U xU1) = (x), x A+
Khi gi l vt ca i sA .Nu :
(x) < , x A+
th gi lvt hu hn.Nu :
(x) = sup{(y)|y x; (y) < }
th gi l vt na hu hn .Nu : (x) = 0, x 0 m suy rax = 0 th gi lvt chnh xc (hayvt ng)Vt gi lchun tc (Normal) nu:
(T) = sup (T)
trong T l dy cc ton t tng tiT.
i s von Neumann gi l hu hn (hay na hu hn) nu vi mix A, x = 0 tn ti vt chun tc hu hn sao cho (x) = 0. Trn is von Neumann na hu hn lun tn ti mt vt chun tc , chnhxc na hu hn.
1.2 Ton t o c theo mt vt
Phn ny ta s nh ngha khi nim v tnh o c theo mt vt trn mt i s von Neumann A v ch ra rng tp A cc ton t oc l mt i s topo y .Gi s A l mt i s von Neumann na hu hn hot ng trnkhng gian Hilber H v l trng thi na hu hn chun ng trn A.
1.2.1 Cc tnh cht c bn ca ph
nh ngha 1.2.1. Gi sA l i s Banach . G = G(A) l tp hptt c cc phn t kh nghch ca i s A . Khi G lp thnh mt
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CH NG 1. KI N TH C CHU N B 8
nhm . Ph (x) ca x A l tp hp tt c cc s phc sao choe x khng c kh nghch . C/(x) c gi l tp hp chnh quy caphn tx.
C/(x) = { : (e x)1}
S(x) = sup{|| : (x)}
c gi l bn knh ph ca phn t x.
Ta lun chng minh c rng (x) = , x A.
nh ngha 1.2.2. Gi s a l ton t tuyn tnh vi min xc nhD(a) . K hiu
D(a) = {g : ! g, < g|af >=< g|f > f D(a)}
vi gi thit D(a) = H. Khi D(a
) l khng gian con v ton tag = g, g D(a), g l phn t duy nht
< g|af >=< g|f >
l mt ton t tuyn tnh. a gi l lin hp ca ton ta.
Nu a a th a gi l ton t i xng . Nu a = a th a gi lt lin hp.Khi a l ton t ng v D(a) = H th khi D(a) = H v a = a.
Ch . i s con A ca i s B(H) gi l chun tc nu n giaohon v nu T A th T A.
nh ngha 1.2.3. Ton t P c gi l ton t chiu nu D(P) =H; P = P = P2
Nh vy , ton t chiu P l b chn v c s tng ng mt - mtgia cc ton t chiu v cc khng gian con ng trong khng gianHilbert.
nh ngha 1.2.4. Xt H l khng gian Hilbert . (, ) l mt khnggian o, l trng. P l tp hp cc ton t chiu trong khnggian HilbertH. nh xE : P c gi l mtkhai trin n vtrn (, ) nu cc iu kin sau c tha mn :
1. E() = 0, E() = I
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CH NG 1. KI N TH C CHU N B 9
2. E(AB) = E(A)E(B)
3. E(A B) = E(A) + E(B) nuAB =
4. x, y H hm tp hp Ex,y xc nh bi cng thc
Ex,y(A) =< E(A)x, y >
l mt o phc trn
nh l 1.2.5. (V biu din Ph) Nu T B(H) v T l ton tchun tc th tn ti ng mt khai trin n v trn cc tp con Borelca ph(T) ca ton tT sao cho
T =
(T)
dE()
NuT l t lin hp th(T) R. Khi
T =
ba
dE
NuT l ton t unitarT T = TT = I . Khi (T) nm trong vngtrn n v . Khi
T =
20
eidE()
1.2.2 Khi nim v ton t khng b chn
Vi cc ton t (tuyn tnh) a, b trn H ta c th nh ngha tng a + bv tch ab l cc ton t trn H vi min xc nh :
D(a + b) = D(a) D(b)
D(ab) = { D(b) b D(a)}
Cc php ton ny c tnh cht kt hp, v th a+b+c v abc l cc tont c nh ngha tt. Hn na ,vi mi a,b,c ta c : (a + b)c = ac + bcv c(a + b) ca + cb
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CH NG 1. KI N TH C CHU N B 10
nh ngha 1.2.6. Mt ton ta trn H l ng nu thG(a) can ng trongH H ;a l trc ng nu bao ngG(a) ca th ca n l th ca mtton t ng no ( gi l bao ng ca a , k hiu l [a]) ;a l xc nh tr mt nu D(a) tr mt trongH.
Nu a, b v ab xc nh tr mt th :
(ab) ba
ng thc xy ra nu a b chn v xc nh khp ni.Ton t ng , xc nh tr mt a c biu din cc:
a = u|a|
y |a| l ton t t lin hp dng , v u l mt ng c ring vi
supp(a) l php chiu u ca n v r(a) , php chiu ln bao ng camin gi tr ca a , l php chiu cui ca n.
nh ngha 1.2.7. Nu tnga + b ca hai ton t xc nh tr mtavb l trc ng v xc nh tr mt , th bao ng [a + b] c gi ltng mnh caa vb. Tng t , tch mnh l bao ng [ab] nu ab ltrc ng v xc nh tr mt.
Ta vit||a|| = sup{||a||
|||| 1}
vi mi ton t xc nh khp ni a trn H, b chn hoc khng. Khi c lng sau y ng :
||a + b|| ||a|| + ||b||; ||ab|| ||a||.||b||
K hiu A l hon tp ca A (hon tp A ca i s von Neumann Al tp tt c cc b trong B(H) giao hon vi a trong A). nh l hontp 2 ln von Neumann khng nh A = A.
nh ngha 1.2.8. Ton t tuyn tnh a trn H c gi l kt hpviA (v ta vitaA) nu:
y A : ya ay
Ta k hiu A l tp tt c cc ton t ng , xc nh tr mtkt hp vi A
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CH NG 1. KI N TH C CHU N B 11
1.2.3 M u v php chiu
K hiu Aproj l dn cc php chiu ( trc giao ) trong A . i vih (pi)iI cc php chiu trc giao trong A, k hiu :
iI
pi; (iI
pi)
l php chiu ln iI
piH; (iI
piH)
Ta c :(iI
pi) =
iI
pi ; (iI
pi) =
iI
pi
y p = 1 p l php chiu trc giao vi p. Hai php chiu p v q
l tng ng nu p = uu v q = uu vi u A no . Ta k hius tng ng l . Cc php chiu tng ng c cng vt.
Mnh 1.2.9. Gi sa l ton t ng, xc nh tr mt kt hp viA. Khi :
supp(a) r(a)
yr(a) k hiuphp chiu ln bao ng min gi tr caa.Vi cc php chiup, q A ta c:
(p q) p q (p q)
ko theo :(p q) (p) + (q)
Tng qut hn :(iI
pi) iI
(pi)
i vi h ty (pi)iI cc php chiu trongA ( nuI hu hn th iuny ko theo bng qui np; i vi trng hp tng qut , s dng tnhchun tc ca ).
Nhn xt 1.2.10.
p,q Aproj : p q = 0 = p 1 q
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CH NG 1. KI N TH C CHU N B 12
( y c ngha tng ng vi mt php chiu con ca ).Tht vy :
p = 1 q = (p q) p
= (p q) p q (p q) q = 1 q
1.2.4 L thuyt v ton t o cnh ngha 1.2.11. Gi s, R+ . Khi ta k hiuD(, ) l tptt c cc ton taA sao cho tn ti php chiup A tha mn :
(i) pH D(a) v ||ap||
(ii) (1 p)
KhipH D(a) th ton tap xc nh khp ni , i hi ||ap|| ko
theo ap b chn .
Mnh 1.2.12. Cho 1, 2, 1, 2 R+ . Khi :
(i) D(1, 1) + D(2, 2) D(1 + 2, 1 + 2)
(ii) D(1, 1).D(2, 2) D(12, 1, 2)
Mnh 1.2.13. Gi s, R+:
(i) Nua l ton t trc ng , th:a D(, ) [a] D(, ).
(ii) Nua l ton t ng , xc nh tr mt vi biu din cc a = u|a|th:
a D(, ) u A, |a| D(, )
B 1.2.14. Cho a A v, R+ . Khi :
a D(, ) (],[(|a|))
( y],[(|a|)
k hiuphp chiu ph ca |a| tng ng vi khong ], [ ).
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CH NG 1. KI N TH C CHU N B 13
Chng minh.
t :p = [0,](|a|)
. Khi : pH D(|a|) v
|| |a|p||
Vi p Aproj no ta c :
|| |a|p||
v (1 p) Gi s:
|a| =
0
de
l phn tch ph ca |a| . By gi: pH ta c:
|| |a|||2 2||||2
v : (1 e)H{0}
ta c:|| |a|||2 > 2||||2
V:
|| |a|||2 =
0
2d(e|) =
],[
2d(e|)
nn:(1 e)HpH
phi l {0} ,tc l (1 e) p = 0Vy 1 e 1 p, do
(1 e)
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CH NG 1. KI N TH C CHU N B 14
Mnh 1.2.15. Cho a A v, R+. Khi :
a D(, ) a D(, )
Chng minh. Gi s a = u|a| l biu din cc ca a. Khi u l mtng c ca
]0,[(|a|) = supp(a)ln
]0,[(|a|) = supp(a) = r(a)
Do tnh duy nht ca phn tch ph suy ra vi mi R+, u l mtng c ca
],[(|a|)
ln ],[(|a|). p dng b trn c iu phi chng minh.
nh ngha 1.2.16. Mt khng gian con E ca H c gi l trmt nu R+ , tn ti php chiu p A sao cho pE E v(1 p) .
Mnh 1.2.17. Gi sE l khng gian con tr mt caH . Khi tn ti mt dy tng (pn)nN cc php chiu trongA vi
pn 1, (1 pn) 0,
n=1pnH E
.
Chng minh. Ly cc php chiu qk A, k N sao cho :
qkH E
v (1 qk) 2k. Vi mi n N , t :
pn =
k=n+1
qk
Khi
pnH =
k=n+1
qkH E
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CH NG 1. KI N TH C CHU N B 15
v
(1 pn) =
k=n+1
(1 qk)
k=n+1
(1 qk)
k=n+1
2k = 2n
. Ko theo: pn 1.Tht vy, k hiu p l supremum ca dy tng pn , ta c
n N : (1 p) (1 pn) 2n
do (1 p) = 0 v p = 1. Hn na
n=1pnH E
Vy nu E l khng gian con tr mt ca H th E tr mt trongH.
B 1.2.18. (i) Cho p0 Aproj .Gi s rng :
R+, p Aproj : p0 p = 0
v(1 p) . Khi : p0 = 0.
(ii) Cho p1, p2 Aproj. Gi s rng
R+, p Aproj : p1 p = p2 p
v(1 p) . Khi : p1 = p2.
Mnh 1.2.19. Cho a, b A vE l khng gian con tr mt ca
H cha trongD(a) D(b). Gi sa|E = b|E . Khi a = bChng minh. Xt trong khng gian Hilbert H2 = H H i s von
Neumann A2 =
A A
A A
c trang b vt na hu hn chun ng 2
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CH NG 1. KI N TH C CHU N B 16
xc nh bi 2
x11 x12
x21 x22
= (x11) + (x22). K hiu pa v pb l cc
php chiu ln th G(a) v G(b) ca a v b .V a v b kt hp vi Ann G(a) v G(b) bt bin di tt c cc phn t ca
A2 = { y 00 y |y A}
v do pa, pb A2 . Gi s R+ khi tn ti mt php chiu p A
vi pH E v (1 p) /2. t p2 =
p 0
0 p
Khi : 2(1 p2) . Hn na
pa p2 = pb p2
V a v b thng nht trn pH E nn
G(a) (pHpH) = {(,a)| pH,a pH}
= {(,b)| pH,b pH} = G(b) (pHpH)
Theo b trn , ta suy ra pa = pb, do a = b.
nh ngha 1.2.20. Mt ton t ng , xc nh tr mt kt hp viA
c gi l o c nu vi mi R+ , tn ti mt php chiup A sao cho pH D(a) v(1 p) .K hiuA ltp tt c cc ton t ng , o c ,xc nhtr mt
Nhn xt 1.2.21. 1. Nu a, b A v a b th a = b.
2. Nu a A, v a l i xng th a t lin hp .
3. Nu a ng v p Aproj tha mn pH D(a) th ton t , xc nh
khp ni ap cng ng v b chn.
nh ngha 1.2.22. Ton t aA c gi tin o c nu vimi R+ tn ti php chiup A sao cho
pH D(a), ||a|| <
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CH NG 1. KI N TH C CHU N B 17
v(1 p) .Hay tng ngGi saA . Khi a l tin o c khi v ch khi
R+, R+ : a D(, )
Mnh 1.2.23. (i) Ta c A
A.(ii) Via A tha A.
(iii) Cho a, b A khi a + b vab xc nh tr mt v tin ng , v[a + b] A, [ab] A.
(iv) A l mt i s i vi tng mnh v tch mnh.
T y ta s b qua k hiu [ ] trong k hiu tng mnh v tch
mnh.nh ngha 1.2.24. Vi mi, R+ , ta t
N(, ) = A D(, )
tc l , N(, ) l tp cc a A, o c sao cho tn ti php chiup A tha mn ||ap|| v(1 p) .
nh l 1.2.25. (i) N(, ), vi , R+ to thnh c s cho cc
ln cn ca 0 i vi topo trn A bin A thnh khng gian vc ttopo.
(ii) A l i s topo Hausdorff y vA l mt tp con tr mtcaA
Chng minh. iu kin (i) l hin nhin . Ta chng minh (ii) :(1) . A l Hausdorff , ta s chng minh rng
,R+
N(, ) = {0}
Lya
,R+
N(, )
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CH NG 1. KI N TH C CHU N B 18
Khi : R+, R+ : (],[(|a|))
V l ng (faithful) , iu ny ko theo
],[(|a|) = 0
, do a = 0.(2). Tip theo ta s chng minh A l i s topo.Theo kt qu trn php ton lin hp l lin tc. Ly a0, b0 A v, R+ . Chn , R+ sao cho
a0 N(, ), b0 N(, )
Khi vi mi a, b A tha mn a a0 N(, ) v b b0 N(, ) ,ta c
ab a0b0 = (a a0)(b b0) + a0(b b0) + (a a0)b0
N(, )N(, ) + N(, )N(, ) + N(, )N(, )
N(2, 2) + N(, 2) + N(, 2)
N(( + + ), 6)
Do (a, b) ab l lin tc.(3).A l tr mt trong A.Tht vy , gi s a A , ly cc php chiu pn A sao cho
pn 1, (1 pn) 0,nN
pnH D(a)
Khi apn A v apn a trong A v
||(apn a)pm|| = 0
vi mi m n v (1 pm) 0 khi n (4). Cui cng ta s chng minh rng khng gian vc t topo A l y.V A c mt c s m c cho cc ln cn ca 0 ( chng hn s dng
N(1/n, 1/m), n, m N
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CH NG 1. KI N TH C CHU N B 19
) ta ch cn chng minh rng mi dy Cauchy (an)nN trong A hi t .V vy gi s (an)nN l mt dy Cauchy trong A v
n N : an+1 an N(2(n+1), 2n)
Ly php chiu pn A sao cho
||(an+1 an)pn|| 2(n+1)
v (1 pn) 2n. Vi mi n N ,t:
qp =
k=n+1
pk
V
(1 qn) = (
k=n+1
(1 pk))
k=n+1
(1 pk)
k=n+1
2k = 2n
vm n + 1, l N : ||(am+l am)qn|| 2
m
V qn pkvi mi k m n + 1 v do
||(am+l am)qn||
m+l1
k=m
||(ak+1 ak)qn||
m+l1k=m
||(ak+1 ak)pk|| m+l1k=m
2(k+1) 2m
Ly
nN
qnH
.Khi qnH vi n N no ,v do dy(am)mN
l dy Cauchy. ta = limmam
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CH NG 1. KI N TH C CHU N B 20
Nh vy ta nh ngha mt ton t a vi
D(a) =nN
qnH
(ch rng D(a) l khng gian tuyn tnh con v (qn)nN l mt dytng cc php chiu). Theo cc xy dng , a l tin o c . Vimi n N ta c
qnH D(a)
v (1 qn) 2n. Ta khng nh rng a cng tin ng. thy iuny , p dng lp lun trn cho (an)nN. Do tn ti mt ton t tin o c b sao cho
bn = limman, D(b)
Khi D(a), D(b) : (a|) = lim(am|) = lim(|a
m) = (|b)
nn a b. Nh vy a tin ng . Vy [a] A. t: a0 = [a], cui cngta chng minh an a0 trong A. Gi s , N0. Ly n0 N sao cho
2(n0+1)
v 2n0 . Khi vi mi m n0 + 1 ta c
||(a0 am)qn0|| 2(n0+1)
v(1 qn0) 2
n0
v H : (a0 am)qn0 = liml(am+l am)qn0
v||(am+l am)qn0|| 2m 2(n0+1)
do b n0 + 1 : a0 am N(, )
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CH NG 1. KI N TH C CHU N B 21
1.3 Khng gian Lp theo mt vt
Trong phn ny chng ta s nh ngha khng gian Lp , Lp = Lp(A, )vi 1 p v xy dng cc tnh cht c bn ca chng. Segal lm vi p = 1, 2, ( khng gian L chnh l A ) , v Kunze nghincu trong trng hp tng qut. Cch tip cn ca chng ti da trn
khi nim hi t theo o trnh by trong phn trn s n ginhn.Cho trc mt vt na hu hn chun ng trn i s von NeumannA trn khng gian Hilbert. Gi s
L2 = {a A : (aa) < }
Cho a trong L2 v b trong A . Khi
(ba)
ba ||b||a
a
nn ba cng trong L2 . V a v b u trong L2 nn a + b cng trong L2 v
(a + b)(a + b) 2(aa + bb)
Do L2 l mt idean tri . Vy n l t lin hp v do l idean haipha (sau ny ta s ng nht L2 vi L2 L) . t L = L22 . Khi Lcng l mt idean hai pha (sau ny ta s ng nht n vi L1 L).Nu a trong A+ v (a) < th a1/2 trong L2 , nn a trong L. Ngc
li, gi s c 0 vi c trong L. Khi c l tng hu hn c = biai vibi, ai trong L2. V c
1
2
(bib
i + a
i ai) nn ta c (c) < . Do L
cha tt c cc t hp tuyn tnh hu hn cc phn t a trong A+ vi(a) < , v cc phn t ny to thnh L A+. Nh vy m rngduy nht thnh mt phim hm tuyn tnh (vn k hiu l ) trn L.Theo biu din cc th (ba) = (ab), v do (ba) = (ab) vi mia, b trong L. V vy ,nu a 0 v a trong L th
(ab) = (a1/2ba1/2)
nn0 (ba) ||b||(a)
vi mi b 0 trong Lv do , theo tnh chun tc v na hu hn ca ,vi mi b trong A+. Kt qu l |(ba)| ||b||(a) vi mi b trong A.
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CH NG 1. KI N TH C CHU N B 22
Ta cng s s dng k hiu ||b|| cho chun ton t ca phn t b trongA, v ta t ||a||1 = (|a|) vi a trong L. V a = u|a| vi ||u|| = 1 nnta c
|(ba)| ||b||||a||1; b A, a L (1.1)
vi 1 p < v a trong L , t ||a||p = (|a|p)1/p.Ta khng nh rngbt ng thc Holder:
|(ba)| ||b||q||a||p (1.2)
ng , vi a, b trong L v (1
p) + (
1
q) = 1. thy iu ny , gi s u v
v trong A vi ||u|| 1, ||v|| 1 v gi s c 0, d 0, c L, d Ltha mn c v d b chn cch xa 0 trn b sung trc giao ca cckhng gian trng ca chng. Do tnh lin tc nn s gii hn ny v saus b qua, v ta khng nhc n na . Khi s (udsvc1s) lin tc
v b chn trn 0 Res 1 v chnh hnh phn trong. Theo v dv nguyn l Phragmn-Lindelof c bit n nh l nh l 3 ngthng ,ta c :
|(udkvc1p)| supRes=1|(udsvc1s)|ksupRes=0|(ud
svc1s)|1k
vi 0 k 1, v theo (1.1) v phi ||d||k1||c||1k1 . p dng iu ny
vic = |a|1/k, d = |b|1/(1k)
y a = u|a|, b = v|b|, k = 1/q, v 1 k = 1/q , iu ny ko theo(1.2).Vi a = u|a| trong L, t
b =|a|p1u
||a||p/qp
y 1/p + 1/q = 1 . Khi d thy ||b||q = 1 v (ba) = ||a||p. Tc l
||a||p = sup||b||q=1|(ba)| (1.3)
v supremum l t c. T y ta d dng thu c bt ng thcMinkowski
||a + b||p ||a||p + ||b||p (1.4)
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CH NG 1. KI N TH C CHU N B 23
vi a, b trong L. V v tri l (c(a + b)) vi c tha mn ||c||q = 1 ,nhngn bng (ca) + (cb), do nh hn v phi theo bt ng thc Holder.Cui cng ,ch rng ||a||p = 0 ko theo a = 0 do tnh ng ca . Dovy L l khng gian tuyn tnh nh chun vi chun ||.||p . Gi Lp lkhng gian Banach tng ng vi L sau b sung cho y .Nu a 0 trong L vi biu din ph
a =
0
de
th
||a||pp = (ap) =
0
pd(e)
nn ||a||pp p(e) (1.5)
vi mi 0. Vy , nu an trong L l dy Cauchy trong Lp th n ldy Cauchy theo o. Do tn ti mt nh x tuyn tnh t nhinlin tc ca Lp vo A. Ta chng minh c rng nh x t nhin ny ln nh.
nh ngha 1.3.1. i vi ton t dng t lin hp a kt hp vi Abt k , ta t
(a) = supnN(
n0
de)
y
a =
0
de
l biu din ph caa . Khi vi mip ]1, [ ta c th nh ngha
Lp(A, ) = {a A(|a|p) < }
v||a||p = (|a|
p)1/p < , a Lp(A, )
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CH NG 1. KI N TH C CHU N B 24
.(Lp(A, ), ||.||p) l cc khng gian Banach trong
I = {x A(|x|) < }
l tr mt , v tt c chng c cha trong ( hoc thm ch l nhnglin tc trong ) A
Li bnh Khi nim v ton t o c c a ra bi I.E.Segal[17] v to thnh c s cho vic nghin cu l thuyt tch phn khnggiao hon , tc l l thuyt tch phn . L(X, ) ( tng ng vikhng gian o (X, ) ) c thay th bi i s von Neumann tng quthn. L thuyt ny ng vai tr chnh trong vic xy dng cc khnggian Lp kt hp vi cc i s von Neumann na hu hn l khng gianc th ca cc ton t ng xc nh tr mt .Trong [18] , E.Nelson a ra mt hng tip cn mi i hi t kin thc v k thut i svon Neumann cho l thuyt ny , da trn khi nim v tnh o ctheo mt vt (bt ngun t khi nim hi t theo o c gii thiubi W.F.Stinespring trong [16] ).Ton t o c bt k cng o dctheo ngha Segal ( trong khi iu ngc li ni chung khng ng ). Tuynhin ,tp hp cc ton t o c ln cha cc khng gianLp theo .
1.3.1 Hi t hu u trong i s von NeumannL thuyt xc sut khng giao hon l nn tng ton hc ca c hclng t, n c th coi l m rng t nhin ca l thuyt xc sut cin. Trong c hc c in, vi mi h ht im vt l c mt a tpkh vi U tng ng .Cc trng thi ca h c biu din bi cc imca U, v cc lng vt l (cc quan st c ) s c m t bi cchm (o c) trn a tp U. Trong c hc lng t, vi mi h vtl c mt khng gian Hilbert H tng ng. Vi h c s bc t do hu
hn , cc trng thi hn hp c cho bi cc ton t lp vt dng (ton t tr mt ) .Cc quan st c s c biu din bi cc ton tt lin hp hot ng trn H. i vi h ht c s bc t do v hn ,ngi ta ng nht trng thi ca h vi trng thi ( ton hc ) trnmt i s ton t A thch hp . Trong hu ht trng hp ta c th lyA l i s ton t von Neumann hot ng trn mt khng gian phc
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CH NG 1. KI N TH C CHU N B 25
(kh ly) . i s tt c cc ton t tuyn tnh b chn trn H l mt is von Neumann. Trng hp c in dn ta n i s von Neumanngiao hon L(U, Bu, ) cc hm o c b chn trn mt khng giano (U, Bu, ) . Khi cc hm o c khng b chn s c gn vo L(U, Bu, ) mt cch t nhin. o sau khi thc trin duynht thnh tch phn
U
f(d)
l mt trng thi chun tc ng trn L. Do trng hp c inc coi l sn ca i s von Neumann (giao hon).
i vi mt khng gian xc sut (, F , ) , gi L(, F , ) l i s(cc lp tng ng ) tt c cc hm nhn gi tr phc, b chn ctyu v F o c trn . N c th coi l mt i s von Neumanngiao hon hot ng trong L2(, F , ) nu ta ng nht hm g Lvi ton t nhn ag : f f g, vi f L2. i s A = L(, F , ) ctrng thi vt chun ng ( cho bi
(f) =
f d
). Theo nh l Ergoroff, s hi t hu chc chn ca dy (fn) tA l tng ng vi s hi t hu u ca n. Tc l ta c th pht
biu li hi t hu chc chn bng chun trong L , trng thi vcc hm c trng.
nh ngha 1.3.2. Gi s A l i s von Neumann vi trng thichun tc ng. Ta ni rng dy(xn) cc phn t caA hi t huu ti phn tx A nu vi mi > 0 ,tn ti mt php chiup Avi(1 p) < tha mn ||(xn x)p|| 0 khin .
Ch . nh ngha trn khng ph thuc vo cch chn do hit hu u tng ng vi hai iu kin sau:(*) Trong mi ln cn mnh ca n v trong A , tn ti php chiu psao cho ||(xn x)p|| 0 , khi n (**) Vi mi trng thi chun ng trn A v , tn ti php chiup A vi (1 p) < tha mn
||(xn x)p|| 0
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CH NG 1. KI N TH C CHU N B 26
Nhn xt. Nu l mt trng thi chun tc ng th topo mnh tronghnh cu n v S trong A c th metric ha bi khong cch
dist(x, y) = [(x y)(x y)]1/2
nh l 1.3.3. Gi s A l mt i s von Neumann vi trng thi
chun ng. Vi dy b chn cc ton t (xn) tA s hi t hu uko theo s hi t mnh ( mnh) ca(xn).
1.3.2 Cc kiu hi t hu chc chn trong i
s von Neumann
Khi nim hi t hu u l s tng qut ca khi nim hi t hu chcchn cho i s von Neumann . Ta c th xt cc phin bn khng giaohon khc ca khi nim ny.
nh l 1.3.4. Gi s A l mt i s von Neumann vi trng thichun ng .Vi mi dy b chn (xn) trong A, cc iu kin sau ltng ng
(i) Vi mi > 0 , tn ti php chiu p trong A vi (1 p) < vs nguyn dngN sao cho
||(xn x)p|| <
vin N.
(ii) Vi mi > 0 , tn ti php chiup A vi(1 p) < sao cho
||(xn x)p|| 0
khin .
(iii) Vi , tn ti dy cc php chiu (pn) trong A tng ti 1.( trongtopo mnh ) sao cho
||(xn x)pn|| <
vin = 1, 2,...
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CH NG 1. KI N TH C CHU N B 27
(iv) Vi mi php chiu khc khng p trong A tn ti php chiu khckhngq A sao cho q p v
||(xn x)q|| 0
khin
R rng trong trng hp i s von Neumann giao hon L(, F , )c 4 iu kin va thnh lp u tng ng vi hi t hu chcchn.
nh l 1.3.5. Nu l mt trng thi vt chun ng ( tc A l is von Neumann hu hn ) th c 4 iu kin trn l tng ng.
Gi s l mt vt ( hu hn hoc na hu hn ) . Xt i s
A cc ton t o c i vi (A, ) theo ngha Segal Nelson. Hi thu u (hay hi t gn u khp ni )cng c th c xt i vi dytrong A (c th l i vi dy (xn) trong L1(A, )).
nh ngha 1.3.6. Mt dy(xn) trongA c gi lhi t hu utix nu vi mi > 0, tn ti php chiup A sao cho
(p) < , (xn x)p A
vin > n0 v ||(xn x)p|| 0
khin .
nh l 1.3.7. Dy(xn) trongA ( trongA nu l mt vt ) c gil hi t hu u hai pha ti x A hay(A) nu vi mi > 0 ,tn tiphp chiup A sao cho (1 p) < v
||(xn x)p|| 0
khin
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CH NG 1. KI N TH C CHU N B 28
1.3.3 Dng khng giao hon ca nh l Egoroff
Mnh 1.3.8. Gi sA l mt i s von Neumann hot ng trongkhng gian Hilbert H. Nu dy (xi) trong A hi t mnh ti x0 th vimi > 0 , tn ti dy(pi) ProjA sao cho pi 1 mnh v
||(xi x0)pi|| <
vii = 1, 2,...
nh l 1.3.9. (nh l Egoroff khng giao hon) Gi sA l mt is von Neumann vi trng thi chun ng ; xn l dy trongA hi tnx theo topo ton t mnh . Khi vi mi php chiup A v mi > 0, tn ti php chiu q p trong A v dy con (xnk) ca (xn) saocho (p q) < v ||(xnk x)q|| 0 khik .
B 1.3.10. Gi s {xn} l mt dy ton t dng t A v {n} lmt dy s dng . Nu
n=1
1n (xn) < 1/2
th tn ti php chiup A sao cho
(p) 1 2
n=1 1
n (xn)
v ||pxnp|| 2n vi min = 1, 2,....
nh l 1.3.11. nh l Rademacher-Menchoft. Gi s 1, 2, ....l dy cc bin ngu nhin trc giao vc1, c1, .... l dy s thc tha mn
k=1c2k(lgk)
2 <
Khi chui
k=1
ckk
hi t vi xc sut1.
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CH NG 1. KI N TH C CHU N B 29
Cc k hiu s dng trong ti : A k hiu 1 i s vonNeumann hot ng trong mt khng gian Hilbert phc H ; A l hontp ca A; l mt trng thi trn A ; A+ l nn cc ton t dngtrong A ; ProjA l tp hp tt c cc php chiu trc giao trong A .Vip ProjA lun lun p = 1 p. Ton t n v trong A l 1, i vitp con Borel Z ca ng thng thc v ton t t lin hp x , k hiu
eZ(x) l php chiu ph ca x tng ng vi Z. Vi x A th |x|2 = xx.A l tp cc ton t ng , o c , xc nh tr mt. A l tp ccton t ng , xc nh tr mt v kt hp vi A
1.3.4 Khi nim v lut s ln
Mt bin c ngu nhin c th xy ra m cng c th khng xy i vimi php th. i lng ngu nhin c th ly mt trong cc gi tr c
th ca n. Nhng khi xt mt s ln nhng bin c ngu nhin hay ilng ngu nhin , ta c th thu c kt lun no m trn thc tc th xem l chc chn. Trong l thuyt xc sut ngi ta gi nhngnh l khng nh dy no nhng i lng ngu nhin hit theo xc sut v hng s l nhng nh l lut s ln . Nhngnh l lut s ln c in ( lut s ln i vi hi t theo xc sut )nh nh l Bernoulli v nh l Chebyshev , Khinchin...v lut s lni vi hi t hu chc chn nh nh l Kolmogorov.
nh ngha 1.3.12. Lut yu s ln cn c gi l nh lKhinchin Xt n bin ngu nhin X1, X2,....,Xn c lp , cng phnphi vi phng sai hu hn v k vng E(X). Khi vi mi s thc dng , xc sut khong cch gia trung bnh tch ly
Yn =X1 + X2 + ... + Xn
n
v k vngE(X) ln hn l tin v0 khin tin v v cc.
limnPX1 + X1 + .... + Xn
n E(X) = 0
nh ngha 1.3.13. Lut mnh s ln Kolmogorov Xtn bin ngunhin c lp X1, X2, ........., Xn cng phn phi xc sut vi phng saiE(|X|) < . Khi trung bnh tch ly
Yn =X1 + X2 + .... + Xn
n
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CH NG 1. KI N TH C CHU N B 30
hi t hu nh chc chn vE(X) .Tc l
P
limnYn() = E(X)
= 1
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Chng 2
Lut mnh s ln trong
i s von Neumann
Trong chng ny chng ta s cp n mt s kt qu c coi lm rng ca nh l c in cho dy bin ngu nhin c lp ( haykhng tng quan ) . D nhin ta s cn khi nim tng qut tng ngv tnh c lp trong i s von Neumann. Vic thit lp li nh nghac in khng kh. Nhng iu cn nhn mnh y l tnh c lplin quan n trng thi l mt iu kin rt hn ch ,c bit khi
khng phi l vt.Khi nim c lp trong xc sut khng giao khngng vai tr qu quan trng nh trong xc sut c in. l l do vsao cc nh l v dy ton t c lp dng nh t quan trng so vicc nh l martingale hay ergodic . Rt may l i vi trng thi vt(tracial state) th cc k thut vn tng t nh trng hp c in.Vvy ta thu c rt nhiu kt qu ng cho c trng hp giao hon vkhng giao hon theo cch lm khng khc nhiu cch lm c in .
Thay cho vic nghin cu tnh c lp, ta s nghin cu tnh trcgiao ( lin quan n trng thi ) vi iu kin km cht hn nhiu. Ccnh l lin quan n dy trc giao dng nh c ng dng nhiuhn, v s c nghin cu trong chng ny. Cc nh l nh vy clin h vi l thuyt tng quan trong cc qu trnh ngu nhin lngt ( xem [14] ) v cho ta mt s thng tin v bin thin tim cn cady quan st c khng tng quan.
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 32
Nu trng thi l vt th ta s thit lp c mt s nh l chody ton t o c. Chnh xc hn, ta s xt dy (xn) A ,vi A l*- i s t p cc ton t o c theo ngha Segal-Nelson . Cc thutng v mt s kt qu lin quan n ton t o c c th xem thmti liu trong phn ph lc.
2.1 Tnh c lp
Cho A l mt i s von Neumann vi trng thi chun tc ng (faithful normal state ) . K hiu A1, A2 l cc i s von Neumanncon ca A . Theo Batty [11] ta c 2 phin bn ca khi nim clp i vi dy ton t .
nh ngha 2.1.1. Cc i s conA1, A2 c gi l c lp ( lin quann ) nu(xy) =(x)(y) vi mix A1, y A2 .
R rng quan h c lp c tnh cht i xng.
nh ngha 2.1.2. Cc phn tx, y A ( hay tA nu l trng thivt ) c gi lc lp nu cc i s von NeumannW(x) vW(y)ln lt sinh ra bi x vy l c lp.
Dy {xn} cc phn t t A ( hay A nu l mt vt ) c gil c lp lin tip nu vi mi n , i s W(xn) c lp viW(x1, x2,....,xn1) .
nh ngha 2.1.3. H {B, } ca cc i s von Neumann concaA ( hay trongA nu l vt ) gi lc lp yu nuB c lpviW{B; {}}
2.2 Hi t hu y trong i s vonNeumann
T y ta s s dng mt s kiu hi t trong ATa cng nhn nh ngha sau:
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 33
nh ngha 2.2.1. Dy {xn} trong A c gi l hi t hu y tix nu vi mi > 0 ,tn ti dy (qn) cc php chiu trongA sao cho
n
(1 qn) <
v ||(xn x)qn|| < ; n = 1, 2, ....
Trc ht ta lu rng nu l vt th hi t hu y ko theohi t hu u.
Chng minh. Tht vy , gi s xn 0 hu y . Khi tn ti dycc php chiu qn sao cho ||xnqn|| < , n = 1, 2,... v
n(qn ) <
t :
pn =
s=n
qn
, ta c :
(1 pn)
s=n
(1 qn) 0
Tc l xn 0 hu khp ni . Theo nh l 1.3.5 th xn 0 hu u.
Khi l mt trng thi , ta c kt qu sau:
nh l 2.2.2. Gi sA l i s von Neumann vi trng thi chuntc ng , v (xn) l dy b chn trong A . Nu xn x hu y thxn x hu u.
Chng minh. Gi s ||xn|| 1 v x = 0 . Cho trc > 0. Ta s tmdy (qn) cc php chiu trong A sao cho :n
(qn ) <
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 34
v ||xnqn|| < vi n = 1, 2,...C nh dy s dng (n) tha mn:n 0 v
n=1
(1 qn)1n < /2
Theo b 1.3.10 , tn ti php chiu p A vi (p
) < v tha mn||pqnp|| = ||qnp
||2 < 2n; n = 1, 2, ....
Khi ta c:
||xnp|| ||xnqnp|| + ||xnqnp|| ||xnqn|| + ||q
np|| < + (2n)
1/2 < 2
vi n > M0()V vy iu kin sau c tha mn:
(*) Vi mi > 0 , tn ti php chiu p vi (p) 1 v tha mn||xnp|| < vi n > n0();Theo nh l 1.3.4 th xn 0 hu u ;
2.3 nh l gii hn mnh cho dy trc
giao
Trong mc ny ta s chng minh nh l gii hn mnh sau y v dytrc giao lin quan n mt trng thi .
nh l 2.3.1. [15] Gi s A l i s von Neumann vi trng thichun tc ng , v (xn) l dy trc giao tng i trong A (tc l(xnxm) = 0 vim = n)Nu :
n=1(lg n
n
)2(|xn|2) < (2.1)
Th cc trung bnh :
Sn =1
n
nk=1
xk (2.2)
hi t hu u ti0
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 35
chng minh nh l ny ta s bt u vi kt qu nh sau .
Mnh 2.3.2. Gi s (yn) l dy trc giao tng i trong A . t:
tn =n
k=1
yn (2.3)
Khi tn ti dy ton t dngBm trongA sao cho :
|tn|2 (m + 1)Bm 1 n 2
m (2.4)
v
(Bm) (m + 1)2mk=1
(|yk|2) (2.5)
Chng minh. Vic chng minh da trn tng ca Plancherel [ 7] vc trnh by trong l thuyt v chui trc giao ( xem [8] ). Ta s btu vi biu din nh phn ca ch s n ; ta chia khong I = (0, 2m]thnh cc khong (0, 2m1] v (2m1, 2m] ,mi khong ny li tip tcc chia i .... v ta s thu c dy phn hoch ca I . Cc phnt ca phn hoch u tin c di 2m1 ,ccphn t ca phn hochth r c di 2mr . i vi s nguyn dng n 2m , ta c biu dinnh phn ca n. Khi khong (0, n] c th vit thnh tng ca nhiunht m khong ri nhau I(n)j mi khong thuc mt phn hoch khc
nhau , tc l :
(0, n] =m
j=0
I(n)
j (2.6)
Vi I(n)j l tp rng hoc l khong c di |I(n)
j | ; (j=1,2,...m) Tac th vit :
tn =m
j=0
kI
(n)j
yk (2.7)
(ng nhin , ta s t kI
(n)j
yk = 0
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 36
nu I(n)j trng ) . By gi ta ch rng vi dy Z1, Z2,....,Zn trong A ,ta c :
|n
k=1
Zk|2 n
nk=1
|Zk|2 (2.8)
iu ny d dng suy ra bng quy np t bt ng thc :
xy + yx xx + yy
t:Bm =
I
|kI
yk|2 (2.9)
y I chy trn tt c cc khong l phn t ca phn hoch ca(0, 2m]. Khi ta c:
|tn|2 (m + 1)
mj=0
| kI
(n)j
yk|2 (m + 1)Bm (2.10)
Hn na Bm khng ph thuc vo n (0, 2] nn (2.5) ng .Mnh c chng minh xong .
* Chnh minh nh l 2.3.1
Chng minh. t :
SN = 1N
Nk=1
xk
. Gi s : 2k < N 2k+1. Khi :
|SN S2k|2 = |(
1
N
1
2k)
2ks=1
xs +1
N
Ns=2k+1
xs|2 (2.11)
2[(
1
N
1
2k )
2
|
2k
s=1 xs|2
+
1
N2 |
n
s=2k+1
xs|
2
]
p dng mnh 2.3.2 , ta c:
|SN S2k|2 212k[|
2ks=1
xs|2 + (k + 2)Bk]
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 37
y Bk l ton t dng , khng ph thuc vo N (2k, 2k+1] v thamn :
(Bk) (k + 2)2k+1
s=2k+1
(|xs|2) (2.12)
Do , vi 2k < N 2k+1, ta c:
|SN S2k|2 Dk (2.13)
Vi Dk A+, v
(Dk) 212k[
2ks=1
(|xs|2) + (k + 2)
2k+1s=2k+1
(|xs|2)] (2.14)
Theo gi thit ca nh l , ta c :
k=1
(Dk)
k=1
212k22k
k2
2ks=1
(|xs|2)(
lg s
s)2
+
k=1
212k22k
k2(k + 2)2
2k+1s=2k+1
(|xs|2)(
lg s
s)2
s=1 (|xs|2
)(
lg s
s )
2
(
k=12
k2 + const) (2.15)
Hn na:
k=1
(|S2k|2) =
k=1
1
22k
2ks=1
(|xs|2)
k=1
22k22k
k2
s=1
(|xs|2)(
lg s
s)2 < (2.16)
Cho trc > 0.T (2.15) ,(2.16) suy ra ta c th tm c dy s dng(k) sao cho k 0 v :
k=1
1k (|S2k|2 + Dk) < /2 (2.17)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 38
Theo b 1.3.10 tn ti php chiu p A vi (p) 1 tha mn:
||p|S2k|2p|| < 2k; ||pDkp|| < 2k (2.18)
Nh vy ,vi 2k < N 2k+1 , ta c c lng sau :
||SNp||2 + ||(SN S2k)p + S2kp||
2
2[||(SN S2k)p||2 + ||S2kp||
2]
= 2[p|SN S2k|2p|| + ||p|S2k|
2p||]
2[||pDkp|| + ||p|S2k|2p||]
< 8k 0 khiN . iu ny c ngha l :
SN =1
N
N
s=1 xshi t hu u v 0nh l c chng minh xong.
Ta s thit lp phin bn r chiu ca nh l 2.3.1
nh l 2.3.3. Gi s (x(i)n ); i = 1, 2,....,r l mt h hu hn cc dytrc giao tng i trong A ( tc l((x(i)n )x
(i)m ) = 0 vi m = n v
i = 1, 2,...,r). Gi thit :
n=1
(lg n
n)2(|x(i)n |
2) < ; i = 1, 2,...,r
Khi , vi mi > 0 , tn ti php chiu p A vi (1 p) < vtha mn:
Max1ir||1
N
Nn=1
x(i)n p|| 0; N
tc l, cc trung bnh1
N
Nn=1
x(i)n
hi t ti0 hu u v u theo 1 i r;
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 39
D nhin nu l mt vt th nh l ny l h qu tm thng canh l 2.3.1. Nu l trng thi chun tc ng tng qut , chng minhc th thu c bng cch xem xt k chng minh ca nh l 2.3.1.
Chng minh. t :
S(i)N =
1
N
N
n=1 x(i)n
V p dng mnh 2.3.2 , ta c c lng :
|S(i)N S
(i)2k
| D(i)k ; (i = 1, 2,...,r)
Vi D(i)k A+ no c tnh cht ging Dk . n y ta c th t :
Dk =r
i=1D
(i)k ; SN = (
r
i=1|S
(i)N |
2)1/2
V tip tc chng minh nh trong nh l 2.3.1
Ta s so snh nh l 2.3.1 vi cc kt qu c in .nh l Rademacher - Menchoft v s hi t hu chc chn ca chuitrc giao [8] cng vi b Kronecker cho ta lut mnh s ln sau y:
nh l 2.3.4. i vi dy(Xn) cc bin ngu nhin khng tng quan, nu :
n
(lg nn
)2var(Xn) <
Th:1
n
nk=1
(Xk EXk) 0
vi xc sut1
R rng nh l 2.3.1 c th c coi l s m rng ca lut mnhs ln ni trn cho trng hp khng giao hon .Vi mt vi iu kin mnh hn trong nh l 2.3.1 ta s thu c shi t tt hn ca dy trung bnh . Do d dng chng minh 2 nhl sau:
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 40
nh l 2.3.5. Gi s(xn) l dy trongA trc giao lin quan ti trngthi . Nu
s=1
as(|xs|2) <
khi0 < as 0 v
s=1
1s2as
<
th1
n
ns=1
xs 0
hu y .
Chng minh. t:
Sn =1
N
Ns=1
xs
, khi :
(|SN|2) =
1
N2
Ns=1
(|xs|2)
1
N2aN
Ns=1
as(|xs|2)
Do vy : (|SN|2) < Vi > 0 , ta t :
qN = e[0,2](|SN|2)
Khi :||SNqN|| < vi N = 1, 2,... Hn na :
N=1
(qN) 2
k=1
(|Sk|) <
Kt thc chng minh.nh l tip theo l kt qu mnh hn ca Batty [11] (nh l 2.3.1)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 41
nh l 2.3.6. Gi s(xn) l dy c lp yu b chn u trong A vi(xk) = 0 . Khi
1
n
ns=1
xs 0
hu y .
Chng minh. Cho trc > 0. t :
SN =1
N
Ns=1
xs
D dng ch ra rng :
(|SN|4) N4(3N2 N)
Do : N
(|SN|4) <
t :qn = e[0,4](|SN|
4)
suy ra: N
(qN) <
v ||SNqN||4 || |SN|4qN|| < 4
Vi N = 1, 2,... suy ra iu phi chng minh.
2.4 M rng khng giao hon ca nh l
Glivenko-Cantelli
Ta cn thm mt nh ngha na.
nh ngha 2.4.1. Gi s v l 2 ton t t lin hp c st nhpviA . Gie(.) ve(.) l cc o ph ca v .Ta ni rng v l cng phn phi nu (e(Z)) = (e(Z)) vi mi tp con Borel Zca ng thng thc.
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 42
Ta s chng minh kt qu tng qut ca nh l Glivenko-Cantelliv phn phi thc nghim.
nh l 2.4.2. [9] Gi s n l dy ton t t lin hp , c lp tngi v cng phn phi , st nhp viA. Khi vi mi > 0 , tn timt php chiup trong A sao cho :
sup
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 43
Nhng t nh ngha ca i,k, ta c :
(i+1,k 0) (i,k + 0) 1
k
V kt qu l :
N(i,k + 0) (i,k + 0) 1
k
N( 0) ( 0) N(i+1,k 0) (i+1,k 0) +1
k(2.19)
Nu 1,k th:
1
k N( 0) ( 0) N(1,k 0) (1,k 0) +
1
k(2.20)
V nu > 1,k th:
N( 0) ( 0) = 0 (2.21)
thun tin ta t :
N(0,k + 0) (0,k + 0) = N(1,k 0) (1,k 0) = 0
Theo (2.19) ,(2.20) ,(2.21) th vi mi s thc v k = 1, 2,... tn timt i gia 0 v k 1 sao cho :
N(i,k + 0) (i,k + 0) 1k
N( 0) ( 0)
N(i+1,k 0) (i+1,k 0) +1
k;
do vy vi php chiu p ty t A ta c:
p[N(i,k + 0) (i,k + 0)]p 1
kp
p[N( 0) ( 0)]p
p[N(i+1,k 0) (i+1,k 0)]p +1
kp
V ta thu c kt qu:
||p[N( 0) ( 0)]p||
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 44
max1ik,=0 ||p[N( + ) (i,k + )]p|| +1
k
Ko theo :
sup
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 45
v:||(sn (sn))(1 q)|| n 1
Chng minh. Ta c th gi s (xn) = 0. Ta s nh ngha cc dy phpchiu pn, qn trong A bng quy np. t q0 = 0 . Cho trc qn1 ,t:
pn = e(2,){(1 qn1)snsn(1 qn1)}
v :qn = pn + qn+1
R rng cc pn trc giao ,
qn =N
r=1
pr
, pn v qn thuc W{xn : r n}
.T tnh c lp v cc tnh cht ca vt th vi r n:
(pr|sn|2pr) = (pr((sn sr) + sr)
((sn sr) + sr)pr)
((sn sr)srpr) + (prs
r(sn sr)) + (prs
rsrpr)
= ((sn sr)srpr) + (prs
r(sn sr)) + (prs
rsrpr)
= (pr(1 qr1)srsr(1 qr1)pr) 2(pr);v:
(qn) 2
nr=1
(snsnpr) = 2(snsnqn)
2(snsn) = 2
nj=1
||xj||22
t :
q =
r=1
pr
, khi :
(q) 2n
j=1
||xj||22
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 46
, v:||(1 q)snsn(1 q)||
||(1 pn)(1 qn1)snsn(1 qn1)(1 pn)||
2
Nh l h qu ca nh l trn, ta nhn c dng suy rng sau yca nh l Kolmogorov.
nh l 2.5.2. Gi sxn l dy c lp lin tip trongL2(A, ) sao chochui
n=1
||xn (xn)||22
hi t , v gi
sn =
n
k=1 xk. Khi dysn (sn) hi t hu u.
Chng minh. Ta c th gi s rng xn l t lin hp v (xn) = 0 . Chns nguyn nk sao cho
n=nk
(x2n) < 8k
Theo nh l 2.5.1 tn ti cc php chiu qk vi (qk) < 2k
tha mn||(sn sm)(1 qk)|| < 2
(k1)
vi m, n nkNn {sn} l dy Cauchy theo o . Gi s l gii hn theo o can . Khi ta c:
||(sm s)(1 qk)|| < 2(k+1)
vi n > nk.V vy sn s trn cc tp ln v do hi t hu u .
S dng b Kronecker ta thu c :
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 47
nh l 2.5.3. Gi s{xn} l dy c lp lin tip trongL2(A, ) saocho
n=1
n2||xn (xn)||22 <
. Khi
n1
n
k=1
(xk (xk))
hi t hu u .
Ch rng t kt qu va nu ta c th suy ra m rng sau y calut mnh s ln Kolmogorov.
nh l 2.5.4. Gi s {xn} l dy cc ton t cng phn phi , clp lin tip v t lin hp trongL1(A, ) . Khi :
n1n
k=1
xk (x1)
hu u.
Ta b qua chng minh ca nh l ny v trong phn sau ta s nura kt qu tng qut hn.
2.6 Lut mnh s ln i vi vt
Ta s bt u vi nh l 2.5.2 t suy ra m rng ca phn
ca nh l ba chui Kolmogorov.
B 2.6.1. Gi s{k} l dy ton t o c c lp lin tip t A.Vid > 0 , t
k = ke[0,d]{|k|}
. Khi cc iu kin :
(i)
k (e(d,){|k|}) <
(ii)
k (kk) <
(iii)
k (k) hi t .
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 48
Ko theo s hi t hu u ca chui
k k;
Chng minh. Theo nh l 2.5.2 v (ii) chuik
(k (k))
hi t hu u , v do theo (iii) chuik
k
hi t hu u . iu ny cng vi (i) cho ta s hi t hu u cak k.
Tht vy, t :
sn =
n
k=1 k;
sn =n
k=1
k
R rng ch cn chng minh {sn} l dy Cauchy hu u , tc l :Vi mi > 0 , tn ti p ProjA (tp hp cc php chiu ca A) vi(p) 1 v s nguyn dng N sao cho
||(sn sm)p|| <
vi n, m N.V chui k
k
hi t hu u nn vi mi > 0 , ta tm c q ProjA v N sao cho
||( sn sm)q|| <
vi n > m N v (q) < /2. t :
p = q k=m+1
e[0,d]{|k|}
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 49
theo (i) ta c:
(p) (p) +
k=m+1
(e(d,){|k|}) <
Vi m ln (m N) . Khi :
||(sn sm)p|| ||( sn sm)q|| <
vi n > m NDo {sn} l dy Cauchy hu u . Kt thc chng minh.
Bin i mt cht nh l trn ta c kt qu sau:
B 2.6.2. Gi s {n} v {n} l cc dy ton t o c t A v{cn} l dy s dng . t:
n = ne[0,cn]{|n|}.
Nu n
(e(cn,){|n|}) <
th chui n
(n + n)
hi t hu u khi v ch khi chuin
(n + n)
hi t hu u.Ni ring nu
n
(e(cn,){|n|}) <
, th chuik k hi t hu u khi v ch khi chuik
k
hi t hu u .
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 50
Chng minh. t :
sn =n
k=1
(k + k)
;
sn =n
k=1(k + k)
V x l cc hiu (sn sm), ( sn sm) nh cch ch ra trong chngminh nh l trn.
nh l 2.6.3. Gi sn : R+ R+ l dy hm khng gim sao chon()/ v2/n() khng gim vi min .Gi s
0 < xn
Nu {k} l dy ton t c lp lin tip lin kt vi A v tha mn(j) = 0 v
k=1
(n|k|)
(xk)< (2.22)
Th chui
k=1
kxk
(2.23)
hi t hu u.
Chng minh. t :
n = ne[0,xn]{|n|} = ne[0,1]{|nxn
|}; n = 1, 2, ....
Khi ta c :(e[xn,]){|n|})
(n|n|)
n(xn)
T (2.22) ta thu c :
n=1
(e(1,){|nxn
|}) < (2.24)
Hn na , v :2
x2n
n()
n(xn)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 51
vi 0 xn nn ta c:
(|n|2)
x2nn(xn)
(
xn0
n()ed{|n|})
x2nn(xn)(n|n|)
V do :
n=1
n
xn
2= n=1
n
xne[0,1]{|
nxn
|}2< (2.25)
V : (n) = 0 nn ta c :
|(n)| (|n|e[xn,){|n|}) (
|
n|)
(xn)xn
(V
n()
n(xn)xn
vi xn). Do vy :
n=1 n
xn
n=1(n|n|)
(xn)< (2.26)
Theo b 2.6.1 v (2.24) ,(2.25) ,(2.26) ta thu c (2.23). Kt thcchng minh.
By gi ta s chng minh lut mnh s ln m khng cn gi thitv tnh hu hn ca cc m men thng ca n.
nh l 2.6.4. Gi sn : R+ R+ l dy hm khng gim tha mn
2
/n() khng gim , v gi s0 < xn
, {n} l dy ton t o c c lp lin tip . t:
n(d) = (ed{|n|})
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 52
Nu :
n=1
0
n()
n(xn) + n()n(d) < (2.27)
Th tn ti dy hng s (Ck) sao cho :
1xn
nk=1
(k Ck) 0 (2.28)
hu u.V trong trng hp ny ta c th t :
Ck = (ke[0,xk]{|k|}) (2.29)
Chng minh. i vi hm khng gim : R+ R+ v ton t vi
|| =
0
e(d)
ta t(.) = (e(.))
th s c c lng sau ( vi x > 0 )
0
()
(x) + ()(d)
1
2(x)
x0
()(d) +1
2
x
(d)
=1
2(x)((||e[0,x]{||})) + (e(x,){||})
=1
2(x)((||e[0,x]{||})) + (e(x,){||})
V vy ta c : n
(n|n|)
(xn)< (2.30)
V: n
(e(xn,){|n|}) < (2.31)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 53
Vi:n = ne[0,xn]{|n|} (2.32)
Hn na :
(|n|2)
x2nn(xn)
(n|n|)
V do : n
(|n|2)
x2n<
iu ny ko theo s hi t ca chui :
n
n (n)
xn
V do vy theo b 2.6.2 ta suy s hi t hu u ca chui
n
(n)
xn
By gi ta ch vic p dng b Kronecker.
Hai nh l sau y chnh sa mt s kt qu ca W.Feller ( i vibin ngu nhin thc)
nh l 2.6.5. Gi s{k} l dy ton t t lin hp , cng phn phi
i xng v c lp lin tip , lin kt vi (A, ) .Gi s rng l hmkhng gim sao cho 2/() khng gim.Gi thit: 0 = x0 < x1 < x2, .... v
k=n
1
(xk)= 0(
n
(xn)) (2.33)
Khi iu kin :
n (e(xn,){|1|}) < (2.34)Ko theo:
1
xn
nk=1
k 0 (2.35)
hu u .
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 54
Chng minh. t:n = ne[0,xn){|n|}
. Khi :
n=1
(|n|)
(xn)=
nk=1
(|1|e[xk1,xk){|1|})
n=k
1
(xn)
C
k=1
k[xk1,xk)
()
(xk)(ed{|1|})
C
k=0
(e(xk,){|1|}) <
Do (2.30) ng. Hn na (2.31) tha mn theo gi thit v (n) = 0.( v n phn phi i xng ). Lp li phn chng minh ca nh l (2.6.4)
sau cng thc (2.31) ta thu c (2.35) . iu phi chng minh.Trong trng hp n khng phn phi i xng ta cn hn ch thm
iu kin i vi {xk}
nh l 2.6.6. Gi s{n} l dy cng phn phi , c lp lin tip, v{xn} c m t nh trong nh l (2.6.4)Nu thm vo
(1) = 0;xkxn
C0k
n; k n (2.36)
th (2.34) ko theo (2.35).
Chng minh. Tng t nh l (2.6.5) ta ch ra rng :
1
xn
nk=1
(k (k)) 0
hu u. V do :1
xn
n
k=1
(k) 0
Ta c c lng sau:
1
xn
nk=1
(n) 1
xn
nk=1
xk
(ed{|1|})
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 55
Do s dng phn tch cc :
k k = uk|k k|
ta c:|(k k)| ||uk||(|k k|) = (|k k|)
v|k k| = |k|e(xk,){|k|}
V vy :1
xn
nk=1
(k) 1
xn
nm=1
mk=1
+
m=n+1
nk=1
xm+1xm
(ed{|1|}) S1 + S2
Vi :
S1 =1
xn
nm=1
mxm+1(e[xm,xm+1){|1|})
S2 =1
xn
nm=n+1
nxm+1(e[xm,xm+1){|1|})
S1 0 v :n
m=0
(m + 1)(e[xm,xm+1){|1|})
=
m
(e(xm,){|1|}) <
Theo gi thit v do ta c th p dng b Kronecker.Lp lun tng t cho S2 . Chng minh c hon thnh.
By gi ta s chng minh kt qu cho trng hp khng giao hon
ca lut s ln Marcinkiewicz. Ta bt u vi 2 mnh sau:Mnh 2.6.7. Gi s{n} l dy c lp lin tip trongA v
n=1
10
(e[,){|n|})d <
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 56
Khi chui
n=1
( (n))
hi t hu u , y :
n = ne[0,1){|n|}
Chng minh. Tch phn tng phn , ta c:
10
(e[,){|n|})d =1
2(e[1,){|n|})
+1
2
1
02(ed{|n|})d =
1
2(e[1,){|n|}) +
1
2(|n|
2)
V do , theo gi thit ta suy ra:n
(|n|2) <
v n
(e[1,){|n|}) <
Bt ng thc th nht cng vi nh l 2.6.2 cho ta s hi t hu uca chui
n
(n (n))
v cng vi bt ng thc th hai v b 2.6.2 suy ra s hi t huu ca
n=1
(n (n))
.
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 57
Mnh 2.6.8. Gi s{n} xc nh nh trn v
n=1
10
(e[bn,){|n|})d <
vi{bn} l dy hng s dng , bn . Khi :
(i)
n=1
b1n (n (n))
hi t hu u , v
(ii)
b1n
n
k=1(k (k)) 0
hu u
yn = ne[0,bn){ n}
Chng minh. Ta c :
1
0(e[bn,){|n|})d =
1
0(e[,){
n
bn })d
V theo mnh 2.6.7 chui
n=1
nbn
n
bn
hi t hu u ,vi :
nbn
=
n
bn e[0,1){|
n
bn |}
=nbn
e[0,bn){|n|}
V vy ta c (i)p dng b Kronecker ta thu c (ii).
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 58
By gi ta s chng minh kt qu tng t ca Marcin Kiewicz theoLuczak [9]
nh l 2.6.9. Gi s{n} l dy ton t c lp lin tip , cng phnphi , tLr(A, ), 0 < r < 2. Khi
n
1/rn
k=1(k k) 0hu u. yk = 0 vir < 1 vr = (1) vi1 r < 2
Chng minh. Vi phn phi xc sut trn na ng thng thc:
F() = (e[0,){||}), 0, A,
ta c c lng sau:
r
n=1
(e[n1/r ,){||}) (||r) 1 + r
n=1
(en1/r,){||}) (2.37)
Chng hn xem [10] . S dng (2.37) ta thu c:
n=1
10
(en1/r,){|1|})d
(|1|r)
1
0
1rd <
Nu r < 2 bt ng trn cng vi mnh 2.6.8 cho ta :
n1/rn
k=1
(k (k)) 0
hu u vi :k = ke[0,n1/r]{|k|} 0
Lp lun tiu chun ch ra rng :
n1/rn
k=1
((k) k) 0, n
(Chng hn , xem [10], chng minh hon tt.)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 59
Ta s kt thc phn ny vi mt s lut yu s ln trong i svon Neumann.
nh l 2.6.10. Cho {n} l dy cc ton t t lin hp o c , cngphn phi , c lp lin tip t A . Nu :
limn
n(e(n,){|1|}) = 0
th :
n1
k=1
(k n) 0
theo o, vi :n = (1e[0,n){|1|})
Chng minh. Gi s A tc ng trong khng gian Hilberte H, t :
Sn =n
k=1
k; (n)k = ke[0,n){|k|}
Sn =n
k=1
(n)k ; mn = (Sn) = n(
(n)1 )
Vi s > 0 bt k , ta c:
p = e[2,)(|Sn mn|) e[0,)(|Sn mn|)
n
k=1
e[0,n){|k|} = 0
Tht vy, nu vi x no c chun bng 1 , px = x th
x e[0,n){|k|}(H)
v do kx = (n)k x
vi k = 1, 2,...n.Ko theo Snx = Snx .V th ta thu c :
2 || |Sn mn|e(2,){|Sn mn|x}||
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 60
= ||(Sn mn)x|| ||(Sn Sn)x|| + ||(Sn mn)x||
= || |Sn mn|e[0,){|Sn mn|x}||
iu ny khng th xy ra, vy p = 0 v do (xem ph lc)
e[2,){|Sn mn|} e[,){|Sn mn|} n
k=1 e[n,){|k|} (2.38)Theo bt ng thc Tchebyshev v t kt qu :
(| ()|2) (||2)
Ta thu c :
(e[2,)(|Sn mn|)) 2(|Sn mn|
2) + (n
k=1e[n,)(|k|))
2n
k=1
(|(n)k |
2) +n
k=1
(e[n,))
= 2n(|(n)1 |
2) + n(e[n,)(|1|)) (2.39)
By gi ly > 0 ty , t (2.39) vi =n
2, ta c:
e[,)(|Snn n|)= e[,)(|Sn mnn |)=
e[n,)(|Sn mn|)
4(|
(n)1 |
2)
2n+ n
e[n,)(|1|)
= 42n1
[0,n)
2
ed(|1|)
+n
e[n,)(|1|)
Tch phn cui cng c th c lng nh sau:
[0,n)
2
ed{|1|}) n1k=0
(k + 1)2(e[k,k+1){|1|})
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 61
=n1k=0
(2k + 1)(e[k,n){|1|}) (e[0,n){|1|})
+3n1k=1
k(e[k,n){|k|}) 1 + 3n1k=1
k(e[k,){|1|})
Do vy cui cng:(e[,){|
Snn
n|})
42n1[1 + 3n1k=1
k(e[k,){|1|})] + n(e[n,){|1|}) 0
khi n Chng minh c hon tt.
S dng k thut tng t ta c th chng minh kt qu sau calut yu s ln.
nh l 2.6.11. Cho {n} l dy phn t t lin hp c lp lin tiptL1(A, ). Nu:
(i)n
k=1
(e[n,){|k k|}) 0
khin
(ii)n1
nk=1
(|k k|) 0
khin ,vi || = ||e[0,n){||} v :
(iii)
n2n
k=1
([|k k|2]) 0
khin vik = (k) th :
n1n
k=1
(k k) 0
theo o.
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 62
2.7 Tc hi t trong lut mnh s ln
Cc kt qu trong phn ny c trong t tng ca cc nh l Spitzer-Hsu-Robbins-Katz i vi bin ngu nhin thc. Chng ta s i theophng php da trn c lng tt ca cc m men ca binngu nhin b cht ct nu ra bi Dugue [12] (xem thm [13])
Trong phn ny {n} s k hiu cho dy ton t c lp lin tip t A .Mt vi kt qu s lin quan ti trng hp n t lin hp v cng phnphi .Trong cc nh l khc ta s gi thit n khng nht thit t lin hpnhng cc ton t (|k|, k = 1, 2, ....) cng phn phi vi r > 0 v t > 0ta t :
(n)k =
(n)k,r,t = ke[0,nr/t]{|k|}
, 1 k n V :
(n)k =
(n)k,r,t =
(n)k,r,t (
(n)k,r,t)
Ta s thit lp hai mnh ph tr sau:
Mnh 2.7.1. Cho {n} c lp v tha mn |k| cng phn phi
(A) Nu0 < t < 2 vr > 0 th iu kin(t1) < ko theo
n=1
nr2(e(nr/t,){|n
k=1
nk |}) < (2.40)
(B) Nu0 < t < 1, r 1 th iu kin(|1|t) < ko theo
(|(n)1 |) = 0(n
r/t1) (2.41)
Chng minh. Gi s
o < t < 2, r > 0, , (|1|t) <
Khi theo bt ng thc Tchebysev ta c :
n=1
nr2(e(nr/t,){|n
k=1
(n)k |})
const
n=1
nr2n2r/tn(|(n)1 |
2)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 63
i vi o xc sut trn [0, ) ,
0
xt (dx) <
v r
t > 0
ko theo:
n=1
1
n+1
nr/t0
xt+ (dx) (2.42)
const
k=1k(r/t)
kr/t
(k1)r/txt (dx)
( so snh [12] ). Trong trng hp ca chng ta th :
(|(n)1 |
2) = nn(r/t)(2t)+1
vi 1
n <
V do ta c (2.40)By gi ta s chng minh (2.41) vi gi thit (B) . Tht vy, t:
||1 =
0
ue (du)
(biu din ph) v (du) = (e(du)). Ta c :
n1r/t(|(n)1 |)
nr/t0
xt( xnr/t
)1t (dx) 0
v r 1 v (||t) <
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 64
Mnh 2.7.2. Gi s {n} c lp vi {|k|} cng phn phi v(||2) < . Khi ta c:
m=n
(e(n,){|n
k=1
(n)k |}) < (2.43)
Chng minh. Hin nhin ta c:
(|n
k=1
(n)k |
4) constk=1
(|(n)k |
2|(n)1 |
2)
+n
k=1
(|(n)k |
4)
Hn na:(|(n)k |4) = nn3 + (|(n)1 |4)
Vi:
n=1
n <
(v chng hn: (|
(n)k |
2(n)k )
(1
2||
(n)k ||) const.n
). S dng cng thc (2.42) ta thu c :
(|(n)1 |
4) n3n
vi
n=1
n <
V vy:
n
k=1
(n)k
4 nn4vi
n=1
n <
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 65
v do :
nn4
nk=1
(n)k
4 (n)k 4e(n,){n1
(n)k
}
const.n4e(n,){n
1 (n)k }
Cui cng ta thu c:
n=1
e(n,){ n
k=1
(n)k
}<
nh l 2.7.3. Cho {n} c lp v |k| cng phn phi , 0 < t < 1.Khi iu kin (|1|t) < ko theo
n=1
n1
e(,){ 1
n1/t
nk=1
k
}< (2.44)vi mi > 0
nh l 2.7.4. Cho{n} c lp , t lin hp v cng phn phi. Githit : (|1|t) < , 1 t 2 v(1) = 0. Khi :
n=1
nt2
e(,){1
n
nk=1
k
}< (2.45)ng vi mi > 0.
Chng minh. By gi ta s thit lp 2 iu kin :
(a) {n} c lp v |k| cng phn phi
0 < t < 1, (|1|t) < , r 1
(b) {n} c lp ,cng phn phi ,t lin hp
1 t 2, (|1|t) < , (1) = 0, r = t
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 66
R rng chng minh cc nh l trn ,ta ch cn chng minh rngmi mt trong cc iu kin (a),(b) ko theo:
n=1
nr2
e(nr/t,){ n
k=1
k
}< (2.46)
lm iu ny ,ta s t vi r > 0, t > 0, n = 1, 2, ....
pn = pn,r,t = e(nr/t,){|n
k=1
k|}
qn = qn,r,t =n
k=1
e(nr/t,){|k|}
n = n,r,t = e((n1)r/t,){|
n
k=1
(n)
k,r,t|}
Ch rng :pn q
n
n = 0
vi mi n. Tht vy, nu c :
0 = x = pn qn
n x
th ta s c :
||n
k=1
kx|| = || n
k=1
k
pnx|| nr/t||x||V cng lc :
||
nk=1
kx|| ||
nk=1
(n)k
x|| < (n 1)r/t||x||
L iu khng th xy ra. V vy, ta c:
pn qn n
v do (pn) (qn) + (n)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 67
n y ta cn ch ra c:
n=1
nr2(qn) < (2.47)
V:
n=1
nr2
(n) < (2.48)
Nu t > 0, r > 0 v (|1|t) < th r rng
n=1
nr2(qn,r,t)
n=1
nr1(e(nr/t,){|1|}) <
V vy ch cn chng minh (2.48) .Gi s (a) ng, ta s chng minh :
Qn = n e[0,(nr/t)/4]{|n
k=1
(n)k |} = 0 (2.49)
Thc ra nu tn ti 0 = x = Qnx th ta s c :
||n
1(n)k x|| (n
r/t)/4||x||
v:
||n1
(n)k x|| (n 1)
r/t||x||
t:(n)k = (
(n)k ), n = max1kn|
(n)k |
p dng phn tch cc ca (n)k v theo mnh (2.7.1.b) ta c :
n (|(n)1 |) < 14
nr/t1
V vy vi n , ta thu c :
||n1
(n)k x|| ||
n1
(n)k x|| + n|n|.||x||
nr/t
2||x||
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 68
iu ny l khng th ,do (2.48) ng. iu ny ko theo:
n e((nr/t)/4,){|n1
(n)k |}
Xem xt ti (2.40) ta thu c (2.48) .
By gi ta gi s (b) ng. Nu 1 t < 2 th (2.40) ng (vi r = t vp dng mnh (2.8.a)) .Trong trng hp t = 2 ta c (2.43) theo mnh (2.7.2).Ch rng
n = |((n)1 )| 0
, ta c th chng minh (2.49) ( vi t = r) v kt thc chng minh nhtrng hp (a).
2.8 Ch v ch thch
Batty a ra rt nhiu khi nim v dy ton t c lp. Chng ta chx l dy c lp yu v c lp lin tip.
nh ngha 2.8.1. H {B, } cc i s von Neumann con caAc gi l c lp mnh nu W{B, 1} c lp vi W{B, 2} vi mi tp con ri nhau 1, 2 ca . Dy ( hay h ) ton t
trong A (hay A nu l vt) l c lp mnh nu cc i s vonNeumannW() c lp mnh.
Trong [11] ta c th tm thy trnh by c th lin quan ti mi quanh gia cc kiu c lp. y ta cp n mt lut s ln mnh c nu ra bi S.M.Goldsteinlin quan n dy ton t tha mn iu kin Rosenblatt.
nh ngha 2.8.2. Cho A l i s von Neumann vi trng thi chun
tc ng . Dy {n} t A c gi l tha mn iu kin Rosenblattnu tn ti dy s dng{an} gim v0 sao cho:
|(xy) (x)(y)| an||x||.||y||
vi mi x W(1, 2, ....k) v y W(k+n, k+n+1, ....); (k, n =1, 2,...)
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CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 69
Dy{n} c gi l tha mn iu kin ||.||2 (||.||) hay tim cn giaohon nu tn ti dy s dng{bn} gim v0 sao cho :
||xy yx||2 bn||x||||y||
(hay tng ng
||xy yx|| ||x||||y||) vi mi
x W(1,....,k), y W(k+n, k+n+1, .....)
(k, n = 1, 2, ....).
nh l 2.8.3. Cho {n} l dy t A tha mn iu kin Rosenblattcng vi dy{an} v t :
bn = sup|ij|n||ij ji|| (n = 1, 2, ....)
Gi s rng() an c1n
1, bn c2n2
vic1, c2, 1, 2 l cc hng s dng, n = 1, 2,... v :
() sup||n|| <
Khi :
n1
n
k=1
(k (k)) 0
hu u .
Chng minh. Ta cng nhn nh l trn.
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Kt lunLut s ln l mnh khng nh trung bnh s hc ca cc bin
ngu nhin hi t theo xc sut. Lut mnh s ln l mnh khngnh trung bnh s hc ca cc bin ngu nhin hi t hu chc chn.
Lut s ln u tin c cng b vo nm 1713 bi Jamer Bernoulli.Sau kt qu c Poisson ,Chebyshev ,Markov ,Liapunov m rng.
Lut mnh s ln c pht hin bi E.Borel nm 1909 v c Kol-mogorov hon thin nm 1926.Lun vn cp n mt vn hon ton mi trong xc sut
hin i , l s quan h cht ch gia i s ton t , vt l lngt v xc sut. Cc kt qu , phng php , m hnh trong xc sut cin c thay th bi cc k thut chng minh, cc khi nim ,nhngha mi. Nghin cu lut mnh s ln da trn cc ton t trc giao,vt , trng thi ,i s von Neumann,...cc dng hi t hu u ,
hu chc chn ...Ni dung ca ti vn ang l hng nghin cu cacc nh khoa hc trn th gii, nhiu cng trnh khoa hc khng ngngc cng b nh:L thuyt tch phn khng giao hon c sng tobi I.Segal (1953), c p dng vo l thuyt biu din cc nhm com-pact a phng ( Ray Kunze....) . Ngy nay n c nhiu ng dng quantrng trong l thuyt trng lng t ,iu ny c d bo trongcc cng trnh ca Segal ...Hay l thuyt v khng gian Lp tru tng
ca J.Dixmier ; Khi nim hi t theo o (tnh o c theo mt vt)c gii thiu bi W.F.Stinespring (1959), E.Nelsen(1972)...V gn y
nht l cc l thuyt v chng minh khi ngun ca Haagerup(1974) vkhng gian tru tng Lp kt hp vi i s von Neumann. Sau s xuthin cc khng gian Lp ca Haagerup, A.Connes a ra nh nghav khng gian Lp da trn khi nim o hm khng gian. Nhng khnggian ny c nghin cu bi M.Hilsum.....
Vn c cp trong lun vn tng i mi v phc tp.V
70
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Kt lun 71
vy lun vn khng trnh khi nhng hn ch .Tc gi mong mun nhnc kin ng ca cc thy c gio ,cc ng nghip b sung ,hon thin ti.
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Ti liu tham kho[1] V Vit Yn-Nguyn Duy Tin (2001) ,L thuyt xc sut, Nh xut
bn gio dc.
[2] Nguyn Vit Ph-Nguyn Duy Tin (2004), C s l thuyt xc sut, Nh xut bn i hc Quc gia.
[3] Nguyn Duy Tin (2000), Gii tch ngu nhin ,tp 3, Nh xut bni hc Quc gia H Ni.
[4] Ryszard Jajte (1984), Strong Limit Theorems in Non-CommutativeProbability, Springer -Verlag, Berlin New York Tokyo.
[5] Marianna Terp(1981), Lp Spaces Associated with von Neumann Al-gebras , Universitetsparken.
[6] Edward Nelson(1972) ,Notes on Non-commutative Integration,Princeton University, New Jersey.
[7] M.Plancherel (1913), Sur la convergence des series de fonctions or-thogonalles, Acad . Sci. Paris.
[8] G.Alexits (1961), Convergence problems of orthogonal series, NewYork- Oxford-Paris.
[9] A.Luczak, Some limit theorems in von Neumann algebras, Studia
Math.
[10] M.Loeve(1960), Probability theory , New Jersey.
[11] K.Batty(1979), The strong law of large numbers for states and tracesof aW algebra, Z.Wahrscheinlichkeitstheorie verw .
72
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T I LI U THAM KH O 73
[12] D.Dugue(1958), Traite de statistique theorique et appliquee, Paris.
[13] S.Goldstein(1981), Theorems in almost everywhere convergence invon Neumann algebras, J.Oper.Theory 6.
[14] L.Accardi(1980), Quantum stochastic processes, Dublin Institute forAdvanced Studies, Ser.A29.
[15] R.Jajte , Strong limit theorems for orthogonal sequences in von Neu-mann algebras, Proc.Amer .Math.Soc.
[16] W.Stinespring(1959), Integration theorems for gages and duality forunimodular groups, Trans.Amer.Math.Soc.
[17] I.E.Segan(1953), A non-commutative extension of abstract integra-tion, Ann.of Math.57.
[18] A.Zygmund (1959), Trigonomtric Series, Vol.II, Cambridge Uni-vesity Press, London, New York.