Το Θε‚ρηmα Μοναδικìτηταc των Stone και von...
Transcript of Το Θε‚ρηmα Μοναδικìτηταc των Stone και von...
-
Stone von Neumann
2011
-
ii
L2(R,m) unitary {S }R {Tt}tR, (S f )(x) = eix f (x) (Tt f )(x) = f (x t). Weyl:
S Tt = eitTtS , , t R,
Heisenberg -. Stone von Neumann , - SOT- unitary {U}, {Vt}t Hilbert H , Weyl , unitary R : L2(R,m) H , U = RS R Vt = RTtR, , t R.
, - . von Neumann., unitary , S (n) , n N {+}. von Neumann, - Stone, SOT- unitary . , unitary , - T (n)t , S
(n) .
n = 1. .
-
iii
Abstract
On the Hilbert space L2(R,m), we define the one parameter unitary groups {S }R and{Tt}tR, where (S f )(x) = eix f (x) and (Tt f )(x) = f (x t). These two groups satisfy theWeyl relations :
S Tt = eitTtS , , t R,
which is a form of Heisenbergs Uncertainty Principle of Quantum Mechanics. The theo-rem of Stone-von Neumann states that for every two jointly irreducible SOT-continuousone parameter unitary groups {U}, {Vt}t on a separable Hilbert space H , which satisfythe Weyl relations, there is a unitary operator R : L2(R,m) H , mapping U to S and Vt to Tt.
In this study we give a proof of the above theorem, using the theory of abelian vonNeumann algebras.In particular, we first prove that after a unitary equivalence, we may assume that U isS (n) , for some n N{+}. The crucial tools here are the multiplicity theory of abelianvon Neumann algebras and Stones theorem, which characterizes the SOT-continuousone parameter unitary groups.The second step is to prove that there is a unitary operator which carries Vt to T (n)t andleaves S (n) fixed. The fact that {U}, {Vt}t act jointly irreducibly on H then forces n = 1.Our main arguments in this step are measure-theoretic.
-
iv
, Hilbert. , , : . - , Q P, :
D(Q) = { f L2(R,m) : id f L2(R,m)} (Q f )(x) = (id f )(x) = x f (x)
D(P) = { f L2(R,m) : f AC(R), f L2(R,m)} 1 (P f )(x) = i f (x)
D(PQ) D(QP) L2(R,m), Schwarz, f :
(PQ QP) f = i f
(canonical commutation relation-CCR).
P,Q, A, B, Hilbert , unitary U : H L2(R,m), UAU1 = P UBU1 = Q. A, B CCR . , :
M f : L2[0, 1] L2[0, 1] : g 7 f g, f (x) = x,x [0, 1]
P1 : D(P1) L2[0, 1] : g 7 ig, D(P1) = { AC[0, 1] : (0) = (1)}
1AC(R) = { f : R 7 C : f }
-
v
D(P1M f )D(M f P1), M f , P1 CCR, P1 {2k : k Z}, , , R.
Stone([11]), 1-1 SOT- unitary . , ([10]) Q, P eiQ = S , (S f )(x) = eix f (x) eitP = Tt, (Tt f )(x) = f (x t), :
S Tt = eitTtS
, SOT- unitary {U}, {Vt}t, Weyl :
UVt = eitVtU , , t R. (1)
A, B , , A B, CCR.
SOT- - unitary , Weyl . Johnvon Neumann Marshall Stone 1931 : unitarily -, {U}, {Vt}t , unitary W : H L2(R,m), WUW1 = S WVtW1 = Tt.
Stone - von Neu-mann , vonNeumann . Lectures on InvariantSubspaces H. Helson. , Banach Fourier .
. . -
-
vi
. . . - , , , , .
-
1 1
1.1 B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 7
2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 . . . . . . . . . . . . . . . . . . . . 16
3 von Neumann 21
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 von Neumann . . . . . . . . . . . . . . . . . . . . . 223.3 . . . . . . . . . . . . . . . . . . . . . 253.4 von Neumann . . . . . . . . . . . . . . . . . . . . . . . 27
4 37
4.1 . . . . . . . . . . . . . . . . . . . . 37
5 Weyl 47
5.1 Stone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Stone-Von Neumann . . . . . . . . . . . . . . 52
vii
-
1
1.1 B(H)
1.1. H Hilbert . x H , px : B(H) R : T 7 T x
B(H). , {px : x H} - B(H), , (Strong Operator Topology, SOT). SOT- T0 B(H), :
V(T0 : x1, . . . , xn; ) = {T B(H) : (T T0)x j < , ( j = 1, . . . ,m)}, x1, . . . , xn H > 0. (Ti)iI T B(H) SOT- Tix T x 0, x H .
Hilbert H . H , :
P Q P(H) Q(H), P,Q B(H).
1.2. {Ni : i I} Hilbert H Ni , - Ni, Ni , Ni.
Ni =iI
Ni, Ni
1
-
2 1.
iI
Ni. - {Pi = pro j(Ni), i I} supremum Pi = pro j(Ni), in f imum Pi = pro j(Ni).
1.1. (E)A Hilbert H E =
A
E. E =SOT-lim
E
. (E(H)) , A
E(H)
H , E(H) = A
E(H). x H > 0. Ex E(H), y E(H), A, Ex y < . b , E Eb E, y E(H) Eb(H) E(H). :
Ex Ebx = E(Ex y) Eb(Ex y) E Eb Ex y < .
1.1. (E)A Hilbert H E =
A
E. A
E = I A
E E =SOT-lim
E
1.2. (E)A , Hilbert H E =
AE. x H Ex =
Ex.
. , . .
F = {F : F A}. F F GF =
F
E =F
E. ({GF : F F },) ,
FF
GF =F
E = E.
: Ex =
FFGF x =
Ex, .
1.3. HilbertH . (weak opera-tor topology,WOT) :
x,y : B(H) C : T 7 T x, y. x,y(T ) = 0, x, y H , T = 0, {x,y : x, y H} B(H). WOT- .
-
1.1. B(H) 3
:V(T0 : x1, . . . , xn, y1, . . . , yn; ) = {T B(H) : |(T T0)x j, y j| < , ( j = 1, . . . , n)},
x1, . . . , xn, y1, . . . , yn H > 0, T0 B(H), WOT-. , (Ti)iI Tix, y T x, y,x, y H .
1.1. : B(H) C WOT- SOT-.
. (i) SOT- WOT, WOT-, SOT-.
(ii) SOT-. SOT- 0,
W = {T B(H) :n
k=1T xk2 < 2},
|(T )| < 1, T W . yk = 2 xk p(T ) =( n
k=1Tyk2
)1/2,
W = {T B(H) : p(T ) < 2}.
: |(T )| ( nk=1Tyk2
)1/2, T B(H).
p(T ) = 0, mT W, |(mT )| < 1 m N, |(T )| = 0 . p(T ) , 0, Tp(T ) W, |( Tp(T ) )| < 1, |(T )| < p(T ), .
K = {(Ty1,Ty2, . . . ,Tyn) : T B(H)} Hn. (Ty1,Ty2, . . . ,Tyn) = 0, , |(T )| = 0.
: K C : (Ty1,Ty2, . . . ,Tyn) 7 (T )
, , Hn. Hn, Riesz ([10]),
-
4 1.
(u1, u2, . . . , un) Hn, :
(Ty1,Ty2, . . . ,Tyn) =(Ty1,Ty2, . . . ,Tyn), (u1, u2, . . . , un)Hn =n
k=1
Tyk, ukH =n
k=1
yk ,uk(T )
(Ty1,Ty2, . . . ,Tyn) K,
(T ) =n
k=1
yk ,uk(T )
T B(H). WOT- WOT- yk ,uk , k = 1, 2, . . . , n.
1.2. S B(H) SOT- WOT-.
. WOT- SOT- B(H), S
S OT S
WOT.
, T < SS OT T < S
WOT.
T < SS OT
. Hahn - Banach ([14]), SOT- , R, Re(T ) > Re(S ) < , S S. WOT-, T < S
WOT
1.3. (B(H))1 B(H) WOT-.
. x, y H Dx,y = {z C : |z| x y}.
x,yH
Dx,y ,
: (B(H))1 [(B(H))1]
x,yHDx,y : T 7 {T x, y : x, y H}
, . Tycho-noff
x,yH
Dx,y ,
[(B(H))1]
x,yHDx,y.
b {b(x, y) : x, y H} [(B(H))1]. x1, x2, y1, y2 H , C > 0
-
1.1. B(H) 5
T (B(H))1, :
| b(x j, yk) T x j, yk| <
|b(x j, yk) T x j, yk| <
|b(x1 + x2, yk) T (x1 + x2, yk| <
|b(x j, y1 + y2) T x j, y1 + y2| <
j, k {1, 2}. , > 0, :
|b(x1 + x2, y1) b(x1, y1) b(x2, y1)| < 3
|b(x1, y1 + y2) b(x1, y1) b(x1, y2)| < 3
, |b(x, y)| x y b(x, y) Dx,y, (x, y) 7 b(x, y) sesquilinear . T0 (B(H))1, b(x, y) = T0x, y, x, y H . b [(B(H))1], [(B(H))1]
x,yH
Dx,y.
-
6 1.
-
2
2.1
2.1. R {Et : t R} HilbertH , :
s t Es Et
SOT- limt
Et = 0
SOT- limt+
Et = I
t 7 Et SOT- , SOT limtt0
Et = Et0 .
{Et : t R} (a, b), Ea = Eb.
2.1. {Et : t R} , Hilbert H , F R.
f =n
k=1
k(k ,k+1]F F, :R
f (t)dEto=
nk=1
k(Ek+1 Ek)=
nk=1
kE((k, k+1]) (2.1)
7
-
8 2.
St(F) F, :
: (St(F), ) (B(H), ) : f 7R
f dE (2.2)
- - . St(F) supremum, .
. (i) -.
f , g (St(F)), C.
, f =n
k=1
k(k ,k+1]|F , g =n
k=1
k(k ,k+1]|F
() ( f + g) =R
( f + g)dE =n
k=1
(k + k)E((k, k+1]) =
nk=1
kE((k, k+1]) +n
k=1
k E((k, k+1]) =R
f dE +R
gdE = ( f ) + (g)
() ( f g) =R
( f g)dE =n
k=1
kkE((k, k+1]) =
=
nk=1
kE((k, k+1])n
i=1
iE((i, i+1]) =R
f dER
gdE = ( f )(g)
, (a, b] (c, d] = , E((a, b])E((c, d]) = 0.
() ( f ) =R f dE =
nk=1
kE((k, k+1]) = n
k=1
kE((k, k+1]) =
=
R
f dE = ( f )
() ( f ) =R
f dE =n
k=1
kE((k, k+1]) =n
k=1
[kE((k, k+1])] =
=
nk=1
kE((k, k+1])
= [R
f dE]
= [( f )]
(ii) . h H , :
( f )h22 =
nk=1
kE(k, k+1])h
2
2
..1
=
nk=1
|k|2E((k, k+1])h22
-
2.1. 9
f 2n
k=1
E((k, k+1])h)22.= f 2
n
k=1
E(k, k+1])h
2
2
6 f 2h22.
( f ) f . |k0 | = max {|k| : k = 1, 2, ..., n} = f . E() F, x ran(E((k0 , k0])),
( f )x22 =
nk=1
kE(k, k+1])x
2
2
= |k0 |22x22, ( f ) |k0 | = f .
, ( f ) = f . 0
- F,
C0(F) =
C(F) F ,{
f : F C : lim|x|
f (x) = 0}
2.2. - A Hilbert H - : A B(H).
2.1. F R. C0(F) .
. . f C0(F) f 0, g =
f C0(F) , f = gg. ,
: 0( f ) = 0(gg) = 0(g)[0(g)
]= |0(g)|2 0.
f C0(F) f 1. 1 f f = 1 | f |2 , (1 | f |2) 0, (I ( f f ))x, x 0, x H . ( f )x2 = ( f )x, ( f )x = ( f f )x, x x2, ( f ) 1.
F R. :W+(F) =
{f : F R+ | f , fn C0(F) : fn
f}
W(F) W+(F ).1
-
10 2.
, W(F) , . f , g W+(F), f g W+(F). , f g . W+(F), fn, gn C0(F), n N, fn
f gn
g. fngn C0(F), n N fngn
f g, f g W+(F).
2.2. - 0 : C0(F) B(H) - :W(F) B(H) :
fn, f W+(F), n N, fn
f ( f ) = SOT- limn
( fn) (2.3)
, .
. (1) 0 W+(F) f W+(F) fn C0(F), fn
f . : fn f 0( fn) = fn f {0( fn)} . Banach - Steinhaus ([14]) SOT-lim
n0( fn)
. +( f ) = SOT- lim
n0( fn), f W+(F)
: + gm C0(F),m N, gm
f . n N :
fn gm : F R+ : x 7 min{ fn(x), gm(x)}. fn gm
m fn f = fn F., > 0 n N, K F fn(x) ,x < K. 0 ( fn gm)(x)
fn(x) , x < K,m N, |( fn gm)(x) fn(x)| < , x < K,m N fn gm
fn , Dini lim
mfn gm = fn .
, n N limm
fn gm = fn F, lim
m0( fn gm) = 0( fn).
, , Banach - Steinhaus
-
2.1. 11
SOT-limm
0( fn gm), n N :SOT-lim
m0(gm) SOT- lim
m0( fn gm) = 0( fn)
SOT- limm
0(gm) SOT- limn
0( fn)
fn, gm , SOT-lim
m0(gm) =SOT-lim
n0( fn)
.
f , g W+(F), fn, gn C0(F) fn
f , gn
g. fn + gn
f + g, :+( f +g) = SOT- lim
n0( fn+gn) = SOT- lim
n0( fn)+SOT- lim
n0(gn) = +( f )++(g),
+( f + g) = +( f ) + +(g). , +(cg) = c+(g), c 0., 2 SOT- :+( f )+(g) = SOT- lim
n0( fn)SOT- lim
n0(gn) = SOT- lim
n[0( fn)0(gn)] =
= SOT- limn
0( fngn) = +( f g), +( f g) = +( f )+(g).
(2) W(F) f : F R W(F). f1, f2 W+(F), f = f1 f2. ( f ) = +( f1) +( f2) , f W(F), : f = Re f + iIm f , Re f , Im f : F R Re f , Im f W+(F). ( f ) = (Re f ) + i(Im f )
(i) . f1, f2, g1, g2 W+(F), f1 f2 = g1 g2. f1 f2 = g1 g2 f1 + g2 = g1 + f2 +( f1 + g2) = +(g1 + f2) +( f1) + +(g2) = +(g1) + +( f2) +( f1) +( f2) = +(g1) +(g2) ( f1 f2) = (g1 g2).
(ii) - ..
2 SOT- , . An, Bn
SOT 0, An < M, M > 0. x H AnBnx An Bnx MBnx 0.
-
12 2.
(iii) . f 0 ( f ) 0. f 0, f W+(F) fn C0(F) 0 fn
f ., 0 0( fn)
SOT ( f ).
(iv) f W(F), 0 f f f 21, 1 : F R : x 7 1. (iii):
0 ( f )2 = ( f )( f ) ( f f ) f 2(1) = f 2 I = f 2
(v) fn, f W+(F) fn
f , ( f ) = SOT- limn
( fn).
n N ( fn,m) C0(F) :fn,m
fn ( fn) = SOT- limm
( fn,m)
gn = max{f1,n, f2,n, ..., fn,n
} C0(F) : gn fn gn
f , n N, n N fn,m
fn fn,m fn, m N, fn
f fi fn f , i n. : fi,m fi fn, i n,m N. max
1in
{fi,n
}= gn fn.
gn gn+1, x F, supn{gn(x)} = f (x).
x. fn
f , n1 N | fn1(x) f (x)| < 2 , fn1,m
fn1 , m1 N | fn1,m1(x) f n1(x)| < 2 ,| fn1,m1(x) f (x)| | fn1,m1(x) f n1(x)| + | fn1(x) f (x)| < 2 +
2 =
:
B n1 m1, gm1 = max1im1{fi,m1
} fn1,m1 , f (x) gm1(x) f (x)B m1 < n1, gn1 = max1in1
{fi,n1
} fn1,n1 fn1,m1 , f (x) gn1(x) f (x)
n0 = max{n1,m1}, gn0(x) ( f (x) , f (x)]. gn gn(x) ( f (x) , f (x)], n n0. gn
f , ( f ) = SOT- limn
0(gn).
-
2.1. 13
0(gn) ( fn) ( f ),n N. , , ( f ) = SOT- lim
n( fn).
(vi) .
2.1. 2.1 2.2 {Et : t R} F, -
: (St(F)), ) (B(H), ) : f 7R
f dE =n
k=1kE((k, k+1])
-0 : C0(F) B(H) : f 7
R
f dE
- :W(F) B(H)
(2.3). f W(F)
R
f dE = ( f ).
2.3. F R. - C0(F) Hilbert H {Et : t R}, F,
R
f dE = ( f ), f C0(F) (2.4)
. , - - :W(F) B(H), (2.3). e = F e = F = 1W(F) (e) = I. t R et = (,t]F W(F). (et)
= Et
:
1. E2t = [(et)]2 = (e2t ) = (et) = Et
2. Et = [(et)] = (et ) = (et) = Et
3. t s, et es 0 (et es) 0 (et) (es) Et Es
4. t s, eet
ees (eet)SOT (ees) IEt
SOT IEs EtSOT Es
-
14 2.
5. t +, et
e (et)SOT (e) Et
SOT I
6. t , et
0 (et)SOT (0) Et
SOT 0
(1) - (6) {Et : t R} Hilbert H . , , R < , :
E E = 0 (, ) F =
{Et : t R} F., 2.1 -
:W(F) B(H) : f 7
Ff (t)dE
(2.3). : ( f ) = ( f ), f W(F) f = e e. ( f ) = I E. (Bn), Bn = (, + n] F, :
(Bn) =R
(,+n]FdE = E((, + n]) = E+n ESOT I E
, Bn
f , (Bn)SOT ( f ).
( f ) = I E = ( f ) .
2.1. x H , Fx : R R 7 Etx, x.
, , limt
Fx(t) = 0. Lebesgue - Stieltjes x : x((, t]) = Fx(t). f W(F), R
f dEx, x =R
f dx.
f =
nk=1k(k ,k+1]F F. :
-
2.1. 15
R
f dx =n
k=1
k
R
(k ,k+1]dx =n
k=1
k((k, k+1]) =
nk=1
k{(, k+1] (, k]} =n
k=1
k[Ek+1 x, x Ek x, x] =
n
k=1
kE((k, k+1])x, x = R
f dEx, x
, f W+(F). fn, fn
f . 2.2 ( fn)
SOT ( f ) =R
f dE. , x H , ( fn)x, x ( f )x, x. ( fn)x, x =
R
fndx, , ( fn)x, x
R
f dx .
2.2 ( ). A (H). {Et : t R} (), :
f (A) =(A)
f (t)dEt, f C((A)) (2.5)
, A =(A)
tdEt.
. A (H) . () R. ,
: P((A)) (H) : p 7 p(A) -, - (A), -:
0 : C((A)) (H) : f 7 f (A) (A),
f (A) =R
f dE, f C((A)) (2.6)
-
16 2.
2.2
2.3. {E() : B(R)} Hilbert H , - , :
1. E() = E()
2. E(1 2) = E(1)E(2)
3. E() = 0 E(R) = I
4. x, y H , x,y : 7 E()x, y B(R)
2.2. (i) (1), (2) E() H . (x, y) 7 xy() sesquilinear,
4xy() =3
k=0
ikx+iky,x+iky(), (4)
:
4. x H , x= x,x : 7 E()x, x
B(R).
, x Borel (R, | |), .
(ii) , .
x H . 1,2 B(R) , :
{E(1) + E(2)} x, x = E(1)x, x + E(2)x, x = x(1) + x(1) =
= x((1 2)) = E(1 2)x, x, E(1) + E(2) = E(1 2)
{n : n N} B(R), E(n) , 0, n N
n+1k
E(k) nk
E(k) = E(n+1) = 1,
Cauchy, n
E(n) .(iii)
n
E(n) SOT-.
-
2.2. 17
=
n=1n. Vn = { : n}.
(ii) E() = E(Vn) + E( \ Vn) =n
k=1E(k) + E( \ Vn).
limn+
E( \ Vn)x = 0. :E( \ Vn)x2 = E( \ Vn)x, E( \ Vn)x = E( \ Vn)x, x = x( \ Vn) 0,
x - .
2.4. Hilbert H E((, t]) = Et, t R, -.
. {Et : t R} H . 3 P(R), E() =
{n
E((an, bn]) :
n=1(an, bn]
}: {E() : B(R)} H .
1. 1.2 E() E() = E()
2. tn t, 0 E() E(t, tn]SOT 0, E() = 0.
I E(R) E((, n]), n N, E((, n]) = EnS OT I,
n +. E(R) = I.
3. x H ,
x : P(R) R+ : 7 E()x, x (2.7)
x ., x() = E()x, x = 0, 1 2 E(1) E(2) E(1)x, x E(2)x, x x(1) x(2) {n : n N} Borel R. 1.1 n N {(an,i, bn,i] : i N}, n
(iE(an,i, bn,i]) x E(n)x 2n .
{(an,i, bn,i] : n N, i N
} ,
n=1
n n,i
(an,i, bn,i]. , (n,iE(an,i, bn,i]) x
n=1E(n)x
.3 P(R) R, R
-
18 2.
, {(an,i, bn,i] : n N, i N
} ,
Jm.
Jm \m1k=1
Jk ,
-. , R, {In}n,
n=1
n
n=1In n,i E(an,i, bn,i]x
n=1E(n)x
= n=1
E(In)x
n=1E(n)x .
x H :
E(
n=1n)x, x
n=1
Inx, x
n=1E(n)x, x + x(
n=1
n)
n=1x(n).
x . - , x - Borel R , x.
4. E(1)E(2) = E(1
2),1,2 B(R). E((a, b])E((c, d]) = E((a, b]
(c, d]).
a c :
E((a, b])E((c, d]) = (Eb Ea)(Ed Ec) = EbEd EbEc EaEd + EaEc =
=Emin{b,d} Emin{b,c} Ea + Ea = Emin{b,d} Emin{b,c} = E((a, b] (c, d])
D = {D B(R) : E(D)E((a, b]) = E(D (a, b]), a, b R} Dynkin.
R D, E(R)E((a, b]) = E() = E(R(a, b]) A, B D, A B, B \ A D. ,
E(B \ A)E((a, b]) = E(B)E((a, b]) E(A)E((a, b]) == E(B
(a, b]) E(A (a, b]) = E((B \ A) (a, b])
{Bn} D, B =
n=1Bn D.
, x H , :
-
2.2. 19
E(Bn)x, x = x(Bn) x(B) = E(B)x, x E(Bn) = E(B). :
E(B)E((a, b])x = limn+
E(Bn)E((a, b])x = limn+
E(Bn (a, b])x = E(B (a, b])x,
E(B)E((a, b]) = E(B
(a, b])
D Dynkin - Q = {(a, b], a, b R}, Dynkin (Q). , Q , (Q) - Q, B(R).
D = { B(R) : E(D)E() = E(D ), D B(R)} Dynkin, B(R)., E(1)E(2) = E(1
2),1,2 B(R),
.
5. E1(), E2() H ,
Et = E1((, t]) = E2((, t]), t R. x H , 1x, 2x , Lebesgue - Stieltjes Fx : R R : t 7 Etx, x. Borel R, 1x() =
2x() E1()x, x = E2()x, x, x H .
E1() = E2(), B(R). , E1(), E2() .
{E() : B(R)} . t R Et = E((, t]). {Et : t R} H .
s t, Es = Et + E((t, s]) Et.
x H . t R {tn} R lim
n+tn = t. Etn x Etx2 = (Etn Et)x, x = x((t, tn]) 0,
-
20 2.
t 7 x((, t]) . t 7 Et SOT- .
SOT- limt
Et = 0 SOT- limt+
Et = I
2.3. {Et}tR F W(F). 2.2, () = ()2 = (), () . E() =
{n
E((an, bn]) :
n=1(an, bn]
} Borel , () = E().
, x H , : x(B) = E(B)x, x, 2x, t 7 Etx, x. , 2.1 :
E()x, x = x() = x() =
dx =
dEx, x = ()x, x
x H . , .
-
3
von Neumann
3.1
3.1. Hilbert H S B(H). S :
S = {T B(H) : TS = S T,S S} (3.1)
S SOT-. , (Ai)iI Ai S, lim
iAix Ax = 0,x H , S S :
S (Ax) = S (limi
Aix) = limi
S Aix = limi
Ai(S x) = A(S x). S norm- .
3.2. Hilbert H . von NeumannM H B(H), M =M.
von Neumann C- , SOT-. S B(H), (SS) von Neumann, S, von Neumann S.
3.3. A B(H). E H - A(E) E. , E - -. , -, AP = PAP P = pro j(E).
21
-
22 3. VON NEUMANN
, - A-. , AP = PA, P {A}.
S B(H), S- S (E) E,S S. , S S-, S, S S.
3.2 von Neumann
3.1. A von Neumann Hilbert H ,E A K = E(H).
AE = {EA|K : A A} A|K = {A|K : A A} von Neumann , K.
. AE, A|K - B(K) , A A, A A, x K :
(EAE)(EAE)x EA=AE= EAAEx = EAAEx = (EAE)(EAE)x EA|KA|K = A|KEA|K .
AE (A|K) A|K (AE).
.
(i) T (A|K). S =[ T 0
0 0] B(H). S A,
X A, :
ES X =
T 00 0X|K 00 X|K
=T X|K 00 0
=X|KT 00 0
=X|K 00 X|K
T 00 0
= XS A = A, S A, T = ES |K AE
(ii) (AE) A|K . (AE) von Neumann, unitary 1.
1 A = A1 + iA2, A1 = A+A
2 , A2 = iAA
2 . A 1. I A2 , U = A + i
1 A2. U unitary, A = U+U2 . ,
von Neumann {A} norm-, U .
-
3.2. VON NEUMANN 23
unitary T (AE), T A|K . S A, S |K = T .
H0 ={
ni=1
Aixi : Ai A, xi K, n N},
H1 = H0 A- . S :
x =n
i=1Aixi H0, S
(n
i=1Aixi
)=
ni=1
AiT xi,
S (H1 ) = 0. ,
ni=1
Aixi =m
i=1Biyi, Ai, Bi A, xi, yi K,
ni=1
AiT xi =m
i=1BiT yi
ni=1
Aixi = 0,
n
i=1AiT xi = 0. :
ni=1
AiT xi2 = n
i=1
AiT xi,n
j=1
A jT x j
=
i, j
AiT xi, A jT x jxi=Exi
=
T E=ET =
i, j
AiET xi, A jET x j =
i, j
(EAjAiE)T xi,T x jT (AE)
=
i, j
T (EAjAiE)xi,T x jT unitary
=
i, j
(EAjAiE)xi, x j = n
i=1
Aixi2
n
i=1Aixi = 0
ni=1
Aixi = 0 n
i=1AiT xi
= 0 ni=1
AiT xi = 0,
S .
ni=1
AiT xi = n
i=1Aixi
S |H0 , H1 S (H1 ) = 0, S
, . , x K S x = T x, S |K = T . S A. A A. S (H1 ) = 0 H1 A-, x H0
-
24 3. VON NEUMANN
S Ax = AS x. x =n
i=1Aixi H0.
S Ax = S A
ni=1
Aixi
= S ni=1
AAixi
= ni=1
AAiT xi
=A
ni=1
AiT xi
= AS ni=1
Aixi
= AS x .
3.2. {Ha} Hilbert. {Aa} von Neumann Aa Ha, A = aAa = {aAa : Aa A, supa{Aa} < +} von Neu-mann, Hilbert H = aHa = {x = (xa)a :
axa < +}.
. A - B(H). S < A, S < A. S = aS a B(H), S < A. S a0 < Aa0 , Aa0 von Neumann, S a0 < Aa0 . Ta0 Aa0 , S a0Ta0 , Ta0S a0 .
T ={
Ta0 , a = a00 ,
, T A, TS , S T ,
S < A.
3.3. Hilbert H , n N A von Neumann , H . A A, A(n) = (Ai j)nn,2 Hilbert Hn =
ni=1H . A(n) =
{A(n) : A A
}
von Neumann, (A(n)) = (A)(n).
. A(n) - B(Hn). , T B(H) S = (S i j) B(Hn), :
T (n)S = (TS i j) S T (n) = (S i jT )
2 Kronecker i j ={
1 , i= j0 ,
-
3.3. 25
T (n) S T S i j. (A(n)) = Mn(A) (A)(n) Mn(A). . S = (S i j) Mn(A).
Ei j ={
IH , i j0 , A A, S AEi j = AEi jS ,
S kiA jl = AkiS jl k, l, S i j = 0 i , j, S iiA = AS j j, i, j, S (A)(n).
3.3
3.1. Hilbert H A - B(H). A SOT- A
. 1. T A, x H . Ax, A- H . P = pro j(Ax), P A, PT x = T Px. I A, x Ax. T x Ax, (An)nN A, T x = lim
nAnx.
2. k : B(H) B(H k) : A 7 A(n) - 3.3 k(T ) (k(A)), k N. - (1), x = (x1, . . . , xk) H k, (Am)mN A, k(T )x = lim
kk(Am), T x j = lim
mAm(x j), j = 1, . . . , k.
3. : T AS OT
W = {A B(H) : Axi T xi < , i = 1, . . . , k0} SOT- . (2), (Am)mN A, k0(T ) = limk
k0(Am) T x j = limm Am(x j), j = 1, . . . , k0. N N AN(x j) T x j < , j = 1, . . . , k0, An W, W A , .
3.1 ( ). A B(H) , A = AS OT = AWOT . , :
() A von Neumann
-
26 3. VON NEUMANN
() A SOT-
() A WOT-
. 3.1 1.2.
3.4. von Neumann norm .
. A von Neumann. A M, A = A1 + iA2, A1, A2 M . M norm . . A [{E() : B((A))]. E() M, B((A)). T M. , f (A), f C((A))., x, y B(H) :
f (A)T x, y = T f (A)x, y f (A)T x, y = f (A)x,T y
fT x,y =
fx,T y, f C((A)).
T x,y, x,T y , T x,y =
x,T y E()T x, y = E()x,T y
E()T x, y = T E()x, y, (A),x, y B(H).
E()T = T E(), B(H).
3.1. A B(H) {Et : t R} H , A =
R
tdEt. - {A} = {Et : t R}.
3.5. M von Neumann, HilbertH , E M , M von Neumann, E =M.
-
3.4. VON NEUMANN 27
. 1.3 B(H) WOT-, {An : n N} WOT- - M 3. , An norm M. , m N En,m = {En,m1 , . . . , E
n,mkn,m} M
1, . . . , k An kn,mi=1iEn,mi < 1m An < En,m >
.
E = nEn, En =
mEn,m.
E M M von Neumann E M. . T E T En, n N, T An = AnT, n N. {An : n N}
WOT= (B(H))1 M, M, -
M, E M, M =M E.
3.4 von Neumann
3.6. von Neumann M, - Hilbert H . A M, {A} = M. , 0 A I
. 3.5, E = {En : n N} M, E = M. B =
n
14n (2En I) =
n
14n S n,
S n = 2En I. B M : 13 I n
14n (I) B
n
14n I
13 I.
T B(H) B, En. B E1, ...En1, :
4n(B n1k=1
14k
S k) = 4n(+k=n
14k
S k) =+k=n
4n
4kS k = S n +
k=1
14k
S n+k = S n + B1,
3 H . = {, : , } , , = {n : n N}.
d : ((B(H))1, (B(H))1) R+ : (T, S ) 7
n=1
12n|n(TS )|n
. WOT- (B(H))1 , , .
-
28 3. VON NEUMANN
B1 =
k=1
14k S n+k
T
12
[3(S n + B1) (S n + B1)3] =32
S n +32
B1 12
S 3n 32
S 2nB1 32
S n B21 12
B31S 2n=I=
= (32 1
2)S n + (
32 3
2)B1
32
S nB21 12
B31 = S n + B2,
B2 = 32S nB21 12 B
31
B2 :
B2 = 3
2S nB21
12
B31 3
2S n B21
12B31
32B12
12B13
59B1,
S n = 1 B1 13 . B2 B1, T - 12 [3(S n + B2) (S n + B2)3] = S n + B3, B3 =
32S nB
22 12 B32,
B3 59B2 (59 )
2B1. Bk Bk ( 59 )k1B1, T S n + Bk, k N. T lim
k+(S n + Bk) = S n, En.
, T E1 = ET1 TS 1 = S 1T .
4B = 4+n=1
14n
S n =+n=1
14n1
S n = S 1 ++n=1
14n
S n+1 = S 1 + A1,
A1 =+n=1
14n
S n+1.
Ak Ak ( 59 )k1A1, T S 1 + Ak, k N., T E1. A = 32 (B +
13 I),
0 A I, {A} = {B}. , A .
3.1. M von Neumann, Hilbert H A B(H) {A} = M. A =
RdE, 3.4
-
3.4. VON NEUMANN 29
M = {E() : B(R}. P = E(0) , x P(H) :
A|P(H)x = AE(0)x = E(0)AE(0)x = E(0)(R
dE
)E(0)x =
R
d(E(0)EE(0))x =R
d(F)x, F = E|P(H)
{F = R P(H). M|P(H) = {E|P(H) : B(R)}
= {A|P(H)}.
3.4. von Neumann Hilbert H (maximal abelian selfadjoint algebra) masa - H . Zorn vonNeumann Hilbert H masa. , von NeumannM M M, masa M =M.
3.7. (X, ) - , M masa.
. M , T (M), g L(X, ), T = Mg.
(i) , (X) < , 1 L2(X, ). : T (M) T = Mg, g = T1. f L(X, ), :
T f = T ( f1) = T M f1 = M f T1 = M f g = f g = g f .
L(X, ) L2(X, ), T Mg L2(X, ). g L(X, ). > 0, U = {t X : |g(t)| > } . : T2U22 TU2 =
|g|2d
|U |2d = U22.
T , T g.
-
30 3. VON NEUMANN
(ii) - , X, Xn . (i), T |Xn = Mgn , gn = Tn, Mgn T . g X, g|Xn = gn|Xn , n N g L(X, ), g supn gn T. Mg T L2(Xn, ), , L2(X, ).
3.5. S B(H). H S, S , 0,S S \ {0}. S, S H .
3.2. (i) S B(H) H S, S
(ii) M von Neumann H M, S
. (i) T S T = 0. S S, T (S ) = S (T) = 0. S = H , T = 0
(ii) P = pro j(M). M M-, P (M) =M. , I M, P = (I P) = 0, I = P, M. M H .
3.3. von Neumann M Hil-bert H . M masa .
. von Neumann M , N masa M N . , N, N -. Zorn = {n : n N},
-
3.4. VON NEUMANN 31
. Pn = pro j(Nn). Nn N-, Pn N = N . : =
n
12nn N .
, H - N, T, S N T, S n = ,T S (2nPn) = 0, T S Pn N ., NNn,n N, , - .
3.6. H ,K Hilbert, S B(H), T B(K) unitarily , unitary U : H K ,
adU : B(H) B(K) : A 7 UAU
S T .
3.4. von Neumann unitarily unitarily .
. () M B(H), N B(K) von Neumann Uunitary adU : B(H) B(K) : A 7 UAU M N . T M, T M = MT,M M. (UTU)(UMU) = UT MU = UMTU = UMUUTU,UMU N . adU(M) = N , UTU N ., S N US U M, adU M N . M,N unitarily .
() (M) = M = M, , () .
3.2. von Neumann unitarily masa, masa.
-
32 3. VON NEUMANN
3.2. adU -. , C- unitarily , -. . , M masa, - - M(2). M(2) M2(M), M(2) masa . M,M(2) unitarily .
3.3. E() . H von Neumann M = {E()}, , () = E(), B(R), E(), . , B(H), () = 0. :
0 = () = E(), = E()2 E() = 0 E() = 0,
, , M, , L(X, ) = L(X, ), X R. :
f 0 L(X, ) ({x X : f (x) , 0}) = 0
({x X : f (x) , 0}) = 0 f 0 L(X, )
3.8. {Et : t R} F Hilbert H Borel , {Et : t R}. 2.1 -
:W(F) B(H) : f 7R
f dE
(F,B(F), ) , - f : F C g W(F), g = f -...M = {Et : t R}, :
: L(F, )M : f 7 (g) :
(i) -
(ii) H M. unitary :
U :M 7 L2(F, ), U( f )|MU = M f , f L(F, )
(iii) - C-.
-
3.4. VON NEUMANN 33
. (i) ,- .
. B(F). 1 B(F), (41) = 0 1 W(F), () = E(1). (41) = 0, E() = E(1), () = E(). E() M, B(F) M norm- , (L(F, )) M., f , g W(F), f = g -.., ( f ) = (g) ( f ) = (g).
- C-.
.
f =n
k=1
ckk . :
( f ) = max1kn{|ck| : E(k) , 0} = max
1kn{|ck| : (k) , 0} = f .
, .
(ii) H0 ={( f ) : f L(F, )
}.
H0 H . , -, M = {Et : t R}
S OT= {((,t]) : t R}
S OT.
M M (n)n H0, n M., H0 M. U0 : H0 L2(F, ) : ( f ) 7 f . U0 . f =
, . :
f 22 = n
k=1
2d = n
k=1
||2() =n
k=1
||2E(), =
nk=1
||2E()2..=
nk=1
E()2 = ( f )2
-
34 3. VON NEUMANN
U0 U :M L2(F, )., U , :
U(M) = U(H0)U
= U0(H0)2
= { f : f }2 = L2(F, ) f L(F, ), U( f )|MU = M f . f . g L2(F, ). :(U( f )|MU)g = (U( f )|M)(Ug) = (U( f )|M)((g)) = U(( f )(g)) =U(( f g)) = f g = M f g.
(iii) T M, T ( f ) UT |MU
M. , M masa, UT |MU
M, f L(F, ) :UT |MU = M f T |M = UM f U = ( f ) T = ( f )
(T ( f )) = 0 T = ( f )
3.4. () P M = {Et : t R}, Borel B(F), P = E(). , P M. 3.8 f L(F, ), E(),
f dE = P,
f dE =
f 2dE =
f dE., f = f 2 = f -.., f (x) {0, 1}, x. = {x F : f (x) = 1}, f = , P = E().
() A B(H) {Et : t R} A. , 3.8 A, f (A) = ( f ), f L((A), ).
3.3. 3.6 vonNeumann M, Hilbert H , A M, {A} = M. {Et : t R}
-
3.4. VON NEUMANN 35
A, {Et : t R} = {A}. 3.8, :
1. M - L((A), ), , A.
2. H M. unitary :
U :M 7 L2((A), ), U f (A)|MU = M f , f L((A), )
3.4. 3.8(ii), M, M = H , unitary
U : H L2(F, ), U( f )U = M f , f L(F, ), M unitarily M . 3.2, M masa. - , . Radon-Nikodym ([15]) -.. f L1(F, ),
() =
f d, B(F).
W : L2(F, ) L2(F, ) : g 7
f g , g L2((AR), ), :
Wg22 =
f |g|2dxi =|g|2dx1 = g22.
, W unitary, W : L2(F, ) L2(F, ) : g 7 1
fg.
, f L(F, ) : UWM f WU = ( f ). M unitarily M, - , {Et : t R}.
3.2. M von Neumann, Hilbert H A M, {A} = M. :
(i) M masa.
(ii) M .
(iii) M unitarily M = {M f : f L(((A), )}, - - A.
-
36 3. VON NEUMANN
. (i) (ii) 3.3
(ii) (iii) {Et : t R} A, {Et : t R} =M. 3.4, M unitarily M.
(iii) (i) 3.7
: , - .
-
4
4.1
4.1. von NeumannM n, n {+, 1, 2, ...}, P M unitary U : P(H) `2(n) H , M|P(H) masa, :
adU((A|P(H))(n)) = U(A|P(H))(n)U = A, A M
M unitarily (M|P(H))(n).
4.1. () M masa 1.
() 3.5, adU [(M|P(H))(n)
] M,
[(M|P(H))(n)
]= (M|P(H)) B(`2(n)) =
(M|P(H)) Mn(C) n < +(M|P(H)) B(`2) n = + 4.1. J B(H) F(H).
. J 1 . J , T J, T x = 0, x H . P 1, y H ,
37
-
38 4.
Pz = (y y)z = z, yy, z H . , S B(H), S (T x) = y. ,
Pz = z, yy = z, S T xS T x = S T T S z, xx = S T (x x)T S z
P J, .
4.2. M von Neumann n, .
. n < +. M - n, P M unitary U : (P(H))n H , , adU (M|P(H)) Mn(C) M . masa M|P(H). :
idn :M|P(H) Mn(C) Mn(C) : (Mi j)nn 7 ((Mi j))nn
Mi j M|P(H). M m. - :
() m < +, Q M unitary V : (Q(H))m H , adV (M|Q(H)) Mm(C) M
J = I|Q(H). CJ M|Q(H). A = CJ Mm(C) (M|Q(H)) Mm(C). : Mm(C) A : (i j) 7 (i jJ) -. :
Mm(C) A adV |A M
adU (M|P(H)) Mn(C)idn Mn(C)
: idn adU adV |A : Mm(C) Mn(C) - . ker Mm(C), ker , Mm(C), (1mm) = 1nn. Mm(C) , , ker = 0, n = m.
() m = +, (), , ker F(H), .
-
4.1. 39
4.1. {Et : t R} F, Hilbert H M = {Et : t R}. M n, unitary U : H (L2(F, )) `2(n), {Et}t, U( f )U = (M f )(n), f L(F, ). , M unitarily (M)(n).
. M n, P M, unitary
U1 : P(H)`2(n) H , U(( f )|P(H))(n)U = ( f ), f L(F, ). M|P(H) masa, 3.4, unitary
U2 : P(H) (L2(F, )), U2 M f U2 = ( f )|P(H), f L(F, ). unitary
U = (U2)(n) U1 : H L2(F, ) `2(n), U( f )U = (M f )(n), f L(F, ).
4.2. M von Neumann. P M n, n {+, 1, 2, . . . } M|P(H) n.
4.3. M von Neumann, Hilbert H .
(i) P M n, P M P P, P n
(ii) Pa M n, P = Pa - n.
. (i) von Neumann M n P M, P n. 3.6, A M, {A} = M, , P M, B((A)), P = E(), E() . M n, unitary
-
40 4.
U : H L2((A), ) `2(n) U f (A)U = (M f )(n), f L((A), ). U |P(H) : P(H) (M L2((A), )) `2(n). U |P(H) , . , :
UP(H) = UE()(H) = U(A)(H) = U(A)U(L2((A), ) `2(n)) == (M)
(n)(L2((A), ) `2(n)) = (M L2((A), )) `2(n), U |P(H) unitary :U |P(H) f (A)|P(H)
(U |P(H)
)= (M f )
(n), f L((A), ), M|P(H) unitarily vonNeumann MM = {M f , f L((A), )}., MM masa. | : B() R+ : B 7 |(B) = (B). J = M L2((A), ) L2(, |) : f 7 f |. J unitary, JMMJ =M| . M| masa, MM masa.
(ii) P =aA
Pa, Pa M. F A, -
aF
Pa M. F = {F A},
{
aFPa : F F
} , -
(1.1),
FF
{aF
Pa : F F}
=aA
Pa M. M. P M, B((A)), P = E()., k N Pak , Pak P < 1k . Pak P < 1k , k N, P =
kN
Pak , M. Pak M, Bk B((A)), Pak = E(Bk). k N, Borel k = Bk \
k1i=1
Bi. i j = ,
i , j,
k=1k =
k=1
Bk = . Pk = E(k) M,
, x H , Px = Pak x = Pkx =
k=1
Pkx.
-
4.1. 41
k N Pk Pak , (i) Pk n. 4.1, unitariesUk : Pk(H) L2(k, |k) `2(n), UkM|Pk(H)Uk (M|k )
(n)
unitary :
U :
k=1
Pk(H) = P(H)
k=1
(L2(k, |k) `2(n)
),
UM|P(H)U =
k=1
(M|k
)(n).
k=1
(M|k
)(n) unitarily
(M|
)(n). unitary
W :
k=1
(L2(k, |k) `2(n)
) L2(, |) `2(n) :
k=1
fk|(n)k 7
k=1
fkk
(n), WM(n)f W =
k=1
(M f |k
)(n), f L(, |),
.
4.4. H Hilbert . von Neu-mannH P M , M|P(H) .
. x1 H M, Q1 - Mx1, Q1 M. x2 H M|Q1 (H), Q2 - M|Q1 (H)x2 =Mx2, Q2 M
. , xn H Qn = pro j(Mxn), xn+1 M|( n
i=1Qi
)(H)
. n N, Qn = {P M : P Qn},
M. : I = Q1 Qn Qn+1,n N. P M, P Q1, Px1 = x1 (I P)x1 = 0, x1 M, I = P, Q1 = I.
-
42 4.
P M, P Qn, Pxn = xn (I P)xn = 0. xn
M|(n1i=1
Qi
)(H)
, (I P) n1
i=1
Qi
= 0 Qn+1
ni=1
Qi
n1i=1
Qi
, (I P)Qn+1 = 0, PQn+1 = Qn+1 P Qn+1, Qn Qn+1, . :
(i) n0 N, Qn0 = I > Qn0+1.
(ii) Qn = I, n 1.
(i), R = I Qn0+1, R M. MR = M|R(H) AR = A|R(H). xn0+1 M|(n0
i=1Qi
)(H)
,
R(
n0i=1
Qi
)xn0+1 = Rxn0+1 = (I Qn0+1)xn0+1 = 0,
R(
n0i=1
Qi
)= 0, R R
(n0i=1
Qi
). R(H) =
n0i=1
RQi(H)
: Rxi MR. M M, M|R(H)(Rxi) = 0. :M|R(H)(Rxi) = 0 MRxi = 0 MRQi(xi) = 0 MR|Qi(H)(xi) = 0. xi M|Qi(H), (M|Qi(H)) M|Qi(H). :
MR|Qi(H) = 0 MRQi = 0 (I MR)Qi = Qi I MR Qi, i {1, ..., n0}. MR = 0 M|R(H) = 0. : RQi(H) =MR(Rxi)() RQix RQi(H). Qix Mxi, (Mn)nN M, Qix lim
nMnxi. RQix = lim
nRMnxi = lim
nMnRxi = lim
nMn|R(H)Rxi.
RQi(H) MR(Rxi).() M M. :RQi (M|R(H)(Rxi)) = RQiMRxi = MRQixi = MRxi = M|R(H)(Rxi) RQi(H) MR(Rxi). 3.3, unitary
Ui : L2((AR), Rxi) RQi(H), : UiM f Ui = f (AR)|RQi(H), f L((AR), Rxi).
-
4.1. 43
xi i {1, 2, . . . , n0} , Radon - Nikodym fi =
dx1dxi L1((AR), xi).
Wi : L2((AR), x1) L2((AR), xi) : g 7
fig unitary.
W = n0
i=1
UiWi(U1 )(n0) : RQ1(H)n0 R(H),
:
W(f (AR)|RQ1(H)
)(n0) W = n0i=1
UiWi (U1 )(n0) ( f (AR)|RQ1(H))(n0) (U1)(n0) n0
i=1
Wi Ui
==
n0i=1
UiWi (U1 ( f (AR)|RQ1(H)U1)(n0) n0
i=1
Wi Ui
= n0i=1
(UiWiM f Wi U
i
)=
=
n0i=1
f (AR)|RQi(H) = f (AR), f L((AR), )
, MR n0. (ii), Qn = I, n N. xn
M,n N. 3.2, P M, Pxn = 0, P = 0. P M Pxn = 0. :
PQn = 0 I P Qn I P Qn = I P = 0., Qn, QiQ j = 0, i , j.
Zorn {xn : n N}, {yn : n N} M, {Qn : Q
n = pro j(Myn)}, Q
nQ
m, n , m.
Q =
n
Q
n
, Q M, Q = {P : P M, P Q}. : Q , I. Q = I. von Neumann M|Qn y Q(H). y M, , P M Py = 0, :
Py = 0 PQ = 0 I P Q I P Q = I P = 0., QMy = MQy = My, M M. My Q(H) pro j(My)Qn, n N, , {Qn : n N}.
-
44 4.
R = I Q, (i),
W =
i=1
Ui (U1 )() : RQ1(H) `2(N) H , W ( f (AR)|R1(H))() W = f (AR),
f L((AR), )
4.1 ( ). M von Neumann, Hilbert H , {Et}t - F, {Et : t R} =M.
() Pn M, 1 n +, , Pn n
1n
-
4.1. 45
Pm
1n
-
46 4.
-
5
Weyl
5.1 Stone
5.1. SOT- unitary , V : R B(H) : t 7 Vt , :
(i) Vts = VtV1s
(ii) t R, Vt unitary.
(iii) SOT-limtt0
Vt = Vt0 , t0 R
5.1. {E : R}, Hilbert H . Vt = (ut) =
R
eitdE, ut() = eit. {Vt}t SOT- unitary .
. {Vt : t R} 5.1.
(i) uts = utu1s ,s, t R Vts = VtV1s , .
(ii) , Vt ut = 1, t R, Vt . , x H , t R : x = VtVtx Vtx x, . Vt unitary, .
47
-
48 5. WEYL
(iii) x H , :Vtx x22 = (ut 1)(ut 1)x, x =
|ut() 1|2dx().
limt0|ut() 1| 0 |ut() 1| 2,
: R R : 7 2 L1(R, x). , : Vtx x22 0, t 0., Vt , :Vtx Vsx = Vs(Vtsx x) = Vtsx x 0, t s. {Vt : t R} SOT-.
5.2 (Stone). SOT- unitary {Vt}tR Hilbert H , {F : R} H , Vt =
R
eitdF.
. Banach L1(R,+, , ), f f , f (t) = f (t), -. sesquilinear . f L1(R). , H , : (, ) =
RVt, f (t)dt.
|(, )| = R
Vt, f (t)dt
R
|Vt, f (t)|dt
R
Vt | f (t)|dt =R
| f (t)|dt = f 1.
, sesquilinear , ( f ) B(H), ( f ), = (, ) ( f ) f 1. : - H .
f , g L1(R), :
( f g), =R
Vt, (R
f (t s)g(s)ds)
dt =R
(R
Vt, f (t s)g(s)ds)
dt =
Fubini=
R
(R
Vt, f (t s)dt)
g(s)ds t=ts=
R
(R
Vt+s, f (t)dt)
g(s)ds =
-
5.1. STONE 49
=
R
(R
Vt(Vs), f (t)dt)
g(s)ds =R
( f )(Vs), g(s)ds =R
(Vs), ( f )g(s)ds = (g), ( f ) = ( f )(g),
-. , f L1(R), , H :
( f ), = ( f ), =R
Vt, f (t)dt =R
Vt, f (t)dtt=t=
=
R
Vt, f (t)dt = ( f ),
: f L1(R) ( f ) f . Fourier Gelfand. Fourier F : L1(R) C0(R) 1-1, ,- ( g(t) = g(t),g C0(R)) ([7]). , Stone-Weierstrass ([16]), F (L1(R)) C0(R). F (L1(R)) R, t R, f L1(R), f (t) , 0. , t , s, f L1(R), f (x) = e|x|eitx, :
f (s) =R
e|x|+i(ts)xdx = 0 e
x+i(ts)xdx + +0 e
x+i(ts)xdx = 1i(ts)+1 +1
i(ts)1 =2
(ts)2+1
: f (t) = 2 , 0. Beurling - Gelfand1
f L1(R) ([4]), :
limn f f f 1/n1 = sup {|| : ( f )}
1/n = sup{|( f )| : M(L1(R))},
M(L1(R)) L1(R). , L1(R) s : s( f ) = f (s) ([5]). :
limn f f f 1/n1 = sup{| f (s)| : s R} = f
f L1(R), f = f , ( f ) , ( f ), = ( f ), = ( f ), , -
1 A Banach. limnxn1/n1 = sup{|| : (x)}, x A.
-
50 5. WEYL
:
( f )n = sup {|| : (( f ))}n = sup {|n| : (( f ))} = ( f )n = ( f n)
f f f 1 ( f ) limn f f f 1/n1 = f .
f L1(R) f1, f2 L1(R) , f = f1 + i f2.
( f ) ( f1) + ( f2) f1 + f2 2 f , . 0( f ) = ( f ), f F (L1(R)). f L1(R), fi = f i , i = 1, 2, f = f1 + i f2, :
0( f )) ( f1) + ( f2) f1 + f2 2 f , 0 , -, F (L1(R)), C0(R), : C0(R) B(H). 2.3, {E : R}, H , :
( f ), =R
f d, f C0(R)
((, ]) = E, . : Vt, =
R
eitd(). f L1(R), :R
Vt, f (t)dt = ( f ), = ( f ), =R
f d (5.1)
t0 R, fn(t) = n2en|tt0 |
R
fndm = 1, n N. Fourier fn:
fn() =R
fn(t)eitdt =R
n2
en|tt0 |eitdttt0=x
=
=n2
R
en|x|ei(x+t0)dx =n2
eit0R
en|x|eixdx =
=n2
eit0( 0
e(ni)xdx + +
0e(n+i)xdx
)=
n2
eit0(
1n i +
1n + i
)=
=n2
eit0(n + i + n i
n2 + 2
)=
n2 2ne
it0
n2(1 + ( n )2
) = eit01 + ( n )2
-
5.1. STONE 51
:R
Vt, fn(t)dt =R
Vt+t0, fn(t + t0)dt Vt0, , n +
(5.2)
g(t) = Vt+t0, , |g(t)| 2. g , > 0, > 0, |g(t)g(0)| < , |t| < . , n0 N,
|t| fn(t + t0)dt < , n n0.
|t|fn(t + t0)dt = lim
M+2 M
n2
en|t|dt = limM+
n M
en|t|dt =
limM+
nent
nM
= limM+
(en enM
)= en <
n. : R
g(t) fn(t + t0)dt g(0) =
R
g(t) fn(t + t0)dt R
g(0) fn(t + t0)dt =
= R
(g(t) g(0)) fn(t + t0)dt
|t||g(t) g(0)| fn(t + t0)dt +
|t|
-
52 5. WEYL
5.2 Stone-Von Neumann
unitary :
S : L2(R,m) L2(R,m) : f 7 u f , u(x) = eix (5.5)
Tt : L2(R,m) L2(R,m) : f 7 Tt f , (Tt f )(x) = f (x t) (5.6)
SOT-. , > 0 g CC(R), [M,M], M > 0. g , 1 > > 0 |g(x) g(y)| 0 .., T f = Thh f , f L2(R,m). q = Thh , q L(R,m), T f = Mq f .
5.2. {Vt}tR SOT- unitary -, Hilbert L2(R,m)`2(n), n {+, 1, 2, . . . }, Weyl {S (n) }R. unitary Q (Mm) B(`2(n)), t R Vt = QT (n)t Q.
. At = VtT (n)t . {S (n) } Weyl {Vt}t {T (n)t }t, :
AtS (n) = VtT(n)t S
(n) = e
itVtS(n) T
(n)t = e
iteitS (n) VtT(n)t = S
(n) VtT
(n)t = S
(n) At
At(S I) = (S I)At, , t R
As+t = Vs+tT (n)st = VsVtT (n)t T (n)s = VtAsT (n)s = AsT (n)s AtT (n)s As+t = AsT (n)s AtT (n)s
, `2(n). :, : L2(R) L2(R) C : ( f , g) 7 At( f ), (g ).
, sesquilinear , :
|,( f , g)| = |At( f ), (g )| At f g At f g
At B(L2(R)), At At ,
At f , g = ,( f , g). t At
WOT-. , t t, f , g L2(R) :(At A
t ) f , g = (At At)( f ), (g ) 0
-
58 5. WEYL
At(S I) = (S I)At, , t R, , `2(n), :
At S f , g = At(S f ), (g ) = At(S I)( f ), (g ) =
= (S I)At( f ), (g ) = At( f ), (S I)(g ) =
= At( f ), (S g ) = At f , S
g = S A
t f , g, f , g L2(R)
S , At , t R. t L(R,m), Mt = A
t .
:
t = Mt = At At ,
At unitary , t R.
x R :
`2(n) `2(n) C : (, ) = t (x) sesquilinear , :
|t (x)| sup{|at (x)| : 1, 1} At
t(x) B(`2(n)), , `2(n) t(x), = t (x). t R , `2(n)
R C : x 7 t(x), , .
: t L(R,m), - , : (t, x) 7 t(x), , . (t, x)
B : R2 C : B(t, x) 7
[0,x]t(y), dy. t, B x, y 7 t(y), [0, x].
-
5.2. STONE-VON NEUMANN 59
t t0, :
B(t, x) B(t0, x) =
[0,x]t(y), dy
[0,x]t0(y), dy =
=
[0,x]
(t(y), t0(y), )dy ==
R
(t(y), [0,x](y) t0(y), [0,x](y)) [0,x](y)dy == (t
t0 )[0,x], [0,x] 0
AtWOT At0 , t t0. B t.
f n : R2 C : (t, x) 7 n[B(t, x + 1n ) B(t, x)] , R2. Borel R2 , (t, x) 3 lim
nf n (t, x).
f : C : (t, x) 7 lim
nf n (t, x).
f ,t : x 7 f (t, x), Lebesgue t R, f t ,
t .- .
At , M f t , . As+t = AsT (n)s AtT (n)s t, s R, Nt,s Borel -
R Lebesgue, x < Nt,s s+t(x) = s(x)t(x s). , x < Nt,s. m N, x [m,m], [m,m] e1 L2(R) `2(n)
as+t(x)e1 = as+t(x)([m,m] e1)(x) = (As+t([m,m] e1))(x) =
= (AsT (n)s AtT(n)s ([m,m] e1))(x) = as(x)[T (n)s (AtT (n)s ([m,m] e1))](x) =
= as(x)at(x s)([m,m] e1)(x) = as(x)at(x s)e1.
{ei : i N, i n} `2(n). i, j
3 fn : (X,A, ) R . lim inf fn, lim sup fn X, R .
= {x X : lim inf fn(x) = lim sup fn(x) , } = {x X : lim fn(x) , } A, f = lim fn : R .
-
60 5. WEYL
gi j(x, t, s) = |(s+t(x) s(x)t(x s))e j, ei|. , t, s R Borel Nt,s , gi j(x, t, s) = 0 x < Nt,s,
R
gi j(x, t, s)dx = 0, t, s. :
R2
R
gi j(x, t, s)dxd(t, s) = 0Fubini
R
R2
gi j(x, t, s)d(t, s)dx = 0
R2
gi j(x, t, s)d(t, s), x < N i j, m(N i j) = 0
N =i, j
N i, j
x 1. {Vt}tR,
-
5.2. STONE-VON NEUMANN 61
Vt = WVtW. 5.2, unitary Q (Mm) B(`2(n)), Vt = QT (n)t Q. S
(n) Q = QS
(n) , S
(n) = QS
(n) Q
. , {ei : i N i n} `2(n), {Vt}, {S (n) } Q(L2(R,m)e1), ., Hilbert L2(R,m), unitary QW : H L2(R,m).
5.2. : {U}R, {Vt}tR SOT- unitary Hilbert H , Weyl. n {+, 1, 2, . . . } unitary R : L2(R,m) `2(n) H , U = RS (n) R
Vt = RT (n)t R, , t R .
-
62 5. WEYL
-
[1] H. Helson : The Spectral Theorem, Springer-Verlag, New York , (1986).
[2] H. Helson : Lectures on Invariant Subspaces, Academic Press, New York , (1964).
[3] K.R. Davidson : Nest Algebras : Triangular Forms for Operator Algebras on Hilbertspace , Longman Scientific & Technical, (1988).
[4] G. J. Murphy : C-algebras and operator theory, Academic Press, Boston, (1990).
[5] R.V. Kadison, J.R.Ringrose : Fundamentals of the theory of Operator Algebras,Academic Press, New York, (1983).
[6] R.Beals : Topics in Operator Theory, University of Chicago Press, Chicago, (1971).
[7] U.Katznelson : An introduction to harmonic analysis, Cambridge University Press,Cambridge, (2004).
[8] M. Anoussis, A. Katavolos : Unitary actions on nests and the Weyl relations, Bull.London Math. Soc. 27 (1995), 265-272.
[9] . : : , (2010).
[10] . : , , (2008).
[11] . : : , (2006).
[12] . : von Neumann - , , , (1997), 147-199.
63
-
64
[13] . : von Neumann nests, (2000).
[14] . : : 2, (2009).
[15] . , . : , , ,(2005).
[16] . , . , . , . : , , , (1997).
-
von Neumann, 21SOT-
unitary , 47unitary , 31
, 9
, 30, 30
, 7 , 16
Stone - von Neumann, 60Stone, 48 , 25, 15 , 44
(masa), 29
, 21
, 37
, 39
Weyl, 52
(WOT), 2 (SOT), 1
, 21
65
Eisagwg'hTopolog'ies ston B(H)
Fasmatik'h Jewr'iaTo Fasmatik'o J'ewrhmaFasmatik'a M'etra kai Anaparast'aseis
'Algebres von NeumannEisagwg'hParag'omenes 'Algebres von NeumannTo Je'wrhma tou De'uterou Metaj'ethAbelian'es 'Algebres von Neumann
Fasmatik'h Pollapl'othtaEisagwg'h sthn Jewr'ia Pollapl'othtas
Oi sq'eseis tou WeylTo Je'wrhma tou StoneTo je'wrhma monadik'othtas Stone-Von Neumann