Annals of Mathematics
On One-Parameter Unitary Groups in Hilbert Space
Author(s): M. H. Stone
Source: Annals of Mathematics, Second Series, Vol. 33, No. 3 (Jul., 1932), pp. 643-648
Published by: Annals of Mathematics
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ON ONE-PARAMETER
IN HILBERT
BY M. H.
UNITARY
SPACE.'
GROUPS
STONE.
In this note, I give the detailed proof of a result(TheoremB below)
whichI announcedin theProceedingsoftheNationalAcademyofSciences,
16 (1930), pp. 173-4 and whichI originallyintendedto publish in my
forthcoming
book, "Linear Transformations
in Hilbert Space and Their
to Analysis" (ColloquiumPublicationsof theAmericanMatheApplications
maticalSociety). The chapter on group theorywas completedin May,
1930, but was laterexcludedfromthe book. Recent applicationsof this
theoremmake furtherdelay in publicationhighlyundesirable. J. v. Neumann,forexample,has foundit necessaryto generalizethe statementof
the theoremand has given a new proof,whichappears in this numberof
the Annals.2 My own methodseems to me to be a "correct" one from
the pointof view of the theoryof groupsand theirlinearrepresentations;
and in any case, as I had observedin my originalmanuscript,
it yields
withoutany modification
the generalizedformof the theoremrequiredby
v. Neumann(TheoremC below). The numbereddefinitions,
theorems,and
lemmasto whichI referin these pages, will be foundin my book.
THEOREM A. If H is a seif-adjointtransformation
in abstractHilbert
< < + ct, is a family of unitary transspace,D, thenU(T;) eiTH
formations
withthegroupproperty
-
(a)
U(@) U(i)
U(O+i?)
and thecontinuity
property
U~r)
(Ai)
U(i)
wvhen a -> v (Def. 2.3).
The existenceof U(r) ei= H is establishedinTh. 6.1, its unitarycharacter
followsfromTh. 6.6. The groupproperty
(a) is a consequenceofTh. 6.1,
(6), (7). The continuity
property
(,8) followsfromthe relation
I(UW()
-
U(T))f12
Wea -
=
eiTXI2diE(I)fI2,
whereE(s) is the resolutionof the identitycorresponding
to H, as we
have pointedout in Th. 6.1 (5) and Th. 6.2.
1ReceivedMarch19, 1932.
2 Vide
supra,pp. 567-573.
643
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644
M. H. STONE.
is a family of unitary transTHEOREM B. If U(r), - o <r < + x,
property(ai), then
formationswith the groupproperty(a) and thecontinuity
eiTH.
thereexists a unique self-adjoinettransformationH such that U(T)
of
property(ai) impliesthe continuity
It is evidentthat the continuity
f and q in D, by virtueof
(U(r)f, g) as a functionof r for arbitrary
Def. 2.3 and Th. 1.3. Hence we may generalizeTheoremB as follows:
is a family of unitary transTHEOREM C. If U(r), - x <r <+ x,
formationswith the group property(a) and the measurabilityproperty
(y) (U(T)f, g) is a measurablefunction of T for arbitraryf and 4 in lS,
H such that U(r) =-ei rr.
thenthereexistsa unique self-adjointtransformation
that thegroupproperty(a) and themeasurability
TheoremA shows,therefore,
property(y) togetherimply the continuityproperty(O).
Theorem
We now proveTheoremC. First, by the Fourier-Plancherel
formulas
(Th. 3.10), the reciprocal
(la)
1)
P;
(lb)
J
21
J
)1l
r
ip(i;
e-iTd
l)etirdT
are valid when el(1) f 0, and lead to the relation
(2)
(l-m)
X+0
f?
(T; 1)tP(-;r
M) d
t/;1)-t(1f;
=
m)
is
transformation
when the unitarycharacterof the Fourier-Plancherel
(1 a) by the
coupled with equation(1 a) and the equationobtainedfromn
substitutionof a - r for T.
V('; 1)
=(T;
0,
i
We find
r>0, P(l) >0;
ie-ilrT, z>0,
t~;)ie
Pt(T;)
y(l)<0;
ilr,
0,
0,J()O
T<0,
PM(<0;
since (I b) is obviouslyvalid with this choice of At. We findalso that
(4)
My(T;1) =
(-T;
1).
A more completediscussionof some of these relationsis givenin the
Xi
proofofTh. 10.9. Secondly,we definea boundedlineartransformation
withdomainQ by the equation
+0
1)J t(~ ) ( U (r) fj ) dr
(Xzf~
(1) +O.
as a Lebesgue integralin accordance
The integralis to be interpreted
property(y). By the unitarycharacterof U(r)
with the measurability
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645
GROUPS IN HILBERT SPACE.
we have I(U(T)f, g) I _ I U(T)fI IgI = If I IgI
this inBy combining
equalitywith(3), we findthat the integralis absolutelyconvergent
and
does not exceed If I IgI/I
apply
aJ(l) in absolutevalue. We can therefore
a theoremof Fre'chet(Ths. 2.27-2.28) to establishthe existenceof the
desiredtransformation
Xi. Thirdly,we obtainessentialpropertiesof Xi
corresponding
to (2), (3), and (4). The firstis
(2 bis)
( -m)XIXm
Xm,
X,
0(l) t 0,
aJ(m)t 0.
of Xi and the groupproperty(a), we have
From the definition
=_
(XI Xmf,g) = (Xmf, Xl'%)
-
t(e; m)(U(e)f, XI*g) de
/(e; m)(Xi U(e)f, g)de
00?(;
-JJ
(e;m)(U(r)U(e)f,g) drde
tV(r;
I)V(e;m)(UMe+,)f, g)drde.
This integralis an absolutelyconvergent
Lebesgue integralby virtueof
the measurabilityproperty(y),3 the relations (3), and the inequality
newvariablesc e=+
J(U(e+?T)f,g) <Ifj IgI. Hence we can introduce
and T = T and can integratein any convenientorder. If we perform
the
integrationwith respect to the new variabler first,we obtainwiththe
aid of (2) the result
(i-m) (XiXmfg )
=J
(p (or;i)-(or; m))(U(') f, g) d
=
((Xi - Xm)fg),
whichevidentlyimplies(2 bis). The second propertyis
(3 bis)
Xmf =
f
0 implies f
From (2 bis) we see that Xmf
and henceimpliesthat
lf(T;
=
0.
0 impliesXjf = 0 for all not-real1
) (U(T)f, g) dr vanishesforall not-realI
and all g in &. By makinguse of the explicitformof V given in (3),
we conclude'that(U(T)f, g) = 0 for almostall r and all g in ). Hence
3
F(e,
It is to be notedthat if f (T) is a measurablefunctionofT, -()
T)
f (Q+ T) is a measurablefunctionof (Q, T).
<r<+OO,
then
4We have to show that the equationJ
e'? F(T) dr = 0, holdingforall I such that
T >O,
wheneverF(r) is bounded and
0 (1)>0, implies F(r) = 0 almost everywhere,
measurable. If we put I = p + i', v>0, and hold v fast,we can obtainthis resultby
a simpleapplicationof the Fourier-Plancherel
Theorem.
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646
M. H. STONE.
0 for almostall T and thatfwe see thatU(T)f
is
unitarycharacterof U(r). The thirdproperty
J() 4 0.
XI - X,
(4 bis)
The group property(a) impliesthat U(O)
I, U(-u)
Hence we obtainfrom(4) the relation
I, g) - (XigfJ)
(XY*
0 by virtueof the
fq(r;
=
U-1 (a)
=
Us (r).
1) (U(r)g, f)d
-JXiw(-,; 1,U*(r) f,y)dr
f
P+m
-
(T;i
)(U(r)f,
g) dT
(Xif, 9),
implies(4 bis). In the
which holds for all f and g in S) and therefore
fourthplace, we observethat, in accordancewith Th. 4.19, the three
propertiesofXI whichwe havejust provedenableus to assertthe existence
H such that Xi (H I)-' for
of a unique self-adjointtransformation
the transformation
V(W)= eiTH
that
show
must
we
Finally
1.
not-real
all
coincideswithU(0). Accordingto TheoremA, the familyV(T) satisfies
the hypothesesof TheoremsB and C. Hence we can introducethe transY1 definedby the relation
formation
(Y f7 g)
By Th. 6.1, we have
(Yif, g)
lim
+tch
J 00(T;
U
a i+X )00
lim 00
p(T;
l(V(
I)(U
)f, g) d.
eikT
d(E(A)f, g))dT
) Ei),
(;)
j4ext
a;tTV(r )( d (i) (A) f,
a-*+oo -a0
lim
Xd-;E(E())f)g)
=_1Je~
;?
e-O
i
d (E (i) f,
g)
~
whereE(X) is the resolutionof the identityassociatedwithH, and the
+ sign or the - sign is taken accordingas el(1)<0 or el(1)> 0. Now
1-1 = Xi. Hence we have
Ths. 5.7 and 6.1 show that YJe(H-
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P+ 0
fo
647
SPACE.
GROUPS IN HILBERT
g)) dr
P(r; 1) ((U(T)f, g)-(V(T)f,
0
and can concludethatU(T) = V(W)for almostall r. If r is fixedand e
varies in the set specifiedby the identity
U(e) -= V(), thenthedifference
o <a<
a
T - e varies over the entire range + oo with the possible
exceptionof a null set. Hence a assumesat least one value such that
that
U(a) = V(a). We see therefore
u(r)-
U(e + d)
=
U(Q) U(a)
V(W)V(a) =- T(Q + a)
=
=
V(r)
for arbitraryi and suitablychosen e and a, by virtue of the group
property(a). This completesthe proof.
and U(r) = eigH,
THEOREM D. If H is a self-aqjointtransformation
thentherelation-U(r)
f* is valid in S if and onlyif f is in
-
thedomainof H and f*
i Hf.
=
J
If f is in the domainof H, the integral
2
d E ()fJ2 exists by
that
Th. 5.9. From Th. 6.1 (5) and Lemma6.1 (6) we see therefore
lim H-eirH-
TAO~
~~
l1
Thisresultmeansthat
domainof H.
1- eir_1
2
212
()
TAO
ei
2T
-o
(t)1J1
_____
U() f f*
-
=
2
dIE
iHf whenever
f is in the
T by
On the otherhand, we can definea transformation
the relationiTf = f*, valid whenever (1(7) U(0) f convergesin to an elementJf. Then T is an extensionof H withthe propertythat
(Tf g)
lim(f, U(
1
( U(T)U(0)
T)- U(O)
f, q)
-(fTg)
whenever
f and g are elementsin its domain. Hence T is a symmetric
we musthave T = H in accordextension
ofH. SinceH is self-adjoint,
ance with Th. 2.13. For a modification
of this theorem,we referto
v. Neumann'spaper in this numberof the Annals.
(Def. 8.1)
T is permutable
THEOREME. A boundedlinear transformation
H if and onlyif it is permutable
with
withtheself-adjoint
transformation
X
<, < + m.
of thefamilyeiTH,
everytransformation
-
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M. H. STONE.
648
(H- 1)-1 witheITH,it is evident
From the integralrelationconnecting
eYTH if and onlyif it is
witheverytransformation
that T is permutable
with(H - )-1 for all not-real1. As we proved in Th. 8.1,
permutable
with (H- 1)-1 for all not-real1 (or, for a singlesuch1)
T is permutable
if and onlyif it is permutablewith H.
Because of lack of space, I do not go intocertainothergeneralizations
of the theoremsprovedhere. I pointout that the resultsobtainedlead
of all r-parameterabelian unitarygroups
easily to a characterization
..
.
.
.
e
whereH,, ..*, H. are permutableselfn,
U( T.
deals
(cf. Def. 8.2). A less obviousgeneralization
adjointtransformations
U(T), 0 ? r< ?+o, where
withthe familyof isometrictransformations
U('?+r) = U(a) U(T) for a > 0 r > 0. I hope to state this problem
in preciseformand to studyits solutionon some otheroccasion.
YALE UNIVERSITY.
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