- Email: [email protected]
A remark on the spectrum of the analytic generator
Vol. 48 (2001)
No. 3
REPORTS ON MATHEMATICAL PHYSICS
A REMARK
ON THE SPECTRUM
OF THE ANALYTIC
PIOTR MIKOLAJ
GENERATOR
SOLTAN*
Department of Mathematical Methods in Physics, University of Warsaw, Hoia 74, 00-682 Warsaw, Poland (e-mail: [email protected]) (Received
April
23,
2001)
It is shown that although spectrum of the analytic generator of a one-parameter group of isometries of a Banach space may be equal to c, a simple operation of amplifying the analytic generator onto its graph squeezes its spectrum into &. Keywords:
1.
One-parameter
group, analytic generator.
Introduction
Let (X, F) be a dual pair of Banach spaces with a pairing denoted by (., .) . The symbol o(X, 3) will denote a weak topology on X given by the pairing with 3 as well as the product of such topologies on X x X. We shall say that the pair (X, TJ has the Krein property if a a(X, F)-closed convex hull of any a(X, F)-compact set in X is again cr(X, F)-compact. Throughout the paper we shall assume that both (X, F) and (F, X) have the Krein property. Let BF(X) c B(X) be the subspace of a(X, F)-continuous linear maps of X into itself. Let U = {UtJtCR be a a(X, 7)-continuous, one-parameter group of isometries in BF(X). By X, we shall denote the a(X, F)-dense subspace of X consisting of entire analytic elements for U. For any z E @ the operator Uz is defined in the following way: There exists a a(X, 3)-continuous function F, defined on the strip (w: ImwImz 2 0, ]Imzu] 5 IImz]}, with values in X, holomorphic inside the strip, such that for all t E R, F,(t)
= U,x and F,(z)
= y
1.
It is known (cf. [2, 41) that U, is a well-defined a(X, F)-closed linear operator in X and that U,, Uz2 = U,, +z2 for all zi, 22 E @. The analytic generator of the one-parameter group U is the operator Ui. *Partially supported by KBN, grant No. 2 POA3 030 14 and the Foundation for Polish Science. [4071
408
P.M. SOLTAN
At first glance it seems that the spectrum of the analytic generator of U should be contained in IR+. In fact, the analytic generator of a strongly continuous group of unitaries on a Hilbert space is a positive, self-adjoint operator. However, as it was first shown in [3], it may happen that Sp Ui = C. Furthermore, it turns out that we have either Sp Ui C R+ or Sp Ui = C (cf. [4]). We shall use some results on integration of vector-valued functions (cf. [1] Section 1). Given a locally compact space Q, a complex regular Bore1 measure of finite variation u on C2 and a a(X, 3)-continuous norm-bounded function !2 3 w H x(w) E X, the Krein property of (X, 3) implies that there is a unique y E X with (Y, 4) = ~(x(o).
q5 E 3.
4) dv(o),
As usual we shall write Y = so x(w) dv(o). Similarly, for any complex regular Bore1 measure of finite variation u on IR the operator x3x++
s
U,xdv(t)
E
x
w
will be denoted by JR U, du(t). This operator is bounded and, in fact, thanks to the Krein property of (3, X) it is 0(X, 3)-continuous. The pairing of X and 3 gives rise to the operation of transposition defined on the set of a(X, 3)-densely defined, 0(X, 3)-closed operators on X. This operation shall be denoted by S H ST. PROPOSITION 1.1. Let (X, 3) be a dual pair of Banach spaces such that both (X, 3) and (3, X) have the Krein property. Let U = (U,},,, be a a(X, S)continuous group of isometries in BF(X) and let S be a a(X, 3)-densely dejked, CT(X, 3)-closed operator such that U,S = SU, for all t E R. Let v be a complex,
regular Bore1 measure of jinite variation on IR and denote A = JR U, dv(t). AS c SA.
Then
Proof: For any 4 E D(sT) = (the domain of ST) and x E D(S) we have (ASx, 4)
= &U,Sx,
4) du(r) = _/&Urx,
4) du(t)
= J&Utx, ST+) dv(t) = (Ax, ST@), which simply means that Ax E D(Sm) AS c SA.
= D(S)
and SAX = SrrAx
= ASx.
Thus 0
In the next section we shall see that a simple operation on Ui can squeeze its spectrum into IF&+.Let A be the amplification of Vi onto its graph: for (z,) , (II) E Graph(Ui) we define
A REMARK
2.
The spectrum
ON THE SPECTRUM OF THE ANALYTIC
409
GENERATOR
of A
Let p E (E \ I&_ be a parameter.
Define an integrable
Fg = F,(t)
function
on R
bY +,X &l+it) F,(t)
= L 2x s _-oo ($
dE = t(-p)“-’
+ p)2
e-2nt
t/Lit-l -
1 =
,nt - prt.
For any fixed p in the indicated range the function Fw has holomorphic continuation onto the region @ \ (fni: n = 1,2,3, . . .}, and for 0 # z, in that region we have F,(t
- 29 + 2pF,&
- i) + p2FJt)
= 0,
(1)
pFp((z) + F,(z - i) = ienz f;+.
Define now a linear operator
Q,:
Qp =
(2)
X -+ X,
s+CC
F,(t)&
dt E BF(X).
(3)
-fx
Take x E X,.
We have
(QpU2i + 2PQpUi + p2QgL)x
s +m
=
-cc
= s co
Fp (t)Ut+zix dt + 2~ F,(t
=
s --03
s co
- 2i)Utx dt + 2p
s c2
where CO, C1 and C2 are oriented
FJt)Ut+;x
dt + p2
Fp (t)Ut+ix dt + p2
s Cl
s +oO
+CC
Fp(t>Ut+2iX dt + 2~
F,(t
s CO
- i)U,x dt + p2
-CO
Fw(t)Utx dt
F,(t)U,x
F,(t)Utx
s co
(4)
dt
dt,
curves in c, as shown below:
\ /
--i Cl
\ / CO
+
410
P. M. SOLTAN
Using (1) we can subtract 0 from both sides of (4) and taking into account holomorphy properties of the integrated functions we obtain
(QpU2i +2~Qe,ui
+ P’Q,>~
= j& F,(t - 2i)U,x d t + 21.~jc, F,(t - i)U,x dt + p2 j& F,(t)U,x
dt
- (lc, F,(t - 2i)U,x dt + 2~ f,, FCL(t- i)U,x dt + p2 lcz F,(t)U,x = 2~ &,_,,
F,(t - i)Utx dt + p2 &+,
= 2~s~ F,(t - i)U,xdt where I is an oriented
the
+p2sr
F,(t)U,x
dt)
dt
F,(t)U,xdt,
curve in C, as shown below: $ --2i
I
I
I
I
I
!
I
I
I
I
I
I I
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
Since the function we have
t I--+ F,(t - i)U,x
is holomorphic
F@(t - i)U,xdt
2P
.
in the strip {z: 0 < Im z < 2),
= 0,
s I-
and thus (4) takes the form (Q,Uzi + 2pQe,Ui + p2Q,)x The function
t H
= h2
U,x is holomorphic ~~sF,(t)Cr,x
F,(t)Ufx
dt = p22ni ~~ssF,(t)U~x.
s l(and therefore
continuous),
so that
=lim(t - i)F,(t)U,x t-+1 = lim(t - i) F@(t) lim U,x t-+i
=
t-i
ResF,(t)Uix t=i
= &Uix.
We have thus proved that
(QFU2.i +&~Qe,ui Now, if x E X is an arbitrary
+ P~Q~)x = Uix,
element one can define +oO
Js rl
x,=
-
7.c -m
l&x&
dt,
x E x,.
(3
41 1
A REMARK ON THE SPECTRUM OF THE ANALYTIC GENERATOR
and it is easily seen that for all n c 1~ we have x~ ~ Xoc and Xn
,'rIX,~)
~ x. It
is also easily verified with the help of Proposition 1.1 that if x ~ D(Ui) then a(X,~) Uix, > Uix. This shows that X ~ is a core for Ui. On the Banach space X x X define a bounded operator
\
IXQ,,
Q,
]
Since Qu is o(X,~-)-continuous so is R~,. We shall see that R u leaves the ~r(X, f ) - c l o s e d subspace Graph(U/) C X x X invariant. Indeed, for x 6 X ~ , we have
R.
Uix
=
#Qu
Qu
Uix
Using (5) and Proposition 1.1, we obtain
(
,1
Ui - Q u + - I #
-QuUi tx
) (x =
(#OuU+ QuUi)x
1,
-QuUi + - u i - - Q u U z i Ix Ix
) x = (ixQu + QuUi) x ,
X
which means that R~, ( uix ) E Graph(U/). Remembering that X ~ is a core for Ui and that R~, is ~r(X,U)-continuous one easily sees that Ru(Graph(Ui)) C Graph(U/). We can therefore consider a new (X, 5t-)-continuous operator R~ = R , [Graph(U/). Define Graph~ = { ( y )
6 Graph(Ui): x, y E X o o } .
It is easily seen that Graph~ is a ~r(X, f)-sequential core for A, i.e. for any (x) y 6 D ( A ) there exists a sequence {(x,, ~'. ) }n~r~ C Graph~ such that
(xn)
~x.~) ( y )
Ytl
n~ ~
Yn
n~oo
(6)
X
LEMMA 2.1. For any ( y ) ~ Grapho~ we have
Ru(A+#I)
Y
=
Y
.
412
P.M. SOtTAN
Proof:
We compute
=
= The formula
(-Q,Ui+~Ui_cLQ~+Z-~Q~U2i_Q,Ui)x (PQe,ui + p2Qp + QpU2i + PQ,Ui)x i
c
(i(Ui-
Q,uU2i -2PQpui
-P2Qp)
+ I),
(Q,Uzi + 2PQe,ui + p2Qp)x
10 x
2
UiX
’
(A + ZJZ)R, (“>=(;J UiX may be derived analogously LEMMA
1.1.
0
E Graph(Ui),
then there is a
with a prior use of Proposition
2.2. We have R,(Graph(Ui))
= D(A).
Proof: STEP 1: “R,(Graph(Ui))
sequence of R,
{(z)}ll,~
with (:I)
s
we have RLc(GI) 3
(z). R,(G),
(A + tiZ)R, Since the operator
(A + PI)
(T)
Take any
C D(A)“.
Furthermore,
by the c (X, F)-continuity
which combined
(‘t::)=(;::)
3(G).
is a(X, F)-closed
we have
with Lemma 2.1 yields
R, (‘z) E WA + pZ> = D(A), (A + pZ)RI, In particular,
R,(Graph(Ui))
STEP 2: “R,(Graph(Ui))
K:: KEN in the set Graph,
(8)
(G)=(l).
c D(A).
> D(A)“. Let (G) E D(A) and take a sequence such that (6) and (7) hold. By Lemma 2.1 we have
&(A+pZ)(;;)=(;;),
(9)
HEN
Taking the limit of both sides of (9) with y1-+ cc one obtains ..(A+.,,(;) In particular,
(f)
E R,(Graph(Ui))
(10)
= (;). and consequently
R,(Graph(Ui))
> D(A).
q
AREMARKONTHE
Combining sition. -p
SPECTRUM OFTHEANALYTIC
Lemma 2.2 and formulae
PROPOSITION 2.1. E @ \ 1w_ i.e. for any
0
The
operator
G
E D(A)
(8) and (lo),
R,
is the
GENERATOR
we get the following propo-
resolvent
we have R,(A
413
+,uZ)
of
the operator
A at
(G)=(C)
and for any
X
0 Y
E Graph(Ui)
we have R,
and (A + pZ)R,
0
;
E D(A)
(:)=(:>.
COROLLARY 2.1. Let (X, 3) be a dual pair of Banach spaces having the Krein property and such that the pair (3, X) also has the Krein property. Let U be a a(X, 3)-continuous one-parameter group of isometries in BF(X) and let Ui be its analytic generator. Denote by A the amplification of Ui onto Graph(Ui),
Then SpAcIk+. Moreover,
the resolvent of A at -,u E @ \ ILL is given by
= R, =
R(A, -II) where the operator
Q, E BF(X)
3.
+ ;I
PQW
is given by (3).
Let SpvT denote the point spectrum COROLLARY
-Qp
of any operator
2.2. With the notation as in Corollary
The CiorGnescu-Zsid6 In the fundemental
T on X. 2.1 we have S%Ui c I&.
formula
paper [2] Ioana Cioranescu
and L&z16 Zsido proved that if
U is a a(X, 3)-continuous one-parameter group of isometries in BF(X) then for any t E R we can find U, using solely the analytic generator of U. In other words,
the analytic generator UfX =
lim
miir OcReacl
determines
the group. Their formula looks as follows:
sinrrcr n
+O” h”-‘(h s0
+ Ui)-‘Uix
dh,
X
E
D(Ui).
414
P. M. SOLTAN
Using the results of the previous section one can derive a similar formula. For any (a) E D(A) (i.e. for any x E D(U2;)) we have U,x = where Pri is the the computational this formula here one should notice
+ca
sin (-inz)
lim
Im:zO
IT
p -iz-’ Pri(A + pZ)-‘A
s0
projection of X x X onto the details seem to be somewhat since it is mainly a repetition that the family of inequalities
(;)
dp,
first coordinate. In our approach, less tedious. We shall not derive of the work contained in [2], but (cf. [2] p. 345)
IIPri(A + @-‘A (c) II = IIPQ~~x+ Qpuixll i CC, (where r is a parameter in IO, l[ and C, is a suitable constant) which is needed to prove the convergence of the Bochner integral +cc, Pi-’ Prl (A + pZ)-’ A (;) dp s (where cx is any complei number such that 0 < Rea! < 1) can be obtained by a simple computation: +CC +cO
WQ,X + Q,uix =
pF,(t)U,x
s -cc
dt +
s --co
+cO =
s --oo
i+cc
pLLFb(t)Ut+i-x
dt
+
pFw(z)Uzx s ir-m
4.
s i-00
F,(t
-
i)U,x
dt
ir+oo
irfco =
Fw(t)U,+ix dt
Fp(z - i)U,x dz
dz + s ir-cc
Acknowledgments
The author is greatly indebted to Professor S. L. Woronowicz whose help in this work can not be overestimated. This paper is a part of the author’s master thesis written under Professor Woronowicz’s supervision. The author would also like to thank Professor L&z16 Zsido for fruitful discussions and help on subjects related to analytic generators. REFERENCES [I] [2] [3] [4]
W. Arveson: J. Funcr. Ana/ysis 1.5 (1974), 217-243. I. Ciorhescu, L. Zsid6: Tohoku Math. J. 28 (1976), 327-362. A. Van Daele: Math. Scud 37 (1975). 307-318. G. A. Elliot, L. Zsid6: J. Operator Theory 8 (1982), 227-277.
No. 3
REPORTS ON MATHEMATICAL PHYSICS
A REMARK
ON THE SPECTRUM
OF THE ANALYTIC
PIOTR MIKOLAJ
GENERATOR
SOLTAN*
Department of Mathematical Methods in Physics, University of Warsaw, Hoia 74, 00-682 Warsaw, Poland (e-mail: [email protected]) (Received
April
23,
2001)
It is shown that although spectrum of the analytic generator of a one-parameter group of isometries of a Banach space may be equal to c, a simple operation of amplifying the analytic generator onto its graph squeezes its spectrum into &. Keywords:
1.
One-parameter
group, analytic generator.
Introduction
Let (X, F) be a dual pair of Banach spaces with a pairing denoted by (., .) . The symbol o(X, 3) will denote a weak topology on X given by the pairing with 3 as well as the product of such topologies on X x X. We shall say that the pair (X, TJ has the Krein property if a a(X, F)-closed convex hull of any a(X, F)-compact set in X is again cr(X, F)-compact. Throughout the paper we shall assume that both (X, F) and (F, X) have the Krein property. Let BF(X) c B(X) be the subspace of a(X, F)-continuous linear maps of X into itself. Let U = {UtJtCR be a a(X, 7)-continuous, one-parameter group of isometries in BF(X). By X, we shall denote the a(X, F)-dense subspace of X consisting of entire analytic elements for U. For any z E @ the operator Uz is defined in the following way: There exists a a(X, 3)-continuous function F, defined on the strip (w: ImwImz 2 0, ]Imzu] 5 IImz]}, with values in X, holomorphic inside the strip, such that for all t E R, F,(t)
= U,x and F,(z)
= y
1.
It is known (cf. [2, 41) that U, is a well-defined a(X, F)-closed linear operator in X and that U,, Uz2 = U,, +z2 for all zi, 22 E @. The analytic generator of the one-parameter group U is the operator Ui. *Partially supported by KBN, grant No. 2 POA3 030 14 and the Foundation for Polish Science. [4071
408
P.M. SOLTAN
At first glance it seems that the spectrum of the analytic generator of U should be contained in IR+. In fact, the analytic generator of a strongly continuous group of unitaries on a Hilbert space is a positive, self-adjoint operator. However, as it was first shown in [3], it may happen that Sp Ui = C. Furthermore, it turns out that we have either Sp Ui C R+ or Sp Ui = C (cf. [4]). We shall use some results on integration of vector-valued functions (cf. [1] Section 1). Given a locally compact space Q, a complex regular Bore1 measure of finite variation u on C2 and a a(X, 3)-continuous norm-bounded function !2 3 w H x(w) E X, the Krein property of (X, 3) implies that there is a unique y E X with (Y, 4) = ~(x(o).
q5 E 3.
4) dv(o),
As usual we shall write Y = so x(w) dv(o). Similarly, for any complex regular Bore1 measure of finite variation u on IR the operator x3x++
s
U,xdv(t)
E
x
w
will be denoted by JR U, du(t). This operator is bounded and, in fact, thanks to the Krein property of (3, X) it is 0(X, 3)-continuous. The pairing of X and 3 gives rise to the operation of transposition defined on the set of a(X, 3)-densely defined, 0(X, 3)-closed operators on X. This operation shall be denoted by S H ST. PROPOSITION 1.1. Let (X, 3) be a dual pair of Banach spaces such that both (X, 3) and (3, X) have the Krein property. Let U = (U,},,, be a a(X, S)continuous group of isometries in BF(X) and let S be a a(X, 3)-densely dejked, CT(X, 3)-closed operator such that U,S = SU, for all t E R. Let v be a complex,
regular Bore1 measure of jinite variation on IR and denote A = JR U, dv(t). AS c SA.
Then
Proof: For any 4 E D(sT) = (the domain of ST) and x E D(S) we have (ASx, 4)
= &U,Sx,
4) du(r) = _/&Urx,
4) du(t)
= J&Utx, ST+) dv(t) = (Ax, ST@), which simply means that Ax E D(Sm) AS c SA.
= D(S)
and SAX = SrrAx
= ASx.
Thus 0
In the next section we shall see that a simple operation on Ui can squeeze its spectrum into IF&+.Let A be the amplification of Vi onto its graph: for (z,) , (II) E Graph(Ui) we define
A REMARK
2.
The spectrum
ON THE SPECTRUM OF THE ANALYTIC
409
GENERATOR
of A
Let p E (E \ I&_ be a parameter.
Define an integrable
Fg = F,(t)
function
on R
bY +,X &l+it) F,(t)
= L 2x s _-oo ($
dE = t(-p)“-’
+ p)2
e-2nt
t/Lit-l -
1 =
,nt - prt.
For any fixed p in the indicated range the function Fw has holomorphic continuation onto the region @ \ (fni: n = 1,2,3, . . .}, and for 0 # z, in that region we have F,(t
- 29 + 2pF,&
- i) + p2FJt)
= 0,
(1)
pFp((z) + F,(z - i) = ienz f;+.
Define now a linear operator
Q,:
Qp =
(2)
X -+ X,
s+CC
F,(t)&
dt E BF(X).
(3)
-fx
Take x E X,.
We have
(QpU2i + 2PQpUi + p2QgL)x
s +m
=
-cc
= s co
Fp (t)Ut+zix dt + 2~ F,(t
=
s --03
s co
- 2i)Utx dt + 2p
s c2
where CO, C1 and C2 are oriented
FJt)Ut+;x
dt + p2
Fp (t)Ut+ix dt + p2
s Cl
s +oO
+CC
Fp(t>Ut+2iX dt + 2~
F,(t
s CO
- i)U,x dt + p2
-CO
Fw(t)Utx dt
F,(t)U,x
F,(t)Utx
s co
(4)
dt
dt,
curves in c, as shown below:
\ /
--i Cl
\ / CO
+
410
P. M. SOLTAN
Using (1) we can subtract 0 from both sides of (4) and taking into account holomorphy properties of the integrated functions we obtain
(QpU2i +2~Qe,ui
+ P’Q,>~
= j& F,(t - 2i)U,x d t + 21.~jc, F,(t - i)U,x dt + p2 j& F,(t)U,x
dt
- (lc, F,(t - 2i)U,x dt + 2~ f,, FCL(t- i)U,x dt + p2 lcz F,(t)U,x = 2~ &,_,,
F,(t - i)Utx dt + p2 &+,
= 2~s~ F,(t - i)U,xdt where I is an oriented
the
+p2sr
F,(t)U,x
dt)
dt
F,(t)U,xdt,
curve in C, as shown below: $ --2i
I
I
I
I
I
!
I
I
I
I
I
I I
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
Since the function we have
t I--+ F,(t - i)U,x
is holomorphic
F@(t - i)U,xdt
2P
.
in the strip {z: 0 < Im z < 2),
= 0,
s I-
and thus (4) takes the form (Q,Uzi + 2pQe,Ui + p2Q,)x The function
t H
= h2
U,x is holomorphic ~~sF,(t)Cr,x
F,(t)Ufx
dt = p22ni ~~ssF,(t)U~x.
s l(and therefore
continuous),
so that
=lim(t - i)F,(t)U,x t-+1 = lim(t - i) F@(t) lim U,x t-+i
=
t-i
ResF,(t)Uix t=i
= &Uix.
We have thus proved that
(QFU2.i +&~Qe,ui Now, if x E X is an arbitrary
+ P~Q~)x = Uix,
element one can define +oO
Js rl
x,=
-
7.c -m
l&x&
dt,
x E x,.
(3
41 1
A REMARK ON THE SPECTRUM OF THE ANALYTIC GENERATOR
and it is easily seen that for all n c 1~ we have x~ ~ Xoc and Xn
,'rIX,~)
~ x. It
is also easily verified with the help of Proposition 1.1 that if x ~ D(Ui) then a(X,~) Uix, > Uix. This shows that X ~ is a core for Ui. On the Banach space X x X define a bounded operator
\
IXQ,,
Q,
]
Since Qu is o(X,~-)-continuous so is R~,. We shall see that R u leaves the ~r(X, f ) - c l o s e d subspace Graph(U/) C X x X invariant. Indeed, for x 6 X ~ , we have
R.
Uix
=
#Qu
Qu
Uix
Using (5) and Proposition 1.1, we obtain
(
,1
Ui - Q u + - I #
-QuUi tx
) (x =
(#OuU+ QuUi)x
1,
-QuUi + - u i - - Q u U z i Ix Ix
) x = (ixQu + QuUi) x ,
X
which means that R~, ( uix ) E Graph(U/). Remembering that X ~ is a core for Ui and that R~, is ~r(X,U)-continuous one easily sees that Ru(Graph(Ui)) C Graph(U/). We can therefore consider a new (X, 5t-)-continuous operator R~ = R , [Graph(U/). Define Graph~ = { ( y )
6 Graph(Ui): x, y E X o o } .
It is easily seen that Graph~ is a ~r(X, f)-sequential core for A, i.e. for any (x) y 6 D ( A ) there exists a sequence {(x,, ~'. ) }n~r~ C Graph~ such that
(xn)
~x.~) ( y )
Ytl
n~ ~
Yn
n~oo
(6)
X
LEMMA 2.1. For any ( y ) ~ Grapho~ we have
Ru(A+#I)
Y
=
Y
.
412
P.M. SOtTAN
Proof:
We compute
=
= The formula
(-Q,Ui+~Ui_cLQ~+Z-~Q~U2i_Q,Ui)x (PQe,ui + p2Qp + QpU2i + PQ,Ui)x i
c
(i(Ui-
Q,uU2i -2PQpui
-P2Qp)
+ I),
(Q,Uzi + 2PQe,ui + p2Qp)x
10 x
2
UiX
’
(A + ZJZ)R, (“>=(;J UiX may be derived analogously LEMMA
1.1.
0
E Graph(Ui),
then there is a
with a prior use of Proposition
2.2. We have R,(Graph(Ui))
= D(A).
Proof: STEP 1: “R,(Graph(Ui))
sequence of R,
{(z)}ll,~
with (:I)
s
we have RLc(GI) 3
(z). R,(G),
(A + tiZ)R, Since the operator
(A + PI)
(T)
Take any
C D(A)“.
Furthermore,
by the c (X, F)-continuity
which combined
(‘t::)=(;::)
3(G).
is a(X, F)-closed
we have
with Lemma 2.1 yields
R, (‘z) E WA + pZ> = D(A), (A + pZ)RI, In particular,
R,(Graph(Ui))
STEP 2: “R,(Graph(Ui))
K:: KEN in the set Graph,
(8)
(G)=(l).
c D(A).
> D(A)“. Let (G) E D(A) and take a sequence such that (6) and (7) hold. By Lemma 2.1 we have
&(A+pZ)(;;)=(;;),
(9)
HEN
Taking the limit of both sides of (9) with y1-+ cc one obtains ..(A+.,,(;) In particular,
(f)
E R,(Graph(Ui))
(10)
= (;). and consequently
R,(Graph(Ui))
> D(A).
q
AREMARKONTHE
Combining sition. -p
SPECTRUM OFTHEANALYTIC
Lemma 2.2 and formulae
PROPOSITION 2.1. E @ \ 1w_ i.e. for any
0
The
operator
G
E D(A)
(8) and (lo),
R,
is the
GENERATOR
we get the following propo-
resolvent
we have R,(A
413
+,uZ)
of
the operator
A at
(G)=(C)
and for any
X
0 Y
E Graph(Ui)
we have R,
and (A + pZ)R,
0
;
E D(A)
(:)=(:>.
COROLLARY 2.1. Let (X, 3) be a dual pair of Banach spaces having the Krein property and such that the pair (3, X) also has the Krein property. Let U be a a(X, 3)-continuous one-parameter group of isometries in BF(X) and let Ui be its analytic generator. Denote by A the amplification of Ui onto Graph(Ui),
Then SpAcIk+. Moreover,
the resolvent of A at -,u E @ \ ILL is given by
= R, =
R(A, -II) where the operator
Q, E BF(X)
3.
+ ;I
PQW
is given by (3).
Let SpvT denote the point spectrum COROLLARY
-Qp
of any operator
2.2. With the notation as in Corollary
The CiorGnescu-Zsid6 In the fundemental
T on X. 2.1 we have S%Ui c I&.
formula
paper [2] Ioana Cioranescu
and L&z16 Zsido proved that if
U is a a(X, 3)-continuous one-parameter group of isometries in BF(X) then for any t E R we can find U, using solely the analytic generator of U. In other words,
the analytic generator UfX =
lim
miir OcReacl
determines
the group. Their formula looks as follows:
sinrrcr n
+O” h”-‘(h s0
+ Ui)-‘Uix
dh,
X
E
D(Ui).
414
P. M. SOLTAN
Using the results of the previous section one can derive a similar formula. For any (a) E D(A) (i.e. for any x E D(U2;)) we have U,x = where Pri is the the computational this formula here one should notice
+ca
sin (-inz)
lim
Im:zO
IT
p -iz-’ Pri(A + pZ)-‘A
s0
projection of X x X onto the details seem to be somewhat since it is mainly a repetition that the family of inequalities
(;)
dp,
first coordinate. In our approach, less tedious. We shall not derive of the work contained in [2], but (cf. [2] p. 345)
IIPri(A + @-‘A (c) II = IIPQ~~x+ Qpuixll i CC, (where r is a parameter in IO, l[ and C, is a suitable constant) which is needed to prove the convergence of the Bochner integral +cc, Pi-’ Prl (A + pZ)-’ A (;) dp s (where cx is any complei number such that 0 < Rea! < 1) can be obtained by a simple computation: +CC +cO
WQ,X + Q,uix =
pF,(t)U,x
s -cc
dt +
s --co
+cO =
s --oo
i+cc
pLLFb(t)Ut+i-x
dt
+
pFw(z)Uzx s ir-m
4.
s i-00
F,(t
-
i)U,x
dt
ir+oo
irfco =
Fw(t)U,+ix dt
Fp(z - i)U,x dz
dz + s ir-cc
Acknowledgments
The author is greatly indebted to Professor S. L. Woronowicz whose help in this work can not be overestimated. This paper is a part of the author’s master thesis written under Professor Woronowicz’s supervision. The author would also like to thank Professor L&z16 Zsido for fruitful discussions and help on subjects related to analytic generators. REFERENCES [I] [2] [3] [4]
W. Arveson: J. Funcr. Ana/ysis 1.5 (1974), 217-243. I. Ciorhescu, L. Zsid6: Tohoku Math. J. 28 (1976), 327-362. A. Van Daele: Math. Scud 37 (1975). 307-318. G. A. Elliot, L. Zsid6: J. Operator Theory 8 (1982), 227-277.