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.