Vol. 45 (2000) REPORTS Oh’ MATHEMATICAL PHYSICS No. I

POSITIVE OPERATOR MEASURES DETERMINED BY THEIR MOMENT SEQUENCES

ANATOLIJ DVURE~ENSKIJ

Mathematical Institute, Slovak Academy of Sciences, SK-814 73 Bratislava, Slovakia (e-mail: dvurecen@mau.savba.sk)

PEKKA LAHTI

Department of Physics, University of Turku, FIN-20014 Turku, Finland (e-mail: pekka.lahti@utu.fi)

and

KARI YLINEN

Department of Mathematics, University of Turku, FIN-20014 Turku, Finland (e-mail: ylinen@utu.fi)

(Received March 29, 1999 - Revised August 6, 1999)

We solve the moment problem for the polar margins of a physically relevant class of phase space observables.

1. Introduction and notations

The representation of an observable as a positive operator measure in the Hilbert space formulation of quantum mechanics raises the question of determining the mo- ment operators of an observable and of the uniqueness of the operator measure in terms of its moment operators. Though the classical moment problem has been thor- oughly investigated, see e.g. [l-4], much less has been published on the operator measure counterpart of the problem. In this paper we investigate the operator mea- sure formulation of the moment problem, and we solve the problem for the unsharp phase and number observables which are obtained as the polar coordinate angle and radial margins of the phase space observables.

Let 3t be a complex Hilbert space, and let C(x) denote the set of bounded operators on ‘Ft. Let 52 be a nonempty set and A a a-algebra of subsets of ,Q. Let E : A -+ ,C(31) be an operator measure, and let E+,,, I+!I, cp E IFI, denote the complex measure defined by E+,+,(X) = (+ 1 E(X)cp). In this paper we always assume that the operator measures are positive, that is, E(X) 2 0 for all X E A, and normalized, that is, E(Q) = I. Let f : 62 -+ @ be an d-measurable function and let D(,CE(f)) denote the set of those vectors p E K for which f is Ei,V-integrable

11391

140 A. DVURE(?ENSKIJ, P. LAHTI and K. YLINEN

for each @ E ‘7f. The integral of f with respect to E is the unique linear operator LE(f), with the domain D(LE(f)), such that (+ ]LE(f)cp) = SD f dEti,, for all

(p E D(LE(f)), @ E X. We denote this operator also by s f dE. The present article is a continuation of the basic study [5] of this operator integral, also developed in [6]. In particular, the explicit calculation of the moment operators of radial margins of the phase observables given in [5] is a crucial ingredient in the solution of the moment problem of Section 6.

Assume now that Q = R, with A being the a-algebra O(R) of the Bore1 subsets of the real line R. Let id denote the identity function on R. Let N denote the set of natural numbers including 0. For any k E N, we call the linear operator LE (idk) the k-th moment operator of the operator measure E. We also write LE (idk) = & Xk d E(x). In general, LE(idk) need not be densely defined; it is even possible

that D(LE(idk)) = {O). We call the number (I++ 1 LE(idk)q) = &idkdE+,cp, + E tit,

CJJ E D(LE(idk)), the k-th moment of the complex measure E+,(p.

2. Operator measures concentrated on bounded sets

Consider an operator measure E : B(R) --+ C(K) (positive and normalized) which is concentrated on a closed interval [a, b] of the real line R. Its moment operators LE(idk) are bounded self-adjoint operators. It is a well-known consequence of the Weierstrass approximation theorem and the uniqueness part of the Riesz represen- tation theorem that the moment operators LE(idk) determine E uniquely. Indeed, the complex measures E*,, are uniquely determined by the integrals j f dE$,, for continuous functions f : [a, b] -+ IR since all complex Bore1 measures on [a, b] are regular, and on the other hand the integrals s f dE*,, are determined by the moments of E@,cp, since the polynomials are dense in C[a, b]. We ask under what conditions a sequence of bounded self-adjoint operators may appear as the sequence of the moment operators of an operator measure concentrated on [a, b]. A character- ization of such operator sequences is given, for instance, in [7, Theorem II, p. 4631. The multitude of the polynomial conditions of that result may, however, be reduced by using the notion of completely monotone operator sequences.

We say that a sequence of bounded self-adjoint operators (Sk)& c L(X) is completely monotone with respect to an interval [a, b] if, for all m, n E N,

Using a simple transformation, the result [S, Theorem 4.21 can easily be adapted to characterize such operator sequences.

THEOREM 2.1. A sequence of bounded selfudjoint operators (Sk)& C L(IFI),

with So = I, is the sequence of the moment operators of an operator measure E concentrated on the interval [a, b] if and only if (&)rzO is completely monotone with respect to the interval [a, b].

POSITIVE OPERATOR MEASURES DETERMINED BY THEIR MOMENT SEQUENCES 141

3. Operator measures with unbounded support

Consider now an operator measure E : B(R) + C(7-l) (positive and normal- ized) which is not necessarily concentrated on a bounded set. Its moment op- erators LE (idk), k E N, are symmetric, that is, (@I LE(idk)p) = (LE(idk)@lcp) for all @, 4p E D(LE (idk)), but they need not be densely defined. Assuming that D & f$&D(LE(idk)), one may ask whether for a given q E 2) the positive mea-

sure E,,, is determinate, that is, if the moment sequence ((cp 1 LE(idk)q))kZa deter- mines the measure E,,,. One may also ask if the totality of the moment sequences

(((p ( LE(idk)q))k,o, p E D, determines the moment operators LE(idk), and, finally, if the moment operators LE (idk) determine the operator measure E.

It is well known that in the unbounded case a moment sequence ((40 1 LE(idk)~))~a, cp E 27, need not determine the measure E,,,. Indeed, consider the operator measures E and F which are concentrated on [O. co) and defined as follows:

E(X) = $1 s

c-‘~“~ du, X

F(X) = &I s

P”i4[1 - sin(u114)]dz.4, X

for all X E B([O, cc)). Then (see, e.g., [9. p. 1261)

s cm

d‘dE&x) = i(4k + 3)! = s

co

.xk dF,,,(x) 0 0

for all q E IFI, k E N. Thus LE(idk) = LF(idk) for all k E N, though E # F: in fact, E,,, # Fp,cp for each p E 7f \ {O).

The classical results concerning the Hamburger and the Stieltjes moment prob- lems (see e.g. [3] or [4]) lead immediately to necessary conditions for a sequence of (unbounded) self-adjoint operators (Sk)pZo to appear as the sequence of the mo- ment operators of an operator measure E concentrated on IR and lR+ , respectively. In view of the applications to be discussed, we give the Stieltjes version of such conditions: a necessary condition that a sequence of (unbounded) self-adjoint op- erators (&)& is the sequence of the moment operators of an operator measure E : B([O, 00)) -+ C(‘Ft) is that

n

c GT?lSl+m 1 0 l.m=O

and n

c m%ns+m+1 > 0 l,m=O

142 A. DVUREtENSKIJ, P. LAHTI and K. YLINEN

for every finite sequence a~, . . . , an of complex numbers, with the understanding that the operator inequalities are considered on the respective domains $!!,D(&)

and f$!!~‘D(&). In the sequel we shall restrict our attention to the case of exponentially bounded

positive Bore1 measures, which appear in the applications to be considered, and which are known to have the following strong properties (see for instance [lo, Theorems 11.4.3 and 11.5.21).

LEMMA 3.1. Let I_L : f3(IR) + [0, 001 be a positive Bore1 measure such that for some a > 0

s ealr' dp(t) -z cm. w

a) All the moments of p are finite, that is, the integrals

Sk = s tk 44th k E N, R

b) The monomials tk, k E N, are complete in L*(iR, p), that is, if f E L*(IR, p) is such that

s f (t)tk dp(t) = 0 w

for all k E N, then f(t) = 0 p-almost everywhere. c) The measure p is determinate, that is, it is uniquely determined by its moment

sequence (sk)rEo.

4. The phase space observables Ai”)

The polar margins of the phase space observables constitute a class of physically interesting examples of bounded and unbounded operator measures for which the moment problems can be solved using the above results. To do that we shall first introduce the relevant phase space observables. In the following we assume that the Hilbert space E is separable.

For any two vectors q, r+Q E ‘FI, we let 19 ) ( pb 1 denote the operator 1~ ) ( $1 (q) = (9 1 v)q. Let (I n))Eo be a fixed orthonormal basis of ‘FI. We call it the number basis. Let

N=Enln)(nl, iI>0

a=~Jn+lln)(n+ll,

?pO

a*=CJn-tlln+l)(nl, il?O

POSITIVE OPERATOR MEASURES DETERMINED BY THEIR MOMENT SEQUENCES 143

be the associated number, lowering, and raising operators, with the appropriate domains. Let h : B(C) + [0, co] be the two-dimensional Lebesgue measure. Let D, = eZ“*--Za, z E @, be the (unitary) shift operator, and let T be any positive trace one operator, that is, a state. The set function

Z?(C) 3 Z H AT(Z) := i s

DJD; d(z) E C(7-0 Z

is then a (positive normalized) operator measure, the phase space observable as- sociated with the state T [ 11, 121. It is known [ 131 that the only phase space observables AT whose polar coordinate marginal measures define number and phase observables are those which arise from the number states T = 1s ) ( s 1, or from their mixtures T = CFco h, 1s ) ( s (, h, 3 0, C,“==. h, = 1. Therefore, from now on we only consider the phase space observables which are defined by the number states Is){ s]. We denote them by AIS).

5. The polar margins of the phase space observable Ai”)

Writing @ 3 z = ]z]eiXsz = reie, (r, 0) E [0, 00) x [0,2~) we may define the polar coordinate marginal measures of AI.‘):

f3([0, co)) 3 R w A”)((R x [0, 27~)) =: i?‘(R) E C(x),

23([0,2n)) 3 X t-+ AIS)([O, co) x X) =: l+)(X) E C(x).

The operator measures EIS) and FIS), s E N, are known to represent an unsharp number observable and the conjugate unsharp phase observable, respectively [14, 151. The unsharp phase observable FIS) is an operator measure concentrated on a bounded interval of the real line, so that its moment problem is completely solved by the discussion in Section 2. On the other hand, the unsharp number observable

EIS) is unbounded, so that it could well happen that its moment operators do

not suffice to determine the operator measure EIS). We shall turn to the moment problem for the unsharp number observables. For notational simplicity, we consider

from now on instead of El”) the operator measures El”) defined by

J+)(X) = i”)(d%, X E a([O, co)),

so that, in particular,

s 00 s 03

Xk dE’qx) = (X2)k &v(x), 0 0

for all k > 0.

144 A. DVURECENSKIJ, P. LAHTI and K. YLINEN

6. The moment problem for the operator measure EiS)

The moment operators of the operator measures El’), s E N, have already been determined [5]. They are:

L”“(id’)=~~(s.k,n)ln)(nl =:a(s,k,N), n=O

D(LE’“‘(idk)) = D(Z@),

where

IV1

(J(.s,k,n)= C(-l)‘+e’ ; ; l,i’=O (>( )(

..y-z’)(k+;-l’)k!

k

= c UijS k-jni , ai,j E N,

i, j=O

with [n, S] := min(n, s). We now have (1 n))rSo C nkzoD(LE”‘(idk)),

plex measures E/$m, associated with the number basis (I n))rzo are be the following:

E/,f;,,,,(X) = (n I E’s)(X)lm)

and the com-

easily seen to

=(nIA’S)(z/jTx[0,2n))Im)=0, for nfm,

E/;;<,JX) = (n I E’S’(X)ln) = (n 1 Ais)@ x [0,2rt))]n)

= s g(‘) (x) dx, x n

where

We denote E/&) =: pf). The probability measure &) is absolutely continu- ous with respect to the Lebesque measure pi, and its Radon-Nikodym derivative d&)/dpL is gfF). In fact, the density gf) has the form

g(s)(X) = /.r(s)e-x n n ’

where

POSITIVE OPERATOR MEASURES DETERMINED BY THEIR MOMENT SEQUENCES 145

This shows immediately that &O” eax ~/J!)(X) < co for any 0 I a < 1. By Lemma 3.1 we may thus conclude.

COROLLARY 6.1. For all integers s, n >_ 0, the monomials xk, k E IV, are complete in L*(&), [0, KI)), and the measure pf’ is determinate.

THEOREM 6.2. For any s > 0, the operator measure El”) is uniquely determined by the moment operator sequence (o(s, k, N))zQ=,.

Proof: For a given integer s, assume that there is an operator measure L7 : a([O, co)) + L(X) for which LE(idk) = a(s, k, N), for all k 1 0. We recall that the operator measure E is uniquely determined by the complex measures E I,?). I,~), n, m > 0. For any n >_ 0, the sequence (a (s, k, n))~?O determines uniquely the

probability measure E/zi,,fl, so that Eln),ln) = E/ii,,n,. Assume now that n # m. Let

(‘) M,m.n and @l,m,n denote the positive measures Ei,n)+i/ln),lm)+i’(n) and Ei,:i)+i,,n).,,~)+i,l,l),

respectively. Using the polarization identity we may write

1 3 E In),lm) = 4 c AQ?l.n.

l=O

By assumption,

s cc

xk dFl,m,n = 0 s 00

0) Xk dkm,n 0

= s co

xk(hj;s’ + hz))e-” dx. 0

The moment sequence (~ooox (h, k cs) + h(sJ)e-n dx)r=“=, determines again uniquely the m fs)

measure pLl,m.n, (s) _ so that pl,m,n - ~l,~,~. But then

3

&),I~) = i c i'k+,n I=0

1 3 .1 (s) =- 4 c E h,m,n = E(:j,,m) 1

l=O

which concludes the proof.

Acknowledgement

A.D. is grateful to the Academy of Finland for organizing his stay at the De- partment of Physics, University of Turku, in October 1998, when the present paper was initiated. The paper was partially supported by the grant VEGA No. 4033/98 of the Slovak Academy of Sciences.

146 A. DVURECENSKIJ, P. LAHTI and K. YLINEN

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