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Владикавказский математический журнал Январь-март, 1999, Том 1, Выпуск 1
УДК 517.98
THE STIELTJES MOMENT PROBLEM IN VECTOR LATTICES A. G. Kusraev and S. A. Malyugin
1. Introduction
The following problem for the first time was posed in the famous memoir by Thomas Stieltjes [1] devoted to continued fractions: given a real sequence (sk)f?=0, find a nonde-creasing function a on a positive half-line R+ such that
oo
Jtkda(t)=sk (k:= 0,1,...). o
He had called it the moment problem having in mind an obvious mechanical interpretation. Since then the moment problem has been developed in different directions as one of the most attractive and important areas of modern analysis. The extended moment problem is posed analogously on the whole real line and called the Hamburger moment problem, while the moment problem posed on a line segment is called the Hausdorff moment problem.
The whole history of the moment problem is quite well known and there is no need in recalling. It should be only noted that several authors explored vector-valued statements of the moment problem, see [2-5]. In the present short talk we will briefly outline the Stieltjes moment problem in vector lattices.
2. Two examples
First of all we consider two examples justifying the statement of the moment problem in vector lattices
2.1. The first example concerns a stochastic setting of the moment problem. Suppose that the moment sequence depends on a measurable parameter. More precisely, let (ft, X, u) be a measure space and sn : ft —> R be a measurable function for each n E N. Assume that the sequence is positive in the Stieltjes sense, i.e. the inequalities
n n
£ °k°isk+i{u) > o, akcriSk+i+1(io) ^ 0 ((fffc)S£=o Cf; n:=0,l,...)
k,l=0 k,l=0
hold almost everywhere in ft. Then, as is well-known, for almost every lo E ft the Stieltjes moment problem has a solution Now a new question arises: is the function :
w ^ measurable for each A E S or not? This new problem is actually the old one
but set in the space of measurable functions L°(v) := L°((l, Indeed, if we define a
© 1999 Kusraev A. G., Malyugin S. A.
vector measure by assigning A /j.^(A) (A E X) then the problem is to find a vector measure /j. : X —> L°(v) such that
sk(-) = tk dm(t) (k := 0,1,... ).
2.2. The second example concerns the spectral resolution of a self adjoint operator. Consider a positive self-adjoint operator .1 in a Hilbert space and suppose that (e^AeR is its spectral resolution. Then
oo oo
I = J de\, A = J X de\. 0 0
Denote by B a Boolean algebra of orthogonal projections in Hilbert space under consideration and (B) the space of all self-adjoint (not necessarily bounded) operators whose spectral resolutions take values in B. Now we can set the following natural question:
Given a sequence (Ak)kefi of pairwise commuting positive self-adjoint operators in (B) with Ao = I, find a spectral resolution or a spectral measure /j. : £>(M+) —> B such that
oo
Ak = j Xk dfi(X) (k:= 0,1,...). o
3. Vector lattices
We recall the basic notion from the theory of vector lattice (= Riesz spaces), see [6, 7].
3.1. An ordered vector space over is a pair (E, where E is a real vector space and ^ is an order relation in E with the following conditions being fulfilled:
(1) x ^ ylku ^v^-x + u^y + v (x,y,v,u £ E);
(2)ï<y^Aï<A|/ (ï,î/GE;0<AGl.
Thus, inequalities in an ordered vector spaces can be summed and multiplied by positive reals.
3.2. Vector lattice is an ordered vector space which is a lattice. Thus, in any vector lattice E there exist least upper bound sup{xi,..., xn} := x\ V ■ ■ ■ V xn and greatest lower bound inf{xi,..., xn} := x\ A ■ ■ ■ A xn for an arbitrary finite subset {xi,..., xn} C E. In particular, every element x E E has the positive part x+ := î VO, the negative part x~ := (—x)+ := —x A 0, and the module \ x |:= x V (—x).
3.3. Disjointness _ in a vector lattice E is introduced by
_L:= {(x,y) E E x E :| x | A | y |= 0}. A band in E is a set of the form
Mx := {x E E : (Vy G M) x _L y},
where M C E. The set of all bands ordered by inclusion is a complete Boolean algebra *8(E) with the following Boolean operations:
LAK:=LnK, LVK=(L U K)x±, L* := Lx (L, K G »(£)).
The Boolean algebra *8(E) is called the basis of E.
3.4. An element 1 G E is said to be (weak) order unit if {1}XX = E, i.e. if there is no nonzero element in E disjoint to 1. A positive element e G E is called a fragment or a component of the unit if e A (1 — e) = 0. The set of all fragments of 1, denoted by B (1), is a Boolean algebra with lattice operations being induced from E. Moreover, e* := 1 — e is the Boolean complement of e.
3.5. A vector lattice E is said to be Dedekind cr-complete if each order bounded countable set in E has supremum and infimum. In this case B(l), is a cr-complete Boolean algebra. A Dedekind cr-complete vector lattice E can be represented as a direct sum {e}x © jej_xx for every e G E. The corresponding projection 1onto the band {e}xx (parallel to (e}x) is called a band projection and can be calculated by
Pex = sup{x A (ne) : n G N} (x G E+).
4. The Freudenthal spectral theorem
The Freudenthal spectral theorem is one of the most powerful tools in the theory of vector lattices, and can be interpreted as one of the first solutions to the moment problem in vector lattices. In the next two sections E is a Dedekind cr-complete vector lattice with a weak order unit 1. The Boolean algebra of all components of 1 will be denoted by B = B (1).
4.1. A resolution of unity in B is a mapping e : R —> B such that
(1) e(A) < e(/i) for A <
(2) V\e , ' (A) = 1, A/(e , ' (//)
(3) \Jtl<xe(p)=e(\) (AGM).
To each x G E we assign a resolution of identity (e^)AeR in ® by setting e\ := Pc(\) 1, where c(A) = (A1 — x) +. This resolution of identity is called the spectral function of x.
4.2. Now, define the Stieltjes integral with respect to an arbitrary resolution of identity e : R —> B. Let / : R —> R be an uniformly continuous function. Take a partition of real axis A := (\k)kez,
—oo An < ■ ■ ■ < A_i < Ao < Ai < ■ ■ ■ < An —> +oo and compose the integral sum
+oo
X] /(¿«Xe^n+i) - e(An)),
— oo
where An < tn < An+i- It is clear that there exists an order limit for integral sums as partitions are refined. This limit is called the Stieltjes integral of / with respect to a resolution of identity e(-) and denoted by
r +oo
f(X)del:= / /(A) dexx := o-lim £ f(tn)(e(Xn+1) - e(An)).
R ^oo 00
Soundness of the above definitions can be easily verified.
The Freudenthal spectral theorem (1936). For every x G E the integral representation holds:
x
-A-
4.3. A spectral measure is a cr-continuous Boolean homomorphism /j. from £>(R) to (£); here cj-continuity means that for any sequence of pairwise disjoint elements (An) C /?(R) we have
U A>< = V
\k=1 / k=1
Theorem (J .D. M. Wright [8]). Each spectral resolution (ef)xeR has a unique extension to a spectral measure, i.e. there exists a unique spectral measure ¡j,x : £>(R) —)• (E) such that c\ = /j.x(—oo,X) (À G M). Moreover,
f(X)det= [ f(\)dn(\).
4.4. In E one can introduce a unique partial multiplication so that 1 is a neutral element. It can be easily seen that if x ^ 0 and all xn exist then the spectral measure ¡j,x is a solution to the Stieltjes moment problem in E for the sequence 1, x, x2,..., xn,..., i.e.
AkdcTx = / Xkd/j(X) (k := 0,1,...).
5. The main results
5.1. A sequence in E is said to be positive if
J2 VkViSk+i ^ o ((<Tk)f=0 CM; n := 0,1,...),
k,l=o
and positive in Stieltjes sense if
n n
okcriSk+i ^ 0, okcriSk+i+((crfe)£l0n := 0,1,...). k,l=0 k,l=0
5.2. Theorem. Given a sequence (sk)f)=0 C E, there exists a positive measure /j. : /?(R) —> E which solves the Stieltjes moment problem if and only if the sequence is positive in Stieltjes sense.
5.3. Theorem. For every positive in Stieltjes sense sequence (sk)f)=0 there exists a sequence of pairwise disjoint principal band projections (tt)^=0 and a band projection ir}t
with TTh o irk = 0 (k := 0,1,...) and ir^ + X^feLo 71 & = ^e ^or which the following statements hold:
(1) for the sequence (noSk)kLo in ttqE the Stieltjes moment problem has a unique solution;
(2) for the sequence (irSk)k)=0 in irE solution to the Stieltjes moment problem is not unique for whatever nonzero band projection ir ^ ir^;
(3) for the sequence (irnSk)kLo in tt„E the Stieltjes moment problem has a unique solution which can be represented as a linear combination of n disjoint spectral measures, whatever n G N.
5.4. The band projections in Theorem 5.3 can be described explicitly. Let V(R+ ) be the vector space of all polynomials defined on IR+. We introduce a positive linear operator U : V(R+) E defined by
n n
U(P) : = ^«fc.s'fc. p(u) = ^2akuk. k=0 k=0
Fix an arbitrary complex number A G C\IR+ and consider the function R\(u) = 5R1 /(ti — A). We define the following vector in E:
a := ini{U(p — q) :p,qEV(R+), q < Rx < p).
Then 7r/j coincides with the band projection onto the band {a}xx. For an arbitrary (ak)k=0 in IR we put
{n
sk+iak&i
k,l=0
Let En = C){E(oq, ..., an) : ctq + ■ ■ ■ + a2 >0}. Then iro coincides with the projection onto the band D^+o ^n and irn coincides with the projection onto the band En-\ fl E^.
5.5. Analogous results are true for the Hamburger and Hausdorff moment problems. The main difficulty is to find an appropriate measure extension in vector lattices, see [9, 10].
References
1. T. Stieltjes. Recherches sur les fractions continue. Ann. de Toulouse. VIII-IX, 1984-1985.
2. M. G. Krein. Infinite J-matrices and matrix moment problem. Dokl. Akad. Nauk.—1949.—V. 69, No. 2.—P. 125-128.
3. L. V. Kantorovich. To general theory of operations in semiordered spaces. Dokl. Acad. Nauk.— l'.KiG. V. 1. !>. 271-274.
4. K. Schmiidgen. On a generalization of the classical moment problem. J. Math. Anal. Appl.—1987.— V. 125, No. :i. !>. 461-470.
5. Yu. M. Berezanskii. A generalized power problem of moments. Trudy Moskovsk. Mat. Obsc.— 1970. V. 21.—P. 47-102.
6. W. A. J. Luxemburg and A. C. Zaanen. Riesz Spaces. I. North Holland, Amsterdam, 1971.
7. A. C. Zaanen. Riesz Spaces. II. North Holland, Amsterdam, 1983.
8. J .D. M. Wright. Vector lattice measures on locally compact spaces. Math. Z.—1971.—V. 120, No. 3.—P. 193-203.
9. A. G. Kusraev and S. A. Malyugin. On the vector Hamburger moment problem. Optimization.— iggg.—y_ 45.^P. 99^107.
10. S. A. Maiyugin. The moment problem in iCj-space. Siberian Math. J.—1993.—V. 34, No. 2.—P. 297306.
