# Hamburger and Stieltjes Moment Problems for Operators

Authors: L. Lemnete-Ninulescu
Journal: ISRN Mathematical Analysis
 (2014)

---

## Abstract

Solutions to some operator-valued, unidimensional, Hamburger and Stieltjes moment problems in this paper are given. Necessary and sufficient conditions on some sequences of bounded operators being Hamburger, respectively, Stieltjes operator-valued moment sequences are obtained. The determinateness of the operator-valued Hamburger and Stieltjes moment sequence is studied.


---

## Body

## 1. Introduction

A functionE(λ), a≤λ≤b, is called a spectral function if(a)
E(λ) is a bounded, positive operator,(b)
E(λ)≤E(μ), for any λ≤μ,(c)
E(λ+0)=E(λ),(c′)
E(a+0)=0 and/or E(b-0)=E(b) in case a=-∞ and/or b=+∞.The spectral function E(λ) is called an orthogonal spectral function if every E(λ) is an orthogonal projection [1, page 322].(1)
A sequence{An}n=0+∞ of bounded self-adjoint operators, acting on an arbitrary, complex Hilbert space H, subject on the condition A0=IdH, is called a Hamburger, unidimensional operator-valued moment sequence, if there exists an orthogonal spectral function E(λ), -∞≤λ≤+∞, such that An=∫-∞+∞‍λndE(t), n=0,1,2,…, or(2)
a sequence{An}n=0+∞, A0=IdH, of bounded self-adjoint operators is called a unidimensional operator-valued Hamburger moment sequence, if there exists a positive operator-valued measure F(λ), -∞≤λ≤+∞, measure generated by a spectral function, such that An=∫-∞+∞‍λndF(t), n=0,1,2,….A sequence{An}n=0+∞ of bounded positive operators is called a Stieltjes unidimensional operator-valued moment sequence, if there exists a positive operator-valued measure F(λ), 0≤λ≤+∞, (generated by a spectral function) such that An=∫0+∞‍λndF(λ), n=0,1,2,…. The passage from the integral representation (1) to an integral representation (2) is done, usually, by applying Naimark’s dilation theorem, or modified forms of it as in [1].In both cases(1) and (2), the operator-valued measures E(λ) or F(λ) are called the representing measures for the sequence {An}n=0+∞. Necessary and sufficient conditions for representing scalar sequences or operator-valued sequences, in one or several variables, as Hamburger or Stieltjes moment sequences with respect to scalar, respectively, operator-valued, positive measures, represent the subject of many outstanding papers such as [1–4]…, to quote only few of them.In the present paper, in Section3, we give a necessary and sufficient condition on a sequence of bounded, self-adjoint operators to be a Hamburger operator-valued, unidimensional moment sequence. In Section 4, we discuss the uniqueness of the representing measures of the operator-valued Hamburger moment sequence both in (1) and (2) forms. In Section 5, we give some necessary and sufficient conditions on a sequence of positive operators to be a Stieltjes operator-valued, unidimensional moment sequence with respect to a positive, operator-valued measure. The positive representing measures in Sections 3 and 5 are obtained by applying Kolmogorov’s theorem of decomposition of the positive definite kernels.


## 2. Preliminaries

Lett∈R denote the real variable in the real Euclidean space; for H an arbitrary complex Hilbert space, L(H) represents the algebra of bounded operators an H; we denote with δi·:N→{0,1}, the function
(1)δij={1,i=j,0,i≠j;
for K a Hilbert space, B(H,K) represents the set of bounded operators from H in K. We consider the C-vector space of vectorial functions: F={f:{0,1,…,n,…,}→H, f(·)=∑n∈N‍δn·f(n), f with finite support, f(n)∈H}. We define also the convolution f*δ1·∈F as
(2)[f*δ1·](n)=∑k∈Z‍(f(n-k)·δ1k),n∈N*,
and make the convention:f*δ1·(0)=0H. We have f*δ1·=∑n∈N‍δ(n+1)·f(n), f with finite support.In Section3, a necessary and sufficient condition on a sequence of self-adjoint operators to be a Hamburger operator-valued moment sequence is given. In Section 5, we give necessary and sufficient conditions on a sequence of positive operators to be a Stieltjes operator-valued moment sequence. In Section 4, the problem of the uniqueness of the represented measures in Sections 3 and 5 is studied. The representing measures in Sections 3 and 5 are obtained by applying Kolmogorov’s theorem on decomposition of the positive kernels. Classical Kolmogorov’s theorem for the decomposition of positive kernels is as follows:“LetΓ:S×S→L(H) be a nonnegative-definite function where S is an arbitrary set and H a Hilbert space, namely, ∑i,j=1n〈Γ(si,sj)xj,xi〉H≥0, for any finite number of points s1,…,sn∈S and any vectors x1,…,xn∈H. In this case, there exists a Hilbert space K (essentially unique) and a function h:S→B(H,K) such that Γ(s,t)=h(t)*h(s) for any s,t∈S.”We apply this theorem for a particular setS and a particular positive-definite operator-valued function to give an integral representation as Hamburger operator-valued moment sequence and Stieltjes operator-valued moment sequence, respectively, to some sequences of self-adjoint and positive operators, respectively.


## 3. An Operator-Valued Hamburger Moment Sequence Main Result

LetΓ={Γn}n∈N be a sequence of bounded self-adjoint operators, acting on an arbitrary complex, separable Hilbert space; that is, Γn∈L(H), Γn=Γn*, for all n∈N, Γ0=IdH, subject on the following conditions: for any finite vectors’ sequence {xn}n∈I⊂N⊂H, there exists another vector sequence {yn′}n∈I⊂N⊂H such that the following two equations are satisfied;(A)(3)∑p,q∈I‍〈Γp+q+2(xq-yq′),(xp-yp′)〉H-2Im⁡∑p,q∈I‍〈Γp+q+1(xq+yq′),(xp-yp′)〉H+∑p,q∈I‍〈Γp+q(xq+yq′),(xp+yp′)〉H=0;
and for any finite vectors’ sequence {xn}n∈I⊂N⊂H, there exists another vectors’ sequence {yn′′}n∈I⊂N⊂H such that(B)(4)∑p,q∈I‍〈Γp+q+2(xq-yq′′),(xp-yp′′)〉H+2Im⁡∑p,q∈I‍〈Γp+q+1(xq+yq′′),(xp-yp′′)〉H+∑p,q∈I‍〈Γp+q(xq+yq′′),(xp+yp′′)〉H=0.Proposition 1.
LetΓ={Γn}n∈N be a sequence of bounded, self-adjoint operators, acting on an arbitrary complex, separable Hilbert space H, subject on the conditions: Γ0= IdH, (A) and (B) satisfied. The following statements are equivalent. (i)
We have:(5)∑n,m∈N‍〈Γn+mxm,xn〉H≥0for all sequences{xn}n∈Hwith finite support.(ii)
There exists a positive operator-valued measureEA(spectral function) defined onBor(R)such that(6)Γn=∫-∞+∞‍tndEA(t),n=0,1,2,…,k,….Proof.
WhenF={f:{0,1,…,n,…}→H, f(·)=∑n∈{0,1,…,n,…}‍δn·f(n), f with finite support} is the C-vector space of functions defined on N with vectorial values, we consider the kernel Γ as a double indexed, symmetric one:
(7)Γ:{0,1,…,n,…,}×{0,1,…,n,…,}⟶B(H),Γn+m=Γ(n,m).
With the aid of Γ, we introduce the Hermitian, square positive functional ΛΓ:F×F→C, ΛΓ(f,g)=∑m,n∈{0,1,…,n,…,}‍〈Γn+mf(m),g(n)〉H. From property (i) of the kernel Γ, as well as from the properties of the scalar product in H, ΛΓ satisfies the following conditions.
(10) ΛΓ is C-linear in the first argument.
(20) ΛΓ(f,g)¯=ΛΓ(g,f), for all f,g∈F.
(30) ΛΓ(f,f)≥0, for all f∈F; and, moreover, being a Hermitian square, positive functional on F×F, ΛΓ satisfies the Cauchy-Buniakovski-Schwarz inequality, respectively:
(40)
(8)|ΛΓ(f,g)|≤ΛΓ(f,f)1/2ΛΓ(g,g)1/2,∀f,g∈F.
Also, from the construction of the Hermitian functional ΛΓ and the symmetry of the kernel Γ, (Γn+1+m=Γ(n+1,m)=Γ(n,m+1)=Γn+m+1), the functional ΛNΓ satisfies the equalities:
(50)
(9)ΛΓ(f*δ1·,g)=ΛΓ(f,g*δ1·),∀f,g∈F.
With these assumptions, F∋f→〈f,f〉F1/2=ΛΓ(f,f)1/2 is a seminorm on F. Let S be the subset in F, defined as S={f∈F with ΛΓ(f,f)=0}. If follows, using the Cauchy-Buniakovski-Schwarz inequality, that if f1,f2∈S, we have also ΛΓ(α1f1+α2f2,α1f1+α2f2)=0; that is, S⊂F is a vector subspace in F. We consider the separated completion space of F with respect to S; that is, in this case, the quotient completion space K=F/S¯∥·∥Γ. Obviously, K is a Hilbert space with the usual norm in the completion Hilbert space and D=F/S is a dense subspace of it (i.e., ∥h^∥K=limn∥hnx∥=limn〈δn·x+S,δn·x+S,〉F1/2, where hx={hnx} is a Cauchy sequence of elements in F/S and hnx=δn·x^=δn·x+S). The Hilbert space K is uniquely defined and it is also described as K=VRanhnx, (the closed linear span of the ranges of the operators hn:H→K, hnx=δn·x^=δn·x+S). From Kolmogorov’s decomposition theorem of positively defined kernels, with the above construction, the decompositions Γn+m=Γ(n,m)=hn*hm hold for any n,m∈N. Let us consider the densely defined subspace of K, D=F/S=∑n∈I⊂N‍hnxn, I finite xn∈H, D⊂K, and the operator A:D→D, defined by A(∑n∈I⊂N‍hnxn)=∑n∈I⊂N‍hn+1xn. We prove that A is correctly defined. Consequently, we consider the elements ∑m∈J⊂N‍hmxm=∑m∈J⊂N‍(δm·xm+S) and ∑n∈I⊂N‍hnxn=∑n∈I⊂N‍(δn·xn+S) such that ∑n∈I⊂N‍δn·xn-∑m∈J⊂N‍δm·xm∈S and show that A(∑n∈I⊂N‍hnxn)=A(∑m∈J⊂N‍hmxm). The above equality is the same as the equality: ∑n∈I⊂N‍hn+1xn=∑m∈J⊂N‍hm+1xm (modulo S). Indeed, from (50), we have
(10)|ΛΓ(∑n‍δn+1·xn-∑m‍δm+1·xm,∑n‍δn+1·xn-∑m‍δm+1·xm)|=|ΛΓ(∑n‍δn·xn-∑m‍δm·xm,∑n‍δn+2·xn-∑m‍δm+2·xm)|≤ΛΓ(∑n‍δn·xn-∑m‍δm·xm,∑n‍δn·xn-∑m‍δm·xm)1/2·ΛΓ(l,l)1/2=0,
where l=∑n‍δn+2·xn-∑m‍δm+2·xm. From the above definition, we have
(60)
(11)〈Ax,y〉K=〈A(∑n∈I⊂N‍hnxn),∑m∈J⊂N‍hmym〉K=〈∑n∈I⊂N‍hn+1xn,∑m∈J⊂N‍hmym〉K=〈∑m,n∈I∪J⊂N‍hm*hn+1xn,ym〉K=∑m,n〈Γm+n+1xn,ym〉‍H=∑m,n‍〈Γn+m+1xn,ym〉H;
and also
(70)
(12)〈x,Ay〉K=〈∑n∈I⊂N‍hnxn,A(∑m∈J⊂N‍hmym)〉K=〈∑n∈I⊂N‍hnxn,∑m∈J⊂N‍hm+1ym〉K=〈∑m,n∈I∪J⊂N‍hm+1*hnxn,ym〉K=∑m,n‍〈Γm+1+nxn,ym〉H=∑m,n‍〈Γn+m+1xn,ym〉H.
From (60) and (70)〈Ax,y〉K=〈x,Ay〉K for x,y∈D arbitrary; we infer that A is a densely defined, symmetric operator. We prove that A has equal deficiency indices in K; consequently A’s Cayley transform is a partial isometry on K+=R(A+iIK) with values in K-=R(A-iIK). Indeed, let K±=R(A±iIK)=Vn∈Ifinite⊂N(hn+1xn±ihnxn), xn∈H arbitrary, be the ranges in K of the operators (A±iIK). We prove that K± are vector subspaces in K. For this request, we consider the elements f±=∑n∈1finite⊂N‍(hn+1xn±ihnxn), g=∑n∈I2finite⊂N‍(hn+1yn±ihnyn) in K± and α,β∈C arbitrary. Let us define the elements:
{
x
~
n
}
n
∈
I
1
∪
I
2
=
I,
(13)x~n={xn,n∈I1,0H,n∈I2-(I1∩I2),
{
y
~
n
}
n
∈
I
1
∪
I
2
(14)
y
~
n
=
{
y
n
,
n
∈
I
2
,
0
H
,
n
∈
I
1
-
(
I
1
∩
I
2
)
,
also the elements f~±=∑n∈1finite⊂N‍(hn+1x~n±ihnx~n), g~=∑n∈I2finite⊂N‍(hn+1y~n±ihny~n) in K± and α,β∈C. It results that f-f~=0K; g-g~=0K and [αf~±+βg~±]=[αf±+βg±]=∑n∈I1∩I2‍[hn+1(αx~n)+ihn(αx~n)+hn+1(βy~n)±ihn(βy~n)]+∑n∈I1-(I1∩I2)‍[hn+1(αx~n)+ihn(βy~n)+hn+1(βy~n)±ihn(αx~n+βy~n]+∑n∈I2-(I1∩I2)‍[hn+1(αx~n)+βy~n±ihn(αx~n)+βy~n)=∑n∈I1∩I2‍[hn+1(αx~n+βy~n)±ihn(αx~n)+βy~n]∈K±. Because A is a symmetric operator, it results also that R(A±iIK)=K± are closed subspaces in K. We prove that, in conditions (A) and (B) for the kernel Γ, we have K+=R(A+iIK)=R(A-iIK)=K-. Indeed, given an arbitrary element f=∑n∈Ifinite⊂N‍(hn+1xn+ihnxn) we look for an element g∈K-=R(A-iIK), g=∑n∈Ifinite⊂N‍(hm+1ym′-ihmym′) such that (f-g)=0K. For a construction of the elements {xn~}n∈I1∪I2=I,{yn~}n∈I1∪I2=I⊂H like the previous one, we have (f-g)=∑‍[hn+1(x~n-y~n′)+ihn(x~n+y~n′)]=0K⇔∑p,q∈I‍Γp+q[∑n‍δ(n+1).(x~n-y~n′)+iδn.(x~n-y~n′)](q), [∑n‍δ(n+1).(x~n-y~n′)+iδn.(x~n-y~n′)](p)>H=0⇔∑p,q∈I,p,q≥1‍Γp+q[(x~q-1-y~q-1′)+i(x~q+y~q′)], (x~p-1-y~p-1′+i(x~p+y~p′)>H=0K⇔∑p,q≥0‍〈Γp+q+2(x~q-y~q′),(x~p-y~p′)〉H-2Im⁡∑p,q∈I〈Γp+q+1(x~q+y~q′),(x~p-y~p′)〉‍H+∑p,q∈I‍(xq+y~q′),(x~p+y~p′)>H=0K. According to condition (A), on the kernel {Γn}n, such an element exists. We have K+⊂K-. Conversely, let f∈K-,f=∑n∈I‍(hn+1xn-ihnxn); we search for an element g∈K+,g=∑n∈I‍(hn+1y~n′′+ihny~n′′) with the property that (f-g)=0K. Consequently, we have to find an element g∈K+ such that ∑‍[δ(n+1).(x~n-y~n′)+ihn(x~n+y~n′)]=0K⇔∑p,q∈I‍Γp+q[∑n‍δ(n+1).(x~n-y~n′′)-iδn.(x~n+y~n′′)](q), ∑n‍δ(n+1).(x~n-y~n′′)-iδn.(x~n+y~n′′)](p)>H=0K⇔∑p,q∈I‍〈Γp+q+2(x~q-y~q′′),(x~p-y~p′′)〉H+2Im⁡∑p,q∈I‍〈Γp+q+1(x~q+y~q′′),(x~p-y~p′′)〉H+∑p,q∈I‍〈Γp+q(x~q+y~q′′),(x~p+y~p′′)〉H=0. We prove with these computations that K+=R(A+iIK)=K-=R(A-iIK)⊂K. That is dim K+⊥=dim⁡K-⊥⇒A’s Cayley transform has equal deficiency indices and ⇒A admits a self-adjoint extension A~. Let EA be the spectral measure of the self-adjoint operator A~. Because hmx=Am(h0x) for all x∈H and Γmx=h0*hmx, it results that Γmx=h0*Amh0x, for all x∈H and the integral representations Γmx=h0*∫R‍tmdEA(t)h0x, for all x∈H, for all m∈N. We consider the positive operator-valued measure FA(t)=h0*EAh0. With respect to this positive operator-valued measure, we have Γmx=h0*Amh0x=∫R‍tmdFA(t)x, for all x∈H and all m∈N. That is Γm=∫R‍tmdFA(t), for all m∈N, the required Hamburger moment integral representations.
Conversely. If the terms {Γm}m admit the integral representations Γm=∫-∞+∞‍tmdFA(t), for all m=0,1,2,…, for a positive operator-valued measure on R, we have
(15)∑n,m‍〈Γn+mxm,xn〉H=∑n,m‍〈∫-∞+∞‍tn+mdFA(t)xm,xn〉H=∫-∞+∞‍d(〈∑m‍tmFA1/2(t)xm,∑n‍tnFA1/2(t)xn〉H)=∫R‍d(∥FA1/2∑n‍tnxn∥2)≥0,
as it is required by (i).


## 4. About the Uniqueness of the Hamburger Operator-Valued Moment Sequences’ Representations

Let us consider a sequence of bounded operators{An}n∈N, An∈L(H), subject on the condition An=An*, A0=IdH, n=0,1,2,…, H an arbitrary complex Hilbert space. For the sequence {An}n∈N, we get two operator-valued integral representing measures (or spectral functions), EA,EB:Bor(H)→A(H), that is,
(16)An=∫-∞+∞‍tndEA(t)=∫-∞+∞‍tndEB(t),
for all n=0,1,2,…. The operator-valued measures allow us to define the scalar measures μx,νx:Bor(R)→[0,+∞], μx(D)=〈EA(D)x,x〉H, respectively, νx(D)=〈EB(D)x,x〉H when x∈H is arbitrary. With respect to these scalar measures we obtain
(17)anx=〈Anx,x〉H=∫-∞+∞‍tndμx(t)=∫-∞+∞‍tndνx(t),∀n∈N.
From [5, page 283], the Hamburger scalar moment problem is indeterminate (the sequence {anx}n does not uniquely determine the scalar representing measure). It follows that the operator-valued representing measure does not uniquely determine the Hamburger operator-valued moment sequence.However, under some additional conditions about the operator-valued representing measure, the Stieltjes (Hamburger) operator-valued moment sequence is determined [3, pages 509, 510, 511].Moreover, if the representing measure is that associated with a self-adjoint extension of a symmetric operator with deficiency indices (0,0), the self-adjoint extension is the canonical closure of the given operator and is defined on the whole space. Indeed, ifS:D(S)→H is symmetric with R(S±i)¯=H and A⊃S, the canonical closure of S, it follows that H⊇R(A±i)⊃R(S±i) are closed subspaces in H; that is R(A±i)=H. In this case the canonical closure of S is the smallest self-adjoint extension of S and is defined on the whole space H (as in Section 3 of this paper, Proposition 1). The same arguments are in [4, page 1267, Lemma 2.1].Proposition 2.
(1) Let{An}n=0+∞, An∈L(H), for all n∈N, H an arbitrary, complex Hilbert space, subject on the conditions An=An*, A0=IdH, and E1,E2:Bor(R)→A(H) two orthogonal spectral functions on R, such that
(18)An=∫-∞+∞‍tnE1(t)=∫-∞+∞‍tnE2(t),n=0,1,2,….
Then E1=E2 on Bor(R).Proof.
BecauseA1∈L(H), and A1=A1*, the existence of the representation A1=∫-∞+∞‍tdEA(t) with EA:Bor(R)→A(R), EA(R)=A0=IdH is the usual one and is unique. The spectral orthogonal measures coincide; that is, EA=E1=E2. The representing measure is the spectral orthogonal measure associated with the self-adjoint operator A1. From EA(λ)’s multiplicative property, it follows that An=∫-∞+∞‍tnEA(t)=(∫-∞+∞‍tEA(t))n for all n∈N. The uniqueness of the integral representations with respect to spectral functions is assured trivially only in case An=An, for all n∈N when the representation is possible.


## 5. Stieltjes Operator-Valued Moment Sequences

A sequence of bounded operatorsΓ={Γn}n, acting on an arbitrary Hilbert space H, is called a Stieltjes operator-valued moment sequence if there exists a positive operator-valued measure EΓ on [0,+∞) such that
(19)Γn=∫0+∞‍tndEΓ(t),n=0,1,2,….Proposition 3.
Let{Γn}n=0+∞ be an operator sequence with Γn∈A(H) for all n∈N and Γ0= IdH, with conditions (A) and (B) in Proposition 1 satisfied. The following assertions are equivalent.
(
j
)
(20)
(
1
)
∑
n
,
m
‍
〈
Γ
n
+
m
x
n
,
x
m
〉
H
≥
0
(
2
)
∑
n
,
m
‍
〈
Γ
n
+
m
+
1
x
n
,
x
m
〉
H
≥
0
for all sequences {xn}n=0+∞⊂H with finite support.
(
j
j
) There exists a positive operator-valued measure EΓ:Bor(R)→A(H) such that
(21)Γn=∫0+∞‍tndEΓ(t),n=0,1,2,….Proof.
We prove that condition(j) [(1) and (2)] is sufficient.
Condition(j), (1) is the same as (i) in Proposition 1. Consequently, there exists a positive operator-valued measure EΓ:Bor(R)→A(H) such that Γn=∫-∞+∞‍tndEΓ(t), n=0,1,2,…. In the statement (j), (2), if we consider the sequence with finite support {xn}n as xn=ξnx,ξn∈C, x∈H arbitrary, for all n∈N, we obtain
(22)∑n,m‍〈Γn+m+1xn,xm〉H=∑n,m‍∫-∞+∞‍tn+m+1ξnξm¯d〈EΓ(t)x,x〉H=∫-∞+∞‍t|∑n∈I‍tnξn|2d〈EΓ(t)x,x〉H≥0,
for all polynomials ∑n‍tnξn with complex coefficients and all x∈H. It follows that the representing measure EΓ is concentrated on [0,+∞).
Conversely. The implication (jj)⇒(j), (1) is the same as (ii)⇒(i) from Proposition 1. Moreover, from (jj) it results that, for any sequence {xn}n⊂H with finite support, we have
(23)∑n,m‍〈Γn+m+1xn,xm〉H=∑n,m‍∫-∞+∞‍tn+m+1d〈EΓ(t)xn,xm〉H=∫0+∞‍td〈EΓ(t)∑n‍tnxn,∑m‍tmxm〉H=∫0+∞‍td∥EΓ1/2(t)∑n‍tnxn∥≥0,
that is, (j), (2).
We give another second characterization on an operator sequence{Γn}n=0+∞ to be an operator-valued Stieltjes moment sequence.Remark 4.
In the sequel, we argue like in [1, page 329]. Between the Hamburger operator-valued moment sequences and Stieltjes moment sequences we can establish the following bijection.
(A) If Γ={Γn}n=0+∞ is a Stieltjes moment sequence with respect to the spectral measure EΓ(t) on [0,+∞) for the homeomorphism π1:[0,+∞)→(-∞,0], π1(t)=-t, there corresponds a spectral measure FΓ1 on (-∞,0] defined by FΓ1(t)=(1/2)[EΓ(∞)-EΓ∘π1-1(t)] such that ∫-∞0‍λ2kdFΓ1(λ)=(1/2)∫0+∞‍λkdEΓ.
For the homeomorphismπ2:[0,+∞)→[0,+∞), π2(t)=t, there corresponds a spectral measure FΓ2 on [0,+∞) defined by FΓ2(t)=(1/2)EΓ(∞)+(1/2)EΓ∘π2-1(t) such that ∫0∞‍λ2kdFΓ2(λ)=(1/2)∫0+∞‍λkdEΓ.
We define(24)FΓ={FΓ1(t),t<0,FΓ2(t),t≥0.
For FΓ(t) we have the representations: ∫-∞+∞‍λ2kdFΓ(λ)=∫0+∞‍λkdEΓ(λ), k=0,1,…,n,… and ∫-∞+∞‍λ2k+1dFΓ(λ)=0, k=0,1,…,n,….
(B)Conversely. If we have the operator-valued Hamburger moment sequence {Γ0,0,Γ1,0,Γ2,0,…} with respect to the spectral representing measure Fλ, respectively, the sequence {Bn}n=0+∞, defined by Bn=(Γ[n/2]+(-1)nΓ[n/2])/2 for all n∈N, admits the integral representation B2n=Γn=∫-∞+∞‍t2ndF(λ) and B2n+1=0=∫-∞+∞‍t2n+1dF(t). We can construct a spectral measure E(t) on [0,+∞); that is, for π:(-∞,+∞)→[0,+∞), π(t)=t2 we have E(t)=F∘π-1(t) with ∫-∞+∞‍tndF(t)=∫0+∞‍(t2)ndE(t)=B2n=Γn, n=0,1,2,….
For obtaining positive operator-valued measures from orthogonal, spectral functions, in both cases, we compose the orthogonal spectral measure associated with a self-adjoin operator with the same projections, respectively,h0, h0* in our case. Summing the above conditions (A) and (B) in Remark 4, for an operator-valued sequence {Γn}n=0∞∈L(H), we construct the operator-valued sequence Bn=(Γ[n/2]+(-1)nΓ[n/2])/2 for all n∈N and, conversely, for a sequence {Bn}n=0+∞∈L(H), we construct the operator-valued sequence {Γn}n=0+∞, with Γn=B2n, n∈N.
With the above construction, we have the following.Proposition 5.
The sequence{Γn}n=0+∞ that satisfies conditions (A) and (B) in Proposition 1 is a Stieltjes operator-valued moment sequence if and only if
(25)∑n,m=0q‍〈(Γ[(n+m)/2]+(-1)n+mΓ[(n+m)/2]2)xn,xm〉H≥0
for all sequences {xn}n=0+∞⊂H with finite support and all q∈N.Proposition5' (reformulated).The sequence{Γn}n=0+∞ is a Stieltjes operator-valued moment sequence if and only if(26)∑n,m=0q‍〈Bn+mxn,xm〉H≥0for all sequences{xn}n=0+∞⊂Hwith finite support and allq∈N,Bn, for alln∈Ndefined above.Proof.
Let{Γn}n=0+∞ be an operators’ sequence, Γn∈L(H), H an arbitrary complex Hilbert space; we define Bn=(Γ[n/2]+(-1)nΓ[n/2])/2, n∈N; that is, B2n=Γn and B2n+1=0, for all n∈N. In the condition of the hypothesis (Propositions 5 and 5′), we have ∑n,m‍〈Bn+mxn,xm〉H≥0 for all sequences {xn}n⊂H with finite support. From Proposition 1, there exists a positive operator-valued measure on R such that
(27)Bn=∫-∞+∞‍tndF(t),n=0,1,2,….
From Remark 4, there exists a positive operator-valued measure E(t) such that
(28)B2n=∫-∞+∞‍t2ndF(t)=Γ[2n/2]=Γn=∫0+∞‍tndE(t),B2n=∫-∞+∞‍t2ndF(t)=Γ[2n/2]=ΓnΓnΓn=0,1,2,….
That is {Γn}n∈N is a Stieltjes operator-valued moment sequence.
Conversely. If Γn=∫0+∞‍tndE(t), n=0,1,2,…, we construct a measure F(t) positively defined on (-∞,+∞), as in Remark 4, with the property that Γn=∫-∞+∞‍t2ndF(t)=∫0+∞‍tndE(t)=B2n; and B2n+1=∫-∞+∞‍t2n+1dF(t)=0. In this case,
(29)∑n,m=0q‍〈(Γ[(n+m)/2]+(-1)n+mΓ[(n+m)/2]2)xn,xm〉H=∑n,m=0q‍〈∫-∞+∞‍tn+mdF(t)xn,xm〉H=∫-∞+∞‍d∥∑n‍tnF1/2(t)xn∥2≥0.Proposition 6.
Let{Γn}n=0+∞ be an operator sequence with Γn∈A(H) for all n∈N and Γ0=IdH. The following assertions are equivalent:
(
j
)
(30)
(
1
)
∑
n
,
m
‍
〈
Γ
n
+
m
x
n
,
x
m
〉
H
≥
0
(
2
)
∑
n
,
m
‍
〈
Γ
n
+
m
+
1
x
n
,
x
m
〉
H
≥
0
for all sequences {xn}n=0+∞⊂H with finite support.
(
j
j
) There exists a positive operator-valued measure EΓ:Bor(R)→A(H) such that
(31)Γn=∫0+∞‍tndEΓ(t),n=0,1,2,….Proof.
We prove that condition(j) [(1) and (2)] is sufficient.
Condition(j), (1) is the same with (i) in Proposition 1. Consequently, there exists a positive operator-valued measure EΓ:Bor(R)→A(H) such that Γn=∫-∞+∞‍tndEΓ(t), n=0,1,2,…. In the statement (j), (2), if we consider the sequence with finite support {xn}n as xn=ξnx, ξn∈C, x∈H arbitrary, for all n∈N, we obtain
(32)∑n,m‍〈Γn+m+1xn,xm〉H=∑n,m‍∫-∞+∞‍tn+m+1ξnξm¯d〈EΓ(t)x,x〉H=∫-∞+∞‍t|∑n∈I‍tnξn|2d〈EΓ(t)x,x〉H≥0,
for all polynomials ∑n‍tnξn with complex coefficients and all x∈H. It follows that the representing measure EΓ is concentrated on [0,+∞).
Conversely. The implication (jj)⇒(j), (1) is the same with (ii)⇒(i) from Proposition 1. Moreover, from (jj) it results that, for any sequence {xn}n⊂H with finite support, we have
(33)∑n,m‍〈Γn+m+1xn,xm〉H=∑n,m‍∫-∞+∞‍tn+m+1d〈EΓ(t)xn,xm〉H=∫0+∞‍td〈EΓ(t)∑n‍tnxn,∑m‍tmxm〉H=∫0+∞‍td∥EΓ1/2(t)∑n‍tnxn∥≥0,
that is (j), (2).



---

*Source: 836839-2014-04-30.xml*