OPERATOR-VALUED TRIGONOMETRIC MOMENT AND L-MOMENT PROBLEMS

AND L-MOMENT PROBLEMS

LUMINIT¸ A LEMNETE-NINULESCU

We give a necessary and sufficient condition for a sequence of complex matrices to be represented as a trigonometric moment sequence. We also give a necessary and sufficient condition for a sequence of complex matrices to generateL-scalar moment sequences with respect to real-valued measurable functions.

AMS 2000 Subject Classification: Primary 47A57, 44A60; Secondary 15A57.

Key words: L-moment problem, positive bounded Borel measure, linear func- tional, support of a function, support of a measure.

1. INTRODUCTION

The present note is related to some finite dimensional complex matrix valued moment problem. Results about the so-called “k-complex moment prob- lem” and “L-moment problem” will be published [8], [11]. Necessary and sufficient conditions for representing sequences of Hilbert space operators as Hausdorff, Stieltjes-Hermite or Hamburger moment sequences with respect to a Borel operator-valued positive measure are given in many remarkable pa- pers such as [12], [13], [14], [15], [17]. The representations of a sequence of Hilbert space operators as a moment sequence is obtained either by study- ing the positivity of precise functionals associated with the moment sequence (the so-called “moment functionals”) ([14], [15], [17]) or by means of inte- gral representations obtained for some analytical operator-valued functions ([12], [13]). These results are not unexpected when compared with the clas- sical results obtained in the scalar case by Akhiezer [1]. For example, in [13], the notions of R-operator, respectively R₊-operator are introduced and inte- gral representations for these operators with respect to Borel operator-valued measures are provided. In the integral representation of an R+-operator, the interval on which the representation holds is [0⁻,+∞) and the integrand is a Cauchy kernel. A strong connection is thus established between the R+- operator and the Stieltjes-Hermite operator valued moment problem. In the same paper, for special Hilbert spaces a necessary and sufficient condition is given for the solution of a truncated moment problem. In the last section of

REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 473–482

[13], a necessary and sufficient condition is also given for a sequence of op- erators to be a Stieltjes-Hermite moment sequence and also a necessary and sufficient condition for the Stieltjes-Hermite moment problem to be determi- nate. In the first section of this note we recall some notation, definitions and a solution of a complex-matrix valued moment problem which was proved in [8]. In Section 2, we use this solution to solve a trigonometric moment prob- lem with values complex k-dimensional matrices. Some connection between the positive defined matrices that appears in the proof of Propisition 1 in [8], respectively Proposition 2 in Section 2 of the present paper, and the positive defined matrices that appear in Theorems 5.4 and 5.5 in [13] (solutions of the truncated and Stieltjes-Hermite operator valued moment problems) can be established. Also, some similar existence conditions for operator valued trigonometric moment sequences are also given in [17]. The proof in [17] is different from the proof given in the present paper (Proposition 2 in Section 2).

In Section 3 we study an L-scalar moment problem generated by two differ- ent complex matrix valued moment sequences, that are in a prescribed order relation. Both moment problems studied in this note are Hausdorff moment problems. They are therefore determinate.

Letz= (z₁, . . . , z_n) denote the complex variable inCⁿandD₁ⁿthe closed n-dimensional unit polydisc. For α= (α1, . . . , αn),β = (β1, . . . , βn)∈Nⁿ, we set z^α =z^α₁¹. . . z^α_nⁿ,z^β =z1β1. . . znβn.

Definition 1. A k-dimensional matrix Λ = (λ_ij)1≤i,j≤k of complex mea- sures is said to be positive defined on D₁ⁿ if

(a) Λ(M) = (λij(M))1≤i,j≤k is a nonnegative matrix for any Borel set M ∈Bor(D₁ⁿ), and

(b) for all 1 ≤ i, j ≤ k, the positive Borel measure |λ_ij| on Dⁿ₁ has complex moments of all orders.

Let{Γ_αβ = (sij(α, β)1≤i,j≤k∈M(k,C), α, β ∈Nⁿ} be a multisequence of k-dimensional matrices with complex entries.

Definition 2. A multisequence {Γ_α,β}_α,β∈Nn of k-dimensional matrices is called a k-complex moment sequence if there exists a k-dimensional ma- trix Λ = (λ_ij)1≤i,j≤k of complex measures positive defined on Dⁿ₁ such that Γ_αβ = (s_ij(α, β))_1≤i,j≤k =

R

Dⁿ₁ z^αz^βdλ_ij(z)

1≤i,j≤k =R

D₁ⁿz^αz^βdΛ(z) for all α, β ∈Nⁿ.

Let ℘ be the C-vector space of polynomials in z, z with complex co- efficients, ℘an = {P

aαz^α, aα ∈ C, α ∈ H ⊂ Nⁿ, Hfinite}, and the C- linear mapping L associated with {Γ_αβ}_α,β∈Nn, L : ℘ → M(k,C), L(p) =

(lij(p))1≤i.j≤k with L(p) = P

α,β∈H

aα,βΓαβ, H a finite set ⊂ Nⁿ for p(z, z) = P

α,β∈H

aα,βz^αz^β.

Definition3. A linear mappingL(·) = (l(·)_ij)1≤i,j≤kfrom℘intoM(k,C) is called positive on D₁ⁿ if P

1≤i,j≤k

l_ij(p)t_it_j ≥0 for all t_i, t_j ∈ Cand all poly- nomials p∈℘ withp(z, z)≥0 for allz∈Dⁿ₁.

We recall the following result from [8].

Proposition 1. Let {Γ_αβ = (sij(α, β))1≤i,j≤k ∈M(k,C) for all α, β ∈ Nⁿ}be a multisequence ofk-dimensional matrices andL= (l_ij)1≤i,j≤k the as- sociated linear map from℘intoM(k,C). The following assertions are equiva- lent:

(α) L is positive defined on the compactDⁿ₁;

(β) {Γ_αβ}_α,β∈Nⁿ is a k-complex moment sequence on D₁ⁿ. 2. A MATRIX-VALUED TRIGONOMETRIC

MOMENT PROBLEM In this section, we use [8, Proposition 1] to prove

Proposition 2. LetKα = (sij(α))1≤i,j≤k∈M(k,C)be ak-dimensional matrix for any α ∈ Zⁿ, k ∈ N^∗, and L = (l_ij)1≤i,j≤k the linear map- ping associated with the sequence of complex matrices {Γ_αβ}_α,β∈Nⁿ, Γ_αβ = (s_ij(α, β))1≤i,j≤k defined by Γ_αβ = Kα−β for all α, β ∈ Nⁿ. The following assertions are equivalent:

(j) the linear map L : ℘ → M(k,C) is positive definite on D₁ⁿ and L((1− |z₁|)^p1. . .(1− |z_n|)^pn|p(z)|²) = 0 for any (p1, . . . , pn) ∈ Nⁿ, |p| = n

P

i=1

pi

6= 0 and any p∈℘an;

(jj)there exists a positive definite matrixΛ = (λij)1≤i,j≤kof measures de- fined on the Borel sets of the complex torusT₁ⁿsuch thatK_α = (s_ij(α))1≤i,j≤k= R

T₁ⁿz^αdλ_ij(z))1≤i,j≤k

=R

T₁ⁿz^αdΛ(z) for allα∈Zⁿ.

Proof. (j) ⇒ (jj) If L:℘ → M(k,C) is positive definite on D₁ⁿ, then it follows from Proposition 1 from [8] that there exists a positive definite matrix Λ = (λij)1≤i,j≤k of measures onDⁿ₁ such that

Γ_αβ=Kα−β= (s_ij(α, β))1≤i,j≤k= Z

D₁ⁿ

z^αz^βdλ_ij(z)

!

1≤i,j≤k

= Z

D₁ⁿ

z^αz^βdΛ(z)

for all α, β∈Zⁿ. By [15], [8], [10], all polynomials p∈℘ with p(z, z)≥0 on Dⁿ₁ can be expressed asp(z, z) =q₁(z, z)−q₂(z, z) with q_i(z, z), i= 1,2, line- ar combinations with positive coefficients of expression of the type

n

Q

i=1

(1−

|z_i|²)^pi|p(z)|², pi ∈ N and p(z) = P

α∈H⊂Nⁿ

aαz^α, H finite, aα ∈ C an ana- lytical polynomial. It follows that if p(z, z) = 0 then on T₁ⁿ there exists an exponent p_k ∈ N with p_k 6= 0 such that (1− |z_k|²)^pk|p(z, z)|= 0. Hence in the decomposition of p(z, z) = q1(z, z)−q2(z, z) the same exponents p_k ∈N from the expressions q_k(z, z) = P

j∈H⊂N

αj|p_j(z)|²

n

Q

i=1

(1− |z_i|²)^pi do appear, that is, (1− |z_k|²)^pk|q_i, i= 1,2. Because L(|p(z)|²

n

Q

i=1

(1− |z_i|²)^pi = 0 for all p∈℘an and all (p1, . . . , pn)∈Nⁿ,|p| 6= 0, we haveL(p(z, z)) = 0 for all p∈℘ with p(z, z) ≥ 0 on Dⁿ₁ and p(z, z) = 0 on T₁ⁿ. Hence, R

D₁ⁿp(z, z)dΛ(z) = 0 for all p ∈ ℘ with p(z, z) ≥ 0 on D₁ⁿ and p(z, z) = 0 on T₁ⁿ, which imply supp Λ⊂ {z∈Cⁿ| |z_i|= 1,1≤i≤n}. The required representations are then

Kα = Z

T₁ⁿ

z^αdΛ(z), ∀α∈Zⁿ.

(jj)⇒ (j). Let Λ = (λij)1≤i,j≤k be a positive definite matrix of measures on the Borel sets of T₁ⁿ. That is, Λ(M) = (λ_ij(M))1≤i,j≤k is positive definite for all M ∈Bor(T₁ⁿ). Let M ∈Bor(D₁ⁿ). We extend Λ : Bor(T₁ⁿ)→ {Positive definite matrices on T₁ⁿ} to Λ : Bor(De ⁿ₁) → {Positive definite matrices on Dⁿ₁}, Λ = (ee λ_ij) with λe_ij(B) = λ_ij(B∩T₁ⁿ) if B ∩T₁ⁿ 6=∅ and λe_ij(B) = 0 if B∩T₁ⁿ=∅. The matrix Λ = (e λeij) is positive definite on D₁ⁿand

Z

T₁ⁿ

z^αdΛ(z) = Z

D₁ⁿ

z^αdeΛ(z) ∀α∈Nⁿ, (1)

Z

T₁ⁿ

z^βdΛ(z) = Z

Dⁿ₁

z^−βdeΛ(z)∀β∈Zⁿ, β= (β₁, . . . , β_n), β_i≤0, ∀1≤i≤n.

(2)

From (1) and (2) we get

Γ_αβ =Kα−β = Z

Dⁿ₁

z^αz^βdΛ(z)e ∀α, β ∈Nⁿ.

Next, from Proposition 1 ((β)⇒(α)) we deduce thatLis positive defined on Dⁿ₁ and L(|p(z)|²

n

Q

i=1

(1− |z_i|²)^pi) = 0 for all p ∈ ℘ and all (p₁. . . p_n) ∈ Nⁿ,

|p| 6= 0, that is (j).

Remark 1. In the “Remark” after Proposition 1 in [8], we have proved that if matrices Γ_αβ ∈M(k,C),α, β∈Nⁿ, are given such that the associated linear map L : ℘ → M(k,C), L(z^αz^β) = Γ_αβ, ∀α, β ∈ Nⁿ, extended by linearity to ℘ is positive defined on Dⁿ₁, then we have Γαβ =^t Γβα for all α, β ∈Nⁿ. In the case of Proposition 2 here, ifL is positive definite, then we have Kα−β = Γαβ =^tKβ−α =^tΓβ−α for all α, β ∈Nⁿ.

3. AN OPERATOR-VALUED L-MOMENT PROBLEM TheL-moment problem consists of characterizing the moment sequence a_n =R

Rtⁿf(t)dt, n ∈N, of a measurable function f which satisfies a boun- dedness condition such as 0 ≤ f ≤ L a.e. dt. In this section, we study a L-scalar moment problem generated by a sequence of moments with values k-dimensional complex matrices, k∈N^∗.

Proposition 3.Let{A_αβ}_αβ∈Nⁿand{B_αβ}_αβ∈Nⁿ be two multisequences of k-dimensional complex matrices. Let also L^A = (l^A_ij)_1≤i,j≤k and L^B = (l^B_ij)1≤i,j≤k, respectively, be the associated linear maps from ℘ into M(k,C).

The following statements are equivalent:

(i) there exists a positive constant M >0 such that

0≤ X

1≤i,j≤k

l_ij^A(p)titj ≤M X

1≤i,j≤k

l^B_ij(p)titj

for all polynomials p∈℘ and all ti, tj ∈C;

(ii)there exists ak-dimensional positive defined matrixΛ^B= (λ^B_ij)1≤i,j≤k

of complex Borel bounded measures on D₁ⁿ and a real valued, bounded, mea- surable function h:D₁ⁿ×C^k→R such that

B_αβ = Z

D₁ⁿ

z^αz^βdΛ^B(z) ∀α, β ∈Nⁿ and

hA_αβt, ti_Ck = Z

Dⁿ₁

z^αz^βh(z, t)dhΛ^B(z)t, ti_Ck ∀α, β ∈Nⁿ∀t∈C^k. Proof. (i)⇒(ii). By (i), condition (α) in Proposition 1 is fulfilled. It follows that there exist positive-definite matrices Λ^A = (λ^A_ij) and Λ^B = (λ^B_ij) of Borel measures on D₁ⁿsuch that the representations

A_αβ = Z

Dⁿ₁

z^αz^βdΛ^A(z) ∀α, β∈Nⁿ

and

B_αβ = Z

D₁ⁿ

z^αz^βdΛ^B(z) ∀α, β∈Nⁿ, do hold. So, we have

X

1≤i,j≤k

l^A_ij(p)t_it_j = X

1≤i,j≤k

Z

D₁ⁿ

p(z, z)dλ^A_ij(z)t_it_j = (3)

= Z

Dⁿ₁

p(z, z)d X

1≤i,j≤k

λ^A_ij(z)t_it_j

!

≤M Z

Dⁿ₁

p(z, z)d

Xλ^B_ij(z)t_it_j

=

=M Z

Dⁿ₁

p(z, z)dΛ^B(z) =M X

1≤i,j≤k

l_ij^B(p)titj

for all p(z, z)≥0 onDⁿ₁. By (3), the positive measure λ^A_t = X

1≤i,j≤k

λ^A_ijt_it_j

is absolutely continuous with respect to the positive measure λ^B_t = X

1≤i,j≤k

λ^B_ijtitj

for all t = (t₁, . . . , t_k) ∈ C^k. Indeed, let the seminorm F : C(Dⁿ₁) → C be defined on the continuous functions of D₁ⁿ as

F(f) =M Z

D₁ⁿ

|f(z)|d

X

1≤i,j≤k

λ^B_ij(z)t_it_j

and the linear functional beL:℘→C,L(p) =R

Dⁿ₁ p(z, z)d P

1≤i,j≤k

λ^A_ij(z)t_it_j . By (i) we have

(4) L(p)≤F(p) ∀p∈℘, p(z, z)≥0.

In particular, forp(z, z) =|q(z)|² Qⁿ

i=1

(1− |z_i|²)^ki,q∈℘_an, inequality (4) holds.

Letf ∈℘ be arbitrary. We have the decomposition (5) f(z, z) =f1(z, z) + if2(z, z)

with f1, f2 polynomials inz, z with real coefficients. If we write

f_j(z, z) =f_j¹(x₁, . . . , x_n, . . . , y₁, . . . , y_n)+if_j²(x₁, . . . , x_n, y₁, . . . , y_n), j= 1,2, f_j^k ∈ R[x₁, . . . , x_n, . . . , y₁, . . . , y_n], k = 1,2, then by Remark 2 we can also write f₁¹(x₁, . . . , y_n) = q₁(z, z)−q₂(z, z), f₁²(x₁, . . . , y_n) = q₃(z, z)−q₄(z, z).

The same decomposition holds for f₂¹ and f₂² with qj, 1 ≤ j ≤ 8, linear

combinations with positive coefficients of expressions of the typeqij, 1≤i≤6, 1≤j ≤n. We then have

(6) |L(f)|=|L(f₁) + iL(f₂)| ≤ |L(f₁)|+|L(f₂)|.

By (4),

0≤L(q_k)≤F(q_k) =M Z

Dⁿ₁

q_kd X

1≤i,j≤k

λ^B_ij(z)t_it_j

!

for all k∈1,8. The above decomposition off implies (7) |L(f₁)| ≤ |L(q₁−q₂)|+|L(q₃−q₄)| ≤2M

Z

D1

|f₁|d X

1≤i,j≤k

λ^B_ij(z)t_it_j

!

and

(8) |L(f₂)| ≤ |L(q₅−q₆)|+|L(q₇−q₈)| ≤2M Z

D1

|f₂|d X

1≤i,j≤k

λ^B_ij(z)t_it_j

! .

It follows from (6), (7) and (8) that

(9) |L(f)| ≤4M

Z

D1

|f|d X

1≤i,j≤k

λ^B_ij(z)titj

! .

The mapping

F₁ :C(D₁ⁿ)→C, F₁(f) = 4M Z

Dⁿ₁

|f|d X

1≤i,j≤k

λ^B_ij(z)t_it_j

!

is a seminorm onC(D₁ⁿ). By the Hahn-Banach theorem we can extendLto the space of continuous complex-valued functions on Dⁿ₁ by preserving inequality (9). We also denote the extension by L.

The functional L is positive, i.e., L(f) ≥ 0 for any f ∈ C_R(D₁ⁿ) with f(z)≥0 for anyz∈D₁ⁿ.Indeed, theR-vector spaceC_R(Dⁿ₁) ={f :D₁ⁿ→R, f continuous on D₁ⁿ} is the closure in the uniform convergence topology on compact Dⁿ₁ of R_n = {P ∈℘ withP(z, z)∈ R} and by Proposition 3 in [4], any polynomialP(z, z) in℘withP(z, z)≥0 onDⁿ₁ is uniformly approximated in D₁ⁿwith polynomials of the form q_ij. But the inequality in (j) asserts that L(qi)≥0, which completes the proof of positivity ofL.

If we consider the uniform normkfk= sup

z∈D₁ⁿ

|f(z)|on C(Dⁿ₁), we obtain

(10) |L(f)| ≤4M

Z

D1

|f|d X

1≤i,j≤k

λ^B_ij(z)titj

! .

This means thatL is a continuous, positive functional on C(Dⁿ₁) andL(f) = R

Dⁿ₁ f(z)d P

1≤i,j≤k

λ^A_ij(z)t_it_j

for all f ∈C(D₁ⁿ).

We now prove that the positive measure d P

1≤i,j≤k

λ^A_ijtitj

is absolutely continuous with respect to the positive measure d

P

1≤i,j≤k

λ^B_ijt_it_j

for allt= (t1, . . . , tk)∈C^k.LetE⊂D₁ⁿbe a measurable set with

P

1≤i,j≤k

λ^A_ij(E)titj

= 0 and X_E the indicator function of E. We approximate X_E pointwise and monotonically by a bounded sequence ϕ_k in C_R⁺(Dⁿ₁). For this sequence, by Lebesgue’s dominated convergence theorem we have

0≤ Z

Dⁿ₁

ϕ_kd X

1≤i,j≤k

λ^A_ij(z)titj

!

≤4M Z

Dⁿ₁

ϕ_kd X

1≤i,j≤k

λ^B_ij(z)titj

!

≤

≤4M Z

D1

limk ϕ_kd X

1≤i,j≤k

λ^B_ij(z)titj

!

=

= 4M Z

D₁ⁿ

XEd X

1≤i,j≤k

λij(z)^B(z)titj

!

= 4M X

1≤i,j≤k

λ^B_ij(E)titj).

Hence, if P

1≤i,j≤k

λ^B_ij(E)titj

= 0, we also have P

1≤i,j≤k

λ^A_ij(E)titj

= 0. By the above, there existsh∈L¹

D₁ⁿ×C^k, P

1≤i,j≤k

λ^B_ijtitj

such that d P

1≤i,j≤k

λ^A_ijtitj

= h(·, t)d P

1≤i,j≤k

λ^B_ijt_it_j

. As in [11], h(·,·) is positive and bounded for all t∈C^k. So, we have

hA_αβt, ti_Ck = Z

Dⁿ₁

z^αz^βh(z, t) d X

1≤i,j≤k

λ^B_ij(z)titj

!

for all α, β ∈Nⁿ. (jj) ⇒ (j). If we have the representations Aαβ = R

Dⁿ₁ z^αz^βdΛ^A(z) and B_αβ =R

D₁ⁿz^αz^βdΛ^B(z) for allα, β ∈Nⁿ, then by Proposition 1 ((β)⇒ (α)) we have P

1≤i,j≤k

l_ij^A(p)t_it_j ≥0 and P

1≤i,j≤k

l^B_ij(p)t_it_j ≥0 for all polinomialsp∈℘ with p(z, z)≥0 ∀z∈D₁ⁿ. Since

hA_αβt, ti_Ck = Z

Dⁿ₁

z^αz^βh(z, t) d X

1≤i,j≤k

λ^B_ij(z)titj

!

with 0≤h(z, t)≤M on Dⁿ₁ withM a positive constant, we have

0≤ X

1≤i,j≤k

l_ij^A(p)t_it_j ≤M X

1≤i,j≤k

l^B_ij(p)t_it_j

for allt= (t1, . . . , t_k)∈C^kand allp∈℘withp(z, z)≥0 onDⁿ₁, that is (j).

Remark2. Let G_D¹_n ={p_1i(x1, . . . , xn, y1, . . . , yn) =^1+x₂ ⁱ, p2i(x1, . . . , yn)

= ^1−x₂ ⁱ, p3i(x1, . . . , yn) = ^1+y₂ ⁱ, p4i(x1, . . . , yn) = ^1−y₂ ⁱ, p5i(x1, . . . , yn) =x²_i + y_i²,p6i(x1, . . . , yn) = 1−x²_i−y²_i, 1≤i≤n}the extremal generators having the usual norm (sup_Dⁿ

1 | · |) equal with 1 on the unit polydiscD₁ⁿ, a semialgebraic compact set.

Let ∆_P =n Q

i,j∈H;r,s1,6

p^k_riⁱ(1p_sj)^lj, H⊂N finite,k_i, l_j ∈N andp_ri, p_sj ∈ GDⁿ₁

o .

Let also Γ_P =n P

i∈H

α_iδ_i, H ⊂N finite,α_i ≥0, δ_i∈∆_Po

be the positive cone generated by ∆P.

By [5], we have R[x₁, . . . , x_n, y₁, . . . , y_n] = Γ_P − Γ_P. As in paper [17] we denote by g1i(z, z) = ^{|zi+1|2+(1−|z}₄ ^i|2) = p1i(x1, . . . , yn), g2i(z, z) =

|1−z_i|²+(1−|z_i|²)

4 =p_2i(x₁, . . . , y_n),g_3i(z, z) = ^{|1−izi|2+(1−|z}₄ ^i|2) =p_3i(x₁, . . . , y_n), g_4i(z, z) = ^{|1+izi|2+(1−|z}₄ ^i|2) =p_4i(x₁, . . . , y_n), g_5i(z, z) =|z|²_i =p_5i(x₁, . . . , y_n), g6i(z, z) = 1− |z_i|² =p6i(x1, . . . , yn).

In this case, every element of ∆_P can be represented as linear combi- nations with positive coefficients of expression of the form qji(z, z) = |p(z)|²

n

Q

i=1

(1− |z_i|²)^ki with k_i ∈N andp∈P_an.

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Received 25 November 2008 “Politehnica” University of Bucharest Department of Mathematics

Splaiul Independent¸ei 313 060042 Bucharest, Romania luminita lemnete@yahoo.com