MARKOV-TYPE AND OPERATOR-VALUED MULTIDIMENSIONAL MOMENT PROBLEMS, WITH SOME APPLICATIONS

MARKOV-TYPE AND OPERATOR-VALUED MULTIDIMENSIONAL MOMENT PROBLEMS,

WITH SOME APPLICATIONS

JANINA MIHAELA MIH ˘AIL ˘A, OCTAV OLTEANU and CONSTANTIN UDRIS¸TE

In Section 2 we prove a general result which gives sufficient conditions for the existence of a solution for a Markov-type moment problem in the space L¹ν(T) (the implication (b)⇒(a) of Theorem 2.1). These sufficient conditions involve the signatures of some quadratic forms having as coefficients the given moments. This general result may be applied to Markov-type moment problems on unbounded, non-semialgebraic subsets of Rⁿ (see Corollary 2.2 and Remark 2.3). An appli- cation to a “non-classical” Markov moment problem on an arc of a hyperbola is given in Corollary 2.4. The last result of Section 2 is the sketch of the construc- tion of the solution h of a classical Markov moment problem on the ellipse. In Section 3 we solve some classical Markov-type moment problems on closed poly- discs inCⁿ, the solutionF applying some spacesX of analytic functions on such polydiscs into a spaceY of self-adjoint commuting operators acting on a Hilbert space H (Y is defined by (3.1)); see also [13] and [20]). Some examples in the spacesL²dt 0,^π₂andL²dt([0,∞[) are given (see Corollaries 3.3 and 3.10).

AMS 2000 Subject Classification: 46A22, 47A57, 28A20, 26D15, 46A40, 53A04, 53A05, 53A07, 46B42, 30B10, 30E05, 32A05, 32A07, 47A10, 47A63.

Key words: Markov-type moment problems in the spaceL¹ν(T) with applications to the Markov moment problem on non-compact non-semialgebraic varieties inRⁿ, construction of the solution hof a Markov moment problem on the ellipse, operator-valued Markov-type moment prob- lems in spaces of analytic functions of several complex variables on closed polydiscs.

1. INTRODUCTION

We recall the (α, β)-Markov moment problem (see [17]). Given real num- bersα, β, α < β, and two finite sequences {y_k}ⁿ₀ ⊂R and {x_k}ⁿ₀ ⊂C([a, b]), the following three problems have to be solved.

(P₁) Find necessary and sufficient conditions on the sequence{y_k}ⁿ₀ for the existence of a Lebesgue measurable functionh on [a, b], α≤h(t)≤β a.e.

REV. ROUMAINE MATH. PURES APPL.,52(2007),4, 405–428

in [a, b] such that

(1.1)

_b

a

x_k(t)h(t)dt=y_k, k= 0,1, . . . , n.

(P₂) Find conditions under which the solution (the linear functionalx→ _b

ax(t)h(t)dt,x∈C([a, b]), if it exists), is unique, and conditions under which it is not unique. In the first case, we say that the moment problem (P₁) is determinate. If two distinct solutions exist, the respective moment problem is said to be indeterminate.

(P₃) If a solutionhdoes exist, how can it be constructed, only using the given momentsy_k,k= 0,1, . . . , n ?

Problem (P₁) was studied in [17], [25], [26], [27], [23], [30], [2], [31], [19], [20], [21] and in many other works.

We can solve a classical moment problem on a semi-algebraic compact K ⊂ Rⁿ, only using the signature of some quadratic forms, since positive polynomials on such compacts K can be written using sums of squares of polynomials multiplied by some other given polynomials (see [5], [8], [37], [34], [33]). The first author who proved a Markov-type result (on the unit sphere in Rⁿ), only using quadratic forms, was Ambrozie ([2], Theorem 4.3). His proof does not use any Hahn-Banach extension type result, the basic tool being the Weierstrass theorem. In [21], we solve such problems on arbitrary compactaK ⊂Rⁿ and also on [0,∞[, only using quadratic forms. The same paper contains a sketch of reducing the “non-compact case” to the “compact case”, for some Borel subsets ofRⁿand an “abstract version” for the result on [0,∞[, when the subspace of polynomials is replaced by an abstract subspace.

The Markov moment problem is also interesting on non-compact, non-semi- algebraic subsets T ⊂Rⁿ, when we do not know how a positive polynomial onT looks, and the Weierstrass theorem does not work.

This is the reason of proving Theorem 2.1, which gives sufficient condi- tions ((b)⇒ (a)) for the existence of the solution of a Markov-type moment problem, in terms of quadratic forms, without any connection with polyno- mials. The applications of this general result are briefly mentioned in the abstract, in the front of the paper.

Another reason of writing this paper is to give some new applications of Theorem 1.1 to some operator-valued Markov-type moment problems in spaces of uniformly convergent power series on closed polydiscs in Cⁿ. The target space of the solution F is a commutative algebra of self-adjoint operators defined in [20], which generalizes the algebra defined in [13] (both of them are also order-complete vector lattices). These results can be proved starting from the general Theorem 1.1 stated below, but using also basic results from

classical and functional analysis. For another type of operator-valued moment problem see [18].

1.1. Theorem (Theorem 1, [25]). Let X be a preordered vector space, X₊ its positive cone, Y an order-comple vector lattice, P :X → Y a convex operator, {x_j;j ∈J} ⊂X, {y_j;j ∈J} ⊂Y given families.

The following statements are equivalent:

(a)there exists a linear positive operator F ∈L₊(X, Y) such that (1.1) F(x_j) =y_j, ∀j∈J,

(1.2) F(x)≤P(x), ∀x∈X;

(b)for any finite subset J₀ ⊂J and any {λ_j;j ∈J₀} ⊂R we have

(1.3)

j∈J0

λ_jx_j ≤x⇒

j∈J0

λ_jy_j ≤P(x) in Y.

The elementsy_j,j∈J, are called the moments whileF is called a solu- tion of the moment problem stated in (a). Theorem 1.1 was first published in [25], without proof. Its proof may be found in [27]. For some of its applications see [25], [27], [28], [23], [29], [19].

1.2. Theorem (Theorem 4, [25]). Let X, Y,{x_j;j ∈ J} ⊂ X and {y_j;j ∈ J} ⊂ Y be as in Theorem 1.1. Let {F₁, F₂} ⊂ L(X, Y). Consider the assertions:

(a)there exists F ∈L(X, Y) such that (1.1) F(x_j) =y_j, ∀j∈J,

(1.4) F₁(ϕ)≤F(ϕ)≤F₂(ϕ), ∀ϕ∈X₊ ;

(b)for any finite subset J₀ ⊂J and any {λ_j;j ∈J₀} ⊂R we have (1.5)

j∈J0

λ_jx_j =ϕ₂−ϕ₁, where ϕ₁, ϕ₂∈X₊ ⇒

j∈J0

λ_jy_j ≤F₂(ϕ₂)−F₁(ϕ₁).

If X is a vector lattice, consider also the assertion

(b) F₁(ϕ) ≤F₂(ϕ), ∀ϕ∈ X₊ and for any finite subsetJ₀ ⊂J and any {λ_j;j∈J₀} ⊂R we have

j∈J0

λ_jy_j ≤F₂

j∈J0

λ_jx_j+

−F₁

j∈J0

λ_jx_j− . Then(b)⇔ (a)and, if X is a vector lattice, then(b)⇔(b)⇔(a).

Theorem 1.2 was first published in [25]. Its proof may be found in [26].

For some of its applications see [26], [27], [23], [30], [2], [19]. This theorem extends the classical Markov moment problem to a sandwich-type moment

problem for linear operators, and this generalization leads to the applications mentioned briefly in the Abstract.

Concerning the determinacy or indeterminacy of the moment problems solved in this paper, it is clear that when Sp{x_j;j∈J}is dense in X andF : X→Y is continuous, the moment problem is determinate. When we consider moment problems on non-compact subsetsT ofRⁿ(even if it is the case of the classical moment problem, whenx_j(t) =t^j = t^j₁¹. . . t^j_nⁿ, t = (t₁, . . . , t_n) ∈ T, j= (j₁, . . . , j_n)∈Zⁿ₊), the question of determinacy of the respective moment problem is sometimes a very difficult one. Important special cases are T = ]− ∞,∞[= R, T = [0,∞[⊂ R. There is a general result which can be extracted from [4] and [6], which says that if F has finite moments of all orders and F is M-indeterminate, then there are infinitely many absolutely continuous distributions and infinitely many discrete distributions with the same moments as those of F. This and other reasons lead to the problem of the construction of a Stieltjes class (see [38], [39]). For other notions related to determinacy, of some multidimensional moment problems, see [16].

The present work does not intend to study the uniqueness of the solutions of all moment problems solved below. However, a natural question is: what can one say about the indeterminacy of the classical moment problems mentioned in Corollaries 2.2 and 2.4? An example of a determinate classical Markov moment problem on [0,∞[ is given in Theorem 2.1 ([21]). It is clear that all the problems solved in Section 3 are determinate.

2. MARKOV-TYPE MOMENT PROBLEMS WITH CONNECTIONS WITH GEOMETRY

We start by an application of Theorem 1.2 to a Markov-type moment problem on an arbitrary measurable space endowed with a σ-finite positive measureν. The main implication is (b) ⇒ (a).

2.1.Theorem. LetT be a measurable space,νa positiveσ-finite measure on T. Let X := L¹_ν(T), X₊ := {x ∈ X;x(t) ≥ 0 a.e. in T}. Endow L¹_ν(T) with the usual norm ₁,x1:=

T |x(t)|dν(t),x∈X. LetJ be an arbitrary family of subscripts, {x_j;j ∈ J} ⊂ X and {y_j;j ∈ J} ⊂ R. Consider the following assertions:

(a)there exists h∈L^∞_ν (T), such that

T

x_j(t)h(t)dν(t) =y_j, ∀j∈J, (2.1)

−1≤h(t)≤1 ν-a.e.;

(2.2)

(b)for any finite subset J₀ ⊂J and any {λ_j;j ∈J₀} ⊂R we have

(2.3)

i,j∈J0

λ_iλ_jy_iy_j ≤

i,j∈J0

λ_iλ_j

T

x_i(t)dν(t)·

T

x_j(t)dν(t);

(c)for any finite subset J₀⊂J and any {λ_j;j∈J₀} ⊂R we have

(2.4)

i,j∈J0

λ_iλ_jy_iy_j ≤

i,j∈J0

|λ_i||λ_j|

T

|x_i(t)|dν(t)·

T

|x_j(t)|dν(t).

Then(b)⇒ (a)⇒ (c).

Proof. (b)⇒ (a). We apply Theorem 1.2 to F₂(x) :=

T

x(t)dν(t), F₁(t) :=−

T

x(t)dν(t) =−F₂(x), x∈L¹_ν(T).

We have to verify implication (1.5). Let J₀ ⊂ J be a finite subset and {λ_j;j∈J₀} ⊂Rsuch that

j∈J0

λ_jx_j =ϕ₂−ϕ₁, ϕ₁, ϕ₂ ∈X₊. Then we have

−

T

ϕ₁(t)dν(t)−

T

ϕ₂(t)dν(t)≤

j∈J0

λ_j

T

x_j(t)dν(t) =

=

T

ϕ₂(t)dν(t)−

T

ϕ₁(t)dν(t)≤

T

ϕ₂(t)dν(t) +

T

ϕ₁(t)dν(t), which lead to

(2.5)

j∈J0

λ_j

T

x_j(t)dν(t) ≤

T

ϕ₂(t)dν(t) +

T

ϕ₁(t)dν(t) =

=

T

ϕ₂(t)dν(t)−

−

T

ϕ₁(t)dν(t)

=F₂(ϕ₂)−F₁(ϕ₁), whereF₂(x) =−F₁(x) :=

Tx(t)dν(t), ∀x ∈L¹_ν(T). On the other hand, the condition (2.3) of hypothesis (b) may be rewritten as

j∈J0

λ_jy_{j 2}

≤

j∈J0

λ_j

T

x_j(t)dν(t) ₂

or, further,

(2.3)

j∈J0

λ_jy_j ≤

j∈J0

λ_j

T

x_j(t)dν(t) .

From (2.3) and (2.5) one deduces

j∈J0

λ_jy_j ≤

j∈J0

λ_jy_j ≤

j∈J0

λ_j

T

x_j(t)dν(t)

^(2.5)≤ F₂(ϕ₂)−F₁(ϕ₁).

Thus (1.5) is verified and, by Theorem 1.2 (b) ⇒ (a), there exists a linear functionalF ∈(L¹_ν(T))^∗ such thatF(x_j) =y_j ∀j∈J, and

−

T

ϕ(t)dν(t)≤F(ϕ) ≤

T

ϕ(t)dν(t) ∀ϕ∈X₊ . This may be rewritten as

|F(ϕ)| ≤

T

ϕ(t)dν(t) ∀ϕ∈X₊ . Now letx∈L¹_ν(T) be arbitrary. We have

|F(x)| ≤ |F(x⁺)|+|F(x⁻)| ≤

T

[x⁺(t) +x⁻(t)]dν(t) =

T

|x(t)|dν(t) =x. Thus F ≤ 1. In particular, F is bounded, hence there exists h ∈ L^∞_ν (T) such that

F(x) =

T

x(t)h(t)dν(t) ∀x∈L¹_ν(T), and

h_∞=F ≤1.

These last relations show thath(t)∈[−1,1]ν-a.e. On the other hand, we have y_j =F(x_j) =

T

x_j(t)h(t)dν(t) ∀j∈J, so that the proof of (b)⇒ (a) is complete.

(a)⇒ (c) is almost obvious. We have

j∈J0

λ_jy_j ^(2.1)=

j∈J0

λ_j

T

x_j(t)h(t)dν(t) ≤

≤

j∈J0

|λ_j| ·

T

|x_j(t)| |h(t)|dν(t)^(2.2)≤

j∈J0

|λ_j| ·

T

|x_j(t)|dν(t).

This obviously leads to (2.4). The proof is complete.

2.2. Corollary. Let T := {t = (t₁, t₂) ∈ R², 0 ≤ t₁ < ∞, 0 ≤ t₂ ≤ exp(−t₁)}and{y_(j1,j2); (j₁, j₂)∈Z²₊} ⊂R. Consider the following assertions:

(a)there exists a Lebesgue measurable function h on T such that (2.1)

T

t^j₁¹t^j₂²h(t₁, t₂)dt₁dt₂ =y_(j1,j2) ∀(j₁, j₂)∈Z²₊,

and

(2.2) −1≤h(t₁, t₂)≤1 a.e. in T;

(b) for any finite subset J₀ ⊂ Z²₊ and any {λ_(j1,j2); (j₁, j₂) ∈ J₀} ⊂ R

we have

(i1,i2),(j1,j2)∈J0

λ_(i1,i2)λ_(j1,j2)y_(i1,i2)y_(j1,j2)≤

≤

(i₁,i₂)(j₁,j₂)∈J0

λ_(i1,i2)λ_(j1,j2) i₁!

(i₂+ 1)ⁱ¹⁺² · j₁! (j₂+ 1)^j1+2;

(c)for any finite J₀ ⊂Z²₊ and any {λ_(j1,j2); (j₁, j₂)∈J₀} ⊂R we have

(i₁,i₂),(j₁,j₂)∈J0

λ_(i1,i2)λ_(j1,j2)y_(i1,i2)y_(j1,j2)≤

≤

(i1,i2),(j1,j2)∈J0

|λ_(i1,i2)||λ_(j1,j2)| i₁!

(i₂+ 1)ⁱ¹⁺² · j₁! (j₂+ 1)^j1+2. Then(b)⇒ (a)⇒ (c).

Proof. We apply Theorem 2.1 tox_j(t) =x_(j1,j2)(t₁, t₂) :=t^j₁¹t^j₂², (t₁, t₂)∈ T, (j₁, j₂)∈Z²₊, dν :=χ_Tdt₁dt₂. We should prove that{x_j;j∈Z²₊} ⊂L¹_ν(T) and compute

T x_j(t)dν(t), j∈Z²₊. We have

T

x_j(t)dν(t) =

T

t^j₁¹t^j₂²dt₁dt₂ = _∞

0

dt₁

_exp(−t1)

0

t^j₁¹t^j₂²dt₂ =

= _∞

0

t^j₁¹· t^j₂²⁺¹ j₂+ 1

exp(−t1)

0



dt₁ = 1 j₂+ 1

_∞

0

t^j₁¹e^−(j2+1)t1dt₁ =

= 1

(j₂+ 1)² _∞

0

e^−u u^j1

(j₂+ 1)^j1du= 1 (j₂+ 1)^j1+2

_∞

0

e^−uu^j1du=

= Γ(j₁+ 1)

(j₂+ 1)^j1+2 = j₁!

(j₂+ 1)^j1+2 <∞.

Thusx_j ∈(L¹_ν(T))₊,∀j= (j₁, j₂)∈Z²₊ and the conclusion follows from Theorem 2.1.

The basic implication is (b) ⇒ (a), which gives a sufficient condition under y_j, j ∈ Z²₊ for the existence of a solution of a Markov classical two dimensional moment problem on an unbounded, non-semialgebraic subset of R², involving only the signatures of some quadratic forms.

2.3. Remark. We can consider a function f : T → R, where T ⊂ R² is a subset with nonempty interior, unbounded, eventually closed and non- semialgebraic (an example of such a set is given in Corollary 2.2). If f ∈ C^k(T^◦)∩C⁰(T) thenS={(t₁, t₂, f(t₁, t₂))|(t₁, t₂)∈T}can be an unbounded surface inR³, which generally may have a non-zero curvature. The application ϕ:T →S,ϕ(t₁, t₂) := (t₁, t₂, f(t₁, t₂)) is bijective , continuous andϕ⁻¹ is also continuous. Thus we can “transport” (using ϕ and ϕ⁻¹) moment problems on S to corresponding moment problems on T, that is, contained in a two dimensional linear variety ofR³. The same remark can be made for varieties of arbitrary dimensionk≤n−1 in Rⁿ.

2.4. Corollary. Let T :={(t₁, t₂) ∈R²;t₁ ≥1, t₂ = (t²₁−1)^1/2} (T is the arc of the hyperbola (H) t²₁ −t²₂ = 1 situated in C₁ := {(t₁, t₂);t₁ ≥ 0, t₂ ≥0}). Let {y_(j1,j2); (j₁, j₂)∈Z²₊} ⊂R. Consider the following assertions:

(a)there exists a Borel measurable function h on T such that (2.1)

T

t^2j₁¹t^2j₂²e^−t1h(t₁, t₂)ds=y_(j1,j2) ∀(j₁, j₂)∈Z²₊, and

(2.2) −1≤h(t₁, t₂)≤1 ds-a.e. in T, whereds is the element of arc on the arc of the hyperbola T;

(b)for any finite J₀⊂Z²₊ and any {λ_(j1,j2); (j₁, j₂)∈J₀} ⊂R we have

(i1,i2),(j1,j2)∈J0

λ_(i1,i2)λ_(j1,j2)y_(i1,i2)y_(j1,j2)≤

≤

(i1,i2),(j1,j2)∈J0

λ_(i1,i2)λ_(j1,j2)m_ds(i₁, i₂)m_ds(j₁, j₂), where

m_ds(j₁, j₂) :=

T

t^2j₁ ¹t^2j₂²e^−t1ds ∀(j₁, j₂)∈Z²₊;

(c)for any J₀, {λ_(j1,j2); (j₁, j₂)∈J₀} ⊂R, with J₀⊂Z²₊ a finite subset,

we have

(i1,i2),(j1,j2)∈J0

λ_(i1,i2)λ_(j1,j2)y_(i1,i2)y_(j1,j2)≤

≤

(i1,i2),(j1,j2)∈J0

|λ_(i1,i2)||λ_(j1,j2)|m_ds(i₁, i₂)m_ds(j₁, j₂), Then(b)⇒ (a)⇒ (c).

Proof. We apply Theorem 2.1 to T endowed with dν := ds, x_j(t) = x_(j1,j2)(t₁, t₂) := t^2j₁ ¹t^2j₂ ²e^−t1 ∀(j₁, j₂) ∈ Z²₊ = J. Since, obviously, x_(j1,j2),

(j₁, j₂) ∈Z²₊, are nonnegative, to prove that x_(j1,j2) ∈ L¹_ds(T) it is necessary and sufficient to prove that

T

x_(j1,j2)(t₁, t₂)ds <∞ ∀(j₁, j₂)∈Z²₊. We have

T

x_(j1,j2)(t₁, t₂)ds= _∞

1

t^2j₁ ¹(t²₁−1)^j2e^−t1

2t²₁−1) t²₁−1

1/2

dt₁ <∞ by elementary properties of the Lebesgue integral. By Theorem 2.1, the con- clusion follows.

Our last result concerns the sketch of the construction of the solution of a classical Markov-type moment problem on the ellipse

K_E =

(t₁, t₂); t²₁ a² +t²₂

b² = 1

, a, b >0.

Given a sequence{y_(j1,j2); (j₁, j₂)∈Z²₊} ⊂R, consider the following problems.

(P₁) Does exist a Borel functionh on K_E such that (2.1)

K_E

t^j₁¹t^j₂²h(t₁, t₂)ds=y_(j1,j2) ∀(j1, j₂)∈Z²₊ and

(2.2) 0≤h(t₁, t₂)≤1 ds-a.e. onK_E?

(P₂) If a solution h exists, is it unique (“modulo” ds-a.e.)?

(P₃) If a solution h exists, how can it be founded?

Problem (P₁) is solved in a more general context by Corollary 2.3 in [21].

Problem (P₂) is clear: the solution is unique since the space of polynomial functions on K_E is dense in C(K_E). So, we have a determinate moment problem.

Next, we sketch a possible answer to (P₃).

2.5. Theorem. Let the sequence {y_(j1,j2); (j₁, j₂) ∈ Z²₊} ⊂ R be arbi- trarily given, and leth be a Borel function on the ellipseK_E such that (2.1) and(2.2) hold. If we define ϕ(θ) := (a²sin²θ+b²cos²θ)^1/2h(acosθ, bsinθ), dθ-a.e. in [−π, π], then we have

h(acosθ, bsinθ) = ϕ(θ)

(a²sin²θ+b²cos²θ)^1/2 dθ-a.e. in[−π, π],

and ϕ is the pointwise limit of a subsequence of the sequence of the partial sums of the Fourier series

(2.3) a₀

2 + ∞ n=1

[a_ncos(nθ) +b_nsin(nθ)],

associated with ϕin L²([−π, π]), where

(2.4)











a₀= _π¹ _π

−πϕ(θ)dθ= _π¹y_(0,0), a_n= _π¹_π

−πϕ(θ) cos(nθ)dθ= ¹_π y_(n,0)

aⁿ −C_n^2y(n−2,2)

aⁿ⁻²b² +· · ·+ +· · ·+ (−1)^kC_n^{2ky(n−2k,2k)}

a^n−2kb^2k +· · ·

(finite sum), n∈Z, n≥1,

(2.5)

















b_n= _π¹_π

−πϕ(θ) sin(nθ)dθ=

= ¹_π

C_n^1y_a^(n−1,1)_n−1b −C_n^3y_a^(n−3,3)_n−3b3 +· · ·+

+ (−1)^kC_n^2k+1y(n−2k−1,2k+1)

a^n−2k−1b^2k+1 +· · · (finite sum), n∈Z, n≥1.

Another conclusion (which implies the first one) is that ϕ is the “sum” of its Fourier trigonometric series but the convergence is only valid inL²([−π, π]), wherea_n,b_n are given by(2.4), (2.5). Thusϕ, hence h, can be “written” using only the moments y_(j1,j2), (j₁, j₂)∈Z²₊.

Proof. Let ϕ₁ : [−π, π]→ [0,1], ϕ₁(θ) := h(acosθ, bsinθ), θ ∈ [−π, π].

Then ϕ₁ is a Borel function on [−π, π], since (acosθ, bsinθ) ∈ K_E, ∀θ ∈ [−π, π], h was supposed to be Borel measurable and θ → (acosθ, bsinθ) is continuous. Sinceh(t₁, t₂)∈[0,1] ds-a.e. onK_E, we haveϕ₁(θ)∈[0,1]dt-a.e.

in [−π, π].

On the other hand, (2.1) implies (2.6)

y_(j1,j2)= _π

−π

a^j1cos^j1θb^j2sin^j2θh(acosθ, bsinθ)(a²sin²θ+b²cos²θ)^1/2dθ=

=a^j1b^j2 _π

−πcos^j1θsin^j2θϕ(θ)dθ, where

(2.7) ϕ(θ) :=ϕ₁(θ)[a²+ (b²−a²) cos²θ]^1/2 ∈[0, b] a.e. in [−π, π].

Thusϕ∈L^∞([−π, π])⊂L²([−π, π]). Using the trigonometric system as a Hilbert base inL²_R([−π, π]), we have

(2.3) ϕ(·) = a₀ 2 +

∞ n=1

[a_ncos(n·) +b_nsin(n·)], where the convergence is in the norm onL²([−π, π]).

By Theorem 3.12 in [35], there exists a subsequence of the sequence of the partial sums of the series from the right hand side of (2.3), which converges pointwise a.e. to ϕ in [−π, π]. To complete the proof, we have to show that a_n, b_n can be expressed as finite sums involving the given moments y_(j1,j2), (j₁, j₂)∈Z²₊, these sums being given by (2.4), (2.5). We have

a₀= 1 π

_π

−πϕ(θ)dθ^(2.6)= 1

πy_(0,0), a_n= 1 π

_π

−πϕ(θ) cos(nθ)dθ, b_n= 1

π _π

−πϕ(θ) sin(nθ)dθ, n≥1.

But

cos(nθ) + i sin(nθ) = (cosθ+ i sinθ)ⁿ=

= cosⁿθ+ iC_n¹cosⁿ⁻¹θsinθ−C_n²cosⁿ⁻²θsin²θ−iC_n³cosⁿ⁻³θsin³θ+

+· · ·+ (−1)^kC_n^2kcos^n−2kθsin^2kθ+ i(−1)^kC_n^2k+1cos^n−2k−1θsin^2k+1θ+

+· · ·+ iⁿ⁻¹C_nⁿ⁻¹cosθsinⁿ⁻¹θ+ iⁿsinⁿθ, which lead to

cos(nθ) = cosⁿθ−C_n²cosⁿ⁻²θsin²θ+· · ·+ (−1)^kC_n^2kcos^n−2kθsin^2kθ+· · · sin(nθ) =C_n¹cosⁿ⁻¹θsinθ−C_n³cosⁿ⁻³θsin³θ+

+· · ·+ (−1)^kC_n^2k+1cos^n−2k−1θsin^2k+1θ+· · · (finite sums),n≥1. These yield

a_n= 1 π

_π

−πcosⁿθϕ(θ)dθ−C_n² _π

−πcosⁿ⁻²θsin²θϕ(θ)dθ+· · ·

+ +· · ·+ (−1)^kC_n^2k

_π

−πcos^n−2kθsin^2kθϕ(θ)dθ+· · ·=

(2.6)

= 1 π

y_(n,0)

aⁿ −C_n²y_(n−2,2)

aⁿ⁻²b² +· · ·+ (−1)^kC_n^ky_(n−2k,2k)

a^n−2kb^2k +· · · ,

n∈Z,n≥1. Thus, (2.4) are proved. Similarly, relations (2.5) follow, so that we “know” somehow ϕ(θ), dθ-a.e. in [−π, π]. This means that we “know”

somehow

h(acosθ, bsinθ) = ϕ(θ)

(a²sin²θ+b²cos²θ)^1/2 dθ-a.e. in [−π, π].

Since every (t₁, t₂)∈K_Ecan be written ast₁ =acosθ,t₂ =bsinθ,θ∈[−π, π], it follows that we “know” somehowh(t₁, t₂) ds-a.e. on the ellipseK_E.

3. OPERATOR-VALUED MARKOV-TYPE MULTIDIMENSIONAL MOMENT PROBLEMS

In this section we consider some Markov classical moment problems in spaces of analytic functions of several complex variables, the target space of the solutionF being the commutative algebra of self-adjoint operators Y defined below (this is also an order-complete vector lattice).

For analytic functions of one variable, Y being another complete vector lattice orY beingR, some earlier results have been proved in [23], [29], [30], [2], [19]. For analytic functions of several variables, we mention the paper [20].

Let H be an arbitrary Hilbert space. Let A be the real vector space of all linear bounded self-adjoint operators acting onH. LetA₁, . . . , A_n∈ Abe commuting elements (A_iA_j =A_jA_i ∀i, j= 1, . . . , n) and put

(3.1)

A1:={U ∈ A; UA_i =A_iU, i= 1, . . . , n},

Y :=Y(A₁, . . . , A_n) :={U ∈ A1; UV =V U, ∀V ∈ A1}, Y₊:={U ∈Y; U(h), h ≥0, ∀h∈H}.

It can be proved thatY is an order-complete vector lattice and a commutative Banach algebra of operators (see [13], pp. 303–305, and [20]).

We next give an application of Theorem 1.1 to the case whenY is defined by (3.1) while X_ρ˜ is the space of all analytic functions x of several complex variables, which can be represented as

x(z₁, . . . , z_n) =

j=(j₁,...,j_n)∈Zⁿ₊

c_(j1,...,jn)z₁^j1. . . z_n^jn, c_j ∈C ∀j∈Zⁿ₊,

the series being uniformly convergent in the closed polydisc ⁿ

k=1

{|z_k| ≤ ρ_k}. Here ˜ρ := (ρ₁, . . . , ρ_n). We organize X_ρ˜ as a real ordered vector space, the order relation being defined by the convex cone

X_ρ,+˜ =

x∈X_ρ˜;x(z) =

j∈Zⁿ₊

c_jz^j, Rec_j ≥0, Imc_j ≥0 ∀j∈Zⁿ₊

.

3.1.Theorem.LetH, Y be as above and additionally assume thatA₁, . . . , A_n are positive. Choose ρ_k > A_k, k = 1, . . . , n. Let X_ρ˜ be the space de- fined above. Let{x_j;j ∈Zⁿ₊} ⊂X, x_j(z) =x_(j1,...,jn)(z₁, . . . , z_n) :=z₁^j1. . . z_n^jn,

z∈ ⁿ

k=1

{|z_k| ≤ρ_k}. Let {B_j;j∈Zⁿ₊} ⊂Y. On X_ρ˜ we consider thesupnorm xρ˜= sup

z∈ ⁿ

k=1{|zk|≤ρk}

|x(z)|.

Consider the following assertions:

(a) there exists a R-linear positive continuous operator F : X_ρ˜ → Y such that

(3.2) F(x_j) =B_j, ∀j∈Zⁿ₊, (3.3) F(x)≤ xρ˜

n k=1

ρ_k(ρ_kI−A_k)⁻¹, ∀x∈X_ρ˜

(in particular,Fρ˜≤ ⁿ

k=1 ρ_k ρ_k−Ak);

(b)we have

(3.4) 0≤B_(j1,...,jn)≤A^j₁¹. . . A^j_nⁿ ∀(j₁, . . . , j_n)∈Zⁿ₊; (c)we have

(3.5) 0≤B_(j1,...,jn)≤ n k=1

ρ^j_k^k+1(ρ_kI−A_k)⁻¹ ∀(j₁, . . . , j_n)∈Zⁿ₊

(in particular we have B_(j1,...,jn) ≤ ⁿ

k=1 ρ^jk_k⁺¹

ρk−Ak ∀(j₁, . . . , j_n) ∈ Zⁿ₊). Then (b)⇒ (a)⇒ (c).

Proof. (b) ⇒ (a). We apply Theorem 1.1, (b) ⇒ (a), to y_j := B_j, j∈J :=Zⁿ₊,P(x) :=xρ˜

n k=1

ρ_k(ρ_kI−A_k)⁻¹, x∈X.

We have to verify implication (1.3). By the definition of the order relation

onX, we have

j∈J0

λ_jx_j ≤x=

Zⁿ₊

c_jx_j, λ_j ∈R, implies

(3.6) λ_j ≤Rec_j ≤ |c_j| ∀j∈J₀, whereJ₀ ⊂Zⁿ₊ is a finite subset. Relations (3.4) lead to

j∈J0

λ_jy_j =

j∈J0

λ_jB_j ^(3.4)≤

j∈J₀⁺

λ_jB_j ^(3.4)≤

j∈J₀⁺

λ_jA^j₁¹. . . A^j_nⁿ ≤