ABSTRACT

In this note a multidimensional Hausdorff truncated operator-valued moment problem, from the point of view of “stability concept” of the number of atoms of the obtained atomic, operator-valued representing measure for the terms of a finite, positively define kernel of operators, is studied. The notion of “stability of the dimension” in truncated, scalar moment problems was introduced in [1]. In this note, the concept of “stability” of the algebraic dimension of the obtained Hilbert space from the space of the polynomials of finite, total degree with respect to the null subspace of a unital square positive functional, in [1], is adapted to the concept of stability of the algebraic dimension of the Hilbert space obtained as the separated space of some space of vectorial functions with respect to the null subspace of a hermitian square positive functional attached to a positive definite kernel of operators. In connection with the stability of the dimension of such obtained Hilbert space, a Hausdorff truncated operator-valued moment problem and the stability of the number of atoms of the representing measure for the terms of the given operator kernel, in this note, is studied.

1. Introduction

The study of scalar truncated moment problems is the subject of many remarkable papers such as: [1-3]. In [2], the problem of finding scalar atomic representing measure for the terms of a finite scalar sequence of complex numbers as multidimensional moment terms is studied.

In [2], with the Hankel matrix

with rank r, the necessary and sufficient existence condition of an atomic representing measure with exactly r atoms for the sequence is the existence of a “flat extension”. A “flat extension” is a rank preserving, nonnegative extension of, associated with a larger moment sequence A main result in [2], establishes some algebraic relations between the condition in which the support of the representing measure of the given sequence is contained in the algebraic variety of zerous of a suitable polynomial and the dependence relations established between the columns of the Hankel matrix. These relations are expressed, also, as zerous of the mentioned polynomial. In [1], the concept of “flat extension” of the Hankel matrix of truncated scalar multidimensional moment sequence is substituted with the concept of “dimension stability” of the algebraic dimension of the Hilbert space obtained as the separation of the space of scalar polynomials with total degree m, with respect to the null subspace of a unital, square positive functional, the Riesz functional. The Riesz functional is in bijection with a moment functional, positive on the cone of sums of squares of real polynomials. The stability condition of the algebraic dimension of such Hilbert space affords in [1] an algebraic condition for obtaining some commutative tuple of selfadjoint operators, defined on the Hilbert space of stable dimension. In the same time, in [1], by extending the Riesz functional on the whole space of polynomials, using the functional calculus of the constructed commutative selfadjoint tuple, the arbitrary powers of it are organized as a algebra of the same stable dimension. The problem of stability of the algebraic dimension of some Hilbert space obtained in this way is naturally connected, via the existence of a commuting tuple of self adjoint operators, with that of solving a scalar truncated multidimensional moment problem. The representing measure of the finite dimensional moment sequence is, in [1], the spectral atomic joint measure associated with the constructed commutative selfadjoint tuple, and has the same number of atoms as the stable algebraic dimension. Truncated operatorvalued problems is the subject in papers [4-7], to quote only a few of them. In [7] a Hausdorff truncated unidimensional operator-valued moment problem is studied. For obtaining the representing measure the Kolmogorov’s decomposition theorem is used. The given positive kernel of operators in [7] acts on an arbitrary, separable Hilbert space, all operators are linear independent, also the number of operators is arbitrary, even or odd.

In the present note, the stability dimension concept in [1], in the following way is adapted: a positive finite operator-valued kernel acting on a finite dimensional Hilbert space is given; a hermitian square positive functional on the space of vectorial functions, via the given kernel, is introduced. The restrictions of the hermitian square positive functional to some subspaces of the vectorial functions are considered. The separation spaces with respect to the null subspaces of these hermitian square positive restricted functionals are obtained. The stability dimensional condition for the obtained Hilbert spaces, in the same way as in [1], affords a construction of a commuting tuple of selfadjoint oparators, defined on the Hilbert space of the stable dimension. The obtained commuting tuple of selfadjoint operators, produced an integral representing joint spectral measure of all powers of the tuple. Via Kolmogorov’s theorem of decomposition of positive operator kernels, a representing positive operator-valued measure as Hausdorff truncated multidimensional moment sequence for all terms in the given kernel is obtained. The first terms of the given kernel, in number equal with “d”—the stable dimension, are linear independent, are integral represented with respect to an atomic operator valued measure with exactely “d” atoms, the remainder terms in the kernel are integral represented with respect to the same measure and the same number of atoms as the first one. The possibility of extension of the given operator sequence with preserving the “stability condition”, as well as the number of atoms of the representing measure is also analysed. In the present note, the number of operators in the given kernel is only even.

In this note, in Section 3 to a positive-definite kernel of operators a square positive functional is attached. The Hilbert spaces obtained as the quotient of some finite dimensional spaces and subspaces of vectorial functions with respect to the null spaces associated with the square positive functional and its restrictions are constructed. The problem of stability of the dimension of the Hilbert spaces in Section 3 and its implications in solving multidimensional, truncated Hausdorff operator-valued moment problems in Section 4, in this note is analysed.

2. Preliminaries

When and the p-dimensional real variable, are arbitrary, we denote with; the addition and substraction in are considered on components. For H an arbitrary Hilbert space, represents the algebra of linear, bounded operators on H, for a commuting tuple of multioperators, for all, we denote with . For an arbitraty the function, is:

with the Kronecker symbol. For we consider the spaces of vectorial functions:

and, for each integer, we denote with

with for all multiindices with at least one indices, the C-vector subspaces of For we also use the same function, defined by:

and, for all

we define the convolutions as

that is

3. Hilbert Spaces Associated with Finite Positive Operator Valued Kernels; Algebraic Prerequisite

We consider for an operator kernel

subject on the condition for all m with, acting on a finite dimensional complex vector space H, positively defined. That is the kernel satisfies the condition

(A)

for all sequences in H. When

is the C-vector space of functions defined on with vectorial values, we consider the kernel as a double indexed, simmetric one:

With the aid of, we introduce the hermitian, square positive functional

the order in is the lexicographical one. From property A of the the kernel, as well as from the properties of the scalar product in H, satisfies the conditions:

1) is C-linear in the first argument.

2), for all

3), for all

Remarks 3.1.

a) is a hermitian, square, positive functional on it results that satisfies the Cauchy-Buniakovski-Schwartz inequality, respectively:

b) Because of the construction of the hermitian functional and the simmetry of the kernel satisfies the equalities:

Definition 3.2. A functional defined on with properties 1)-3) is called a hermitian, square positive functional on.

Let be the subset in, defined as If follows, using the Cauchy-Buniakowski-Schwarz inequality, that if , we have also, that is is a vector subspace in. The map

is a seminorm on. Because H is a finite dimensional complex vector space, is the same. We consider the separated space of with respect to, that is, in this case, the quotient space Obviously, in finite dimensional case for, is a finite dimensional Hilbert space with the norm for an arbitrary. In the special case of the space

, we have

where represents the class in of the function

,

the usual Kroneker symbol, the basis of H and V represents the linear span of these elements. When, we consider the vector subspaces, with

and the restriction, restriction denoted with

. The functional is also a hspf on, consequently it has properties 1), 2), 3) and a), b) from Remarks 3.1. Setting

it results that is a vector subspace in and in. We denote with the Hilbert space with the norm

for all. Because, it results that also and there is the natural inclusion map

The inclusion map is injective one. Indeed, when

, it results

, that is The elements are in, consequently

that is or and,

We are interested in conditions in which is an isomorphism of vectorial spaces and . Because all Hilbert spaces, obtained in this way are finite dimensional, in case is an isomorphism of vectorial spaces, and have the same algebraic dimension. In connection with the problem of the stability of the algebraic dimension for the Hilbert spaces obtained as quatient of some vectorialvalued spaces of functions, we are interested in finding operator-valued atomic representing measure for the terms of. For such operator-valued, representing atomic measure, the stable number of atoms for the representations of the operatorsis the same with that in s integral representations. Such studies are the subject of truncated Hausdorff operator-valued moment problems. The concept of stability of the algebraic dimension of the Hilbert space obtained by separating the space of scalar polynomials with finite total degree with respect to an unital square positive functional (the Riesz functional), was introduced in [1]. The concept of stability of the algebraic dimension appears in [1] in the frame of extending some commuting tuple of selfadjoint operators which were intended to get the joint spectral representing measure for the terms of a Hausdorff truncated scalar moment sequences. The concept of stability of the algebraic dimension in [1] is an alternate, geometric aspect of that of “flatness” in Fialkow’s and Curto’s paper [2,3] regarding the truncated scalar moment problems.

We adapt and reformulate the concept of stability of the dimension concerning unital square positive functionals on space of scalar polinomyals in [1], to hermitian, square positive functionals associated with positive operator-valued kernels in order to solve operator-valued, truncated, Hausdorff moment problems via Kolmogorov’s theorem of decomposition of such kernels. The problem to obtain operator-valued positive operator representing measure for truncated, trigonometric and Hausdorff operator-valued moment problems via Kolmogorov’s theorem of decomposition of positive operator kernels were solved in [7].

The classical Kolmogorov’s theorem of the decomposition of positive kernels states:

“Let a nonnegative-definite function where S is a set and H a Hilbert space, namely

for any finite number of points

and any vectors. In this case there exists a Hilbert space K (essentialy unique) and a function such that for any.”

4. Dimension Stability and Consequences in Truncated, Hausdorff, Multidimensional, Operator-Valued Moment Problems

Let

be an operator kernel with and, positively defined, acting on the finite dimensional Hilbert space H, that is satisfies condition (A) from Section 3; we consider the vector space

and

the associated hspf with. We consider the set Because satisfies Cauchy-Buniakovski-Schwarz inequality, is a vector subspace in, and also

and is a finite dimensional Hilbert space, with the norm

The space is refered as the Hilbert space obtained via the hspf.

For every with, and

we consider the restriction The functional is also a hspf on and satisfies conditions 1), 2), 3) and a), b) in Remark 3.1. The subset

is obviously a subspace in and also in. Consequently, the Hilbert space with respect to the norm is defined via the hspf

and because is a vector subspace in T_N, the natural inclusion map, ,

for is an isometry. For l = N, we have In the same way, for all we have and consequently there exists the naturaly isometries.

If we have an operator-valued kernel acting on a finite dimensional Hilbert space H, subject on for all, such that

for all sequences

with finite support, considering the vector space

,

the associated functional

, with finite, the map is naturally a hspf on, with properties 1), 2), 3) and a), b) from Remark 3.1. Similar constructions of the Hilbert spaces as well as for the isometries can be done.

Definition 4.1. Let , be a positive definite kernel, the hspf associated with and the Hilbert spaces built via and the associated isometries. If for some the injective map is also surjective, that is we say that and the kernel is dimensionally stable (stable) at k.

The kernel is called dimensionally stable if there exists integers such that the kernel is stable at respectively is stable at l.

Remark 4.2. c) Let, , be a positive definite kernel, the hspf associated with stable at and the Hilbert spaces built via; (are bijective maps,) In this case, the maps,

when, andfor all are correctely defined.

Indeed, let be such that

; we shall prove that If

and is stable at

, is an isomorphism of vectorial spaces, exists then such that. In this case, using property a) and b) in Remarks 3.1 for the kernel we have:

(4)

Also, using the Cauchy-Schwartz inequality,

(5)

where we have denoted with.

From (4), (5), we have:

It results, that The maps

are correctely defined.

d) If we consider the subspaces

, also the null spaces

of it. We have

it results,

.

The same

and the null subspace

of it. We have also for the Hilbert quotient spaces we obtain

In the same way, by recurrence, we obtain the vectorial subspaces

the null subspaces

of it and the required quotient Hilbert space

for all From the above construction, the inclusions of the Hilbert spaces

and the naturally isometries

are obtained, for all Because, respectively are stable at, it results

Consequently all the isometries are surjective one. Let as consider the operators:

,

and

by

.

and

by. By recurrence the operators

and

From the construction, immediately, it follows that

(6)

The operators are correctely defined.

Indeed, let we show that also; from the stability condition, it exists such that and, from CauchyBuniakovski-Schwartz and property b) of the kernel, we have:

That is the operators are correctly defined. and extend the operators to We apply (6) for computing

Consequently, it results

(7)

where we have denoted

e) We consider the maps defined by for. With the given isomorphism of vectorial spaces in case of, stable at and with, the linear operators in c), the obtained operators are linear, correctely defined too.

Proposition 4.3. The linear operators, , are selfadjoin on and the tuple is a commuting multioperator on.

Proof. From Remark 4.1. e) The operators , are linear, correctely defined on the Hilbert space. We show that are selfadjoint one and commute; that is we verify

and for all

Let arbitrary,

From the stability of and at, is an isomorphism of vectorial spaces; it results that there exist

with

We have

(8)

where we have denoted by (the class of with respect to. The last statement is due to Kolmogorov’s theorem of decomposition of positive kernels.

We have also,

(9)

From (8) and (9), it results that for all that is

Commutativity. We shall prove that for all Following definitions, it is sufficient to verify in this order that

for all. Let

be the vectorial function, we have:

Because is stable at, is an isomorphism of vectorial spaces, it exists

such that:

case in which

(10)

We compute also

Because is stable at, is an isomorphism, it exists such that and

(11)

We prove that results in (10), (11) are equal; that is:

Indeed, let us consider the element

In these conditions, we can define Because is stable at, we can find such that. In these conditions, from Cauchy-Buniakowski-Schwartz inequality and property b) in Remark 3.1,

(12)

Similarly, if we denote with

we can define the extension

From the stability of, respectively is an isomorphism, it exists with From the same calculation as previous,

that is Consequently, we have:

(13)

From (12) and (13),

That is results in (10) and (11) are equal; commutativity occurs.                                   

Remark 4.4. In conditions of Proposition 4.3, we have for all, where stands for, respectively,

and

Proof. Indeed, because is stable at, if, from the definition of

we have

Let with we consider with and, or. In this case, from Remark 4.2 d) and first assertion of above,

From the definitions of the isomorphisms, , it results

;

that is

for all

Theorem 4.5. Let

with the property for any sequences Let the vector space of vectorial functions and the hspf associated with, as in Remark 3.1, stable at. Then there exists a unique extension of which is a hspf on and has property b) in Remark 3.1.

Proof. From Proposition 4.1, with the same notations as in Section 3, using the stability of at, we can define a p commuting tuple of selfadjoint operators, ,. For an arbitrary and, we define the element

when Let us consider the functional defined by

when with arbitrary.

We prove in the sequel that is an extension of, it is a hspf on and it has property b) in Remark 3.1.

From the properties of the scalar product on and definition of above, we have

for all arbitrary. Obviously, using the same properties, we have:

and, ,

arbitrary. It results that is a hspf on.

We verify that is an extension of For any vector-valued functions

we have:

that is is an extension as hspf to of (the above results uses Remark 4.4 and Kolmogorov’s theorem).

We verify that has also property b) in Remark 3.1, respectively it fullfiels

for all arbitrary. Indeed,

the required property. We prove that is the unique extension of with the mentioned properties. Suppose that are two extensions to of, both of them with the specified properties. We prove by recurrence, that for every and any vectorial function, there exists such that For and, because is an isomorphism, exists then an element such that; that is the required assertion, in case, is true. We consider the statement satisfied in case and prove it for that is for any, with, , there is an element, such that

. This relation means that

We compute:

where we have denoted with:

The same inequality is true for; it results that

The vectorial function

,

is an isomorphism (is stable at), there exists such that

that is We have:

For every, and any, it results, from above, that we can find the elements such that Moreover,

showing that The integer is arbitrary choosen, we obtain that

; that is the extension with such properties is unique.                               

Remark 4.6. Let be defined by

, for any.

In this case, the null space is

.

It results that, for any there exists such that; we prove that for any.

Let such that

, it follows

that is for any consequently, for any

Proposition 4.7. Let be an operator kernel, positively defined and the hspf associated with the kernel as previous, stable at and , the unique extension of to as hspf and property b) in Remark 3.1. Then is stable at any

Proof. Let be the Hilbert spaces built via and be the associated isometries. We prove, by induction, that for all The assertion is true for (is an isomorthism, is stable at N). Assume that the assertion is true for some and prove it for We fix an element and prove thatwe can find an element

such that, with the null space of For the element because is an isomorphism, it exists then

such that

that is

Using the property b) in Remark 3.1 for, we compute:

where we have denoted with

That is is a surjective isometry, insures that is an isomorphism of vectorial spaces. By recurrence, , the unique extension of as a hspf with property b) in Remark 3.1 is stable at any

Corollary 4.8. Let be an operator-valued kernel, positively defined and the hspf associated with as above. If is stable at one indices, then is stable at any

Proof. The unique extension of, as a hspf and with property b) in Remark 3.1 is the extension which is stable at any that is is stable at. It follows by recurrence that is stable at any

In the sequel, we argue like in [1], Remark 2.9.

Remark 4.9. Let be an operator kernel, positively defined, the hspf associated with the kernel as previous, stable at and, the unique extension of to as hspf and with property b) in Remark 3.1, defined in Proposition 4.5. Let be the Hilbert spaces constructed via because is stable at any the isometries, , are bijective one, that is We denote with We may construct the p-tuple of commuting selfadjoint operators on the space, , as in Proposition 4.2. We define as in Remark 4.1 (A) the operators by

We have immediately:

modulo for all It results that imply; respectively

Consequently

consequently we obtain:

A recurrence argument leads to the formula:

Let

subject on the same conditions as in Remark 4.2. We denote with with C-linear vectors in a basis of H,.

Proposition 4.10. Let be an operator kernel, stable at, as above. We consider that:

1) for all, we have , with vectors in a basis of, and for all,. In this case, for all the elements are linear independent in

2) Moreover, we consider in addition, that the kernel is such that for all, as in 1), the elements

are C-linear independent in and for any, , and any the elements

are linear dependent in K_N (in stands for).

Proof. 1) Indeed, let us show, that, if we have such that (14)

it results We have

(14)

Immediately, from above, we have:

(15)

In the same time, from Cauchy-Buniakowski-Schwarz inequality, we have:

(16)

Equatities (15) can be satisfied, in case of (16), only when

(17)

are true. In condition of Proposition 4.10. 1), the only case in which (17) can happen is; that is are linear independent elements in

Theorem 4.11. Let be an operator kernel, positively defined, stable at, such that its terms satisfy conditions 1) and 2) in Proposition 4.10, that is: for all, there exists { with at least one } such that

with the hspf associated with the kernel as previous. Then, there exists a d-atomic positive operator-valued representing measure, with atoms, on a compact set in such that:

Proof. As in Proposition 4.3, in the same conditions about the kernel, stable at and with the same notations, we obtain a commuting p-tuple of selfadjoint operators, ,. From [7], when we aplly Kolmogorov’s decomposition theorem to the positive definite kernel, we get the representations: for every with the operators,

Accordingly to Remark 4.4, we have: for every and every arbitrary vector-value. That is the representations occur, and by replacing in the above representations, we obtain

In these conditions, with respect to the joint spectral measure associated to the commuting tuple A, acting on the finite dimensional Hilbert space, the join spectrum with the spectrum of the bounded operator defined on. The set consists only of isolated, in finite number, principal values of; consequently is an atomic set and the joint spectral measure of A is an atomic one. With respect to the joint spectral measure, we have

We denote with for any a positive, operator-valued atomic measure and obtained the representations:

We consider the vector space

and define the map when

and

To check the definition is correct, we shall use again Remark 4.4 and show that, if, with, we have. Indeed, means that

That is From the definition, is linear; we prove that is also injective. Let us consider

such that that is:

that is is an injective map. We show that

is also a surjective map. We consider the element and prove that there exists such that Indeed, is stable at any, by recurrence, in Proposition 4.7, we have proved that there exists such that that is

The map, above defined, is an isomorphism of vectorial spaces, consequently:

Let also

with. Obviously, is a subspace in (the subspace generated in by). Because of property 2) Proposition 4.10. of the kernel, there exists scalars such that

;

that is

It follows that

Because is an isomorphism of vectorial spaces, the obtained representation is uniquely determinated (modulo T_N) and it results easily from Remark 4.4). From property 1) of the kernel, we have whenever, and also from 2), for all. We have proved in this way that (represents the direct sum.) With the usual operator’s multiplication for all, and from Remark 4.9, endowed with an echivalent norm induced on by the norm on via the map, the subspaces have all a structure of unital commuting C^* algebra with dimension, and. In these conditions, the joint spectral measure of

has precisely s characters, therefore, the joint spectrum has exactely s atoms. Because of the representation, it follows that the joint spectrum of A has exactely atoms. Consequently the measure, in’s representations, has the same number of atoms.                                    

5. Conclusion

We adapt the concept of “stability of the dimension”, in [1], of some Hilbert spaces obtained as the qotient spaces of scalar polynomials of finite degree with respect to the null space of the Riesz functional, to that of “stability of the dimension” of some Hilbert spaces obtained as the quotient spaces of some vectorial-valued functions with respect to the null space of some hermitian square positive functional associated with a positive defined kernel of operators. The stability of this dimension is considered in connection with a truncated operator valued moment problem. The stability of the dimension of the obtained Hilbert space, represents the conditrion for stability of the number of atoms of the obtained operator-valued atomic representing measure for the given kernel.

REFERENCES