Stability of Operator-Valued Truncated Moment Problems

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.


Introduction
The study of scalar truncated moment problems is the subject of many remarkable papers such as: [1][2][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   , N p is the existence of a "flat extension".A "flat extension" is a rank preserving, nonnegative extension M   of M  , 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 M  .These relations are expressed, also, as zerous of the mentioned polynomial.In [1], the concept of "flat extension" of the Hankel matrix of truncated sca-lar 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 C  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 commuta-tive 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.

Preliminaries
When   1 N , N , , , the p-dimensional real variable, are arbitrary, we denote with 1 , , p q q q t t t   ; the addition and substraction in N p are considered on components.For H an arbitrary Hilbert space, represents the algebra of linear, bounded operators on H, for for all 1 i p   , we denote with we consider the spaces of vectorial functions: for all multiindices with at least one indices , the C-vector subspaces of we also use the same function,   defined by:   ,1 ,

Hilbert Spaces Associated with Finite Positive Operator Valued Kernels; Algebraic Prerequisite
We consider for for all m with 0 H , acting on a finite dimensional complex vector space H, positively defined.That is the kernel is the C-vector space of functions defined on  


With the aid of N  , we introduce the hermitian, square positive functional is the lexicographical one.From property A of the the kernel N  , as well as from the properties of the scalar product in H, N   satisfies the conditions: 1) N   is C-linear in the first argument. 2) Because of the construction of the hermitian functional N  and the simmetry of the kernel , Because H is a finite dimensional complex vector space, N F is the same.We consider the separated space of N F with respect to N T , that is, in this case, the quotient space .
In the special case of the space the usual Kroneker symbol, 1 k the basis of H and V represents the linear span of these elements.When , it results that k is a vector subspace in k T F and in N T .We denote with k K the Hilbert space and there is the natural inclusion map , k N : , We are interested in conditions in which , k N J is an isomorphism of vectorial spaces and , obtained in this way are finite dimensional, in case J is an isomorphism of vectorial spaces, k K and N K 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 N  .For such operator-valued, representing atomic measure, the stable number of atoms for the representations of the operators is the same with that in N   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 be an operator kernel with m m and 0 H     , positively defined, acting on the finite dimensional Hilbert space H, that is N  satisfies condition (A) from Section 3; we consider the vector space T is a vector subspace in N F , and also The space N K is refered as the Hilbert space obtained via the hspf N   .For every with 0 l l N   , and H with 0 for all , , with at least one indices , , is obviously a subspace in N T and also in l F .Consequently, the Hilbert space and because is a vector subspace in T N , the In the same way, for all we have k T T and consequently there exists the naturaly isometries , , acting on a finite dimensional Hilbert space H, subject on for all , such that with finite support, considering the vector space

 
: N H, with finite support the associated functional  , with properties 1), 2), 3) and a), b) from Remark 3.1.Similar constructions of the Hilbert spaces 0 l l  as well as for the isometries can be done.
is called dimensionally stable if there exists integers such that the kernel In this case, the maps : for all are correctely defined.
; we shall prove that .
In this case, using property a) and b) in Remarks 3.1 for the kernel  we have: Also, using the Cauchy-Schwartz inequality, where we have denoted with .4), ( 5), we have: , , 0 , , The same and the null subspace , , 0 for the Hilbert quotient spaces we obtain .
In the same way, by recurrence, we obtain the vectorial subspaces , , 0 for all From the above construction, the inclusions of the Hilbert spaces and the naturally isometries : , : are surjective one.Let as consider the operators: : : : . By recurrence the operators 12 , 1 ; : , ; : ; From the construction, immediately, it follows that , 1 .
The operators , ,1 ; and, from Cauchy-Buniakovski-Schwartz and property b) of the kernel N  , we have: , , That is the operators are correctly defined.and extend the operators 1 : We apply (6) for computing , 1; , , . .
where we have denoted .
We consider the maps : for 1  .With the given isomorphism of vectorial spaces in case of  , stable at and with 1 N  j M , the linear operators 1  in c), the obtained i p   j A operators are linear, correctely defined too.
Proposition 4.3.The linear operators : is a commuting multioperator on N K .Proof.From Remark 4.1.e) The operators : are linear, correctely defined on the Hilbert space N K .We show that j A are selfadjoint one and commute; that is we verify ˆˆˆ, , an isomorphism of vectorial spaces; it results that there exist , , ; where we have denoted by with respect to N T .The last statement is due to Kolmogorov's theorem of decomposition of positive ernels.k We have also,

T y T h x h y h h x y
x y From ( 8) and ( 9), it results that ˆ, be the vectorial function , we have: We compute also .
Indeed, let us consider the element In these conditions, we can define . In these conditions, from Cauchy-Buniakowski-Schwartz inequality and property b) in Remark 3.1, we can define the extension .
1 , , , : In this case, from Remark 4.2 d) and first assertion of above, .
From the definitions of the isomorphisms   with the property F the vector space of vectorial functions and N the hspf associated with Then there exists a unique extension    of Proof.From Proposition 4.1, with the same notations as in Section 3, using the stability of that is   is an extension as hspf to F F    of N   (the above results uses Remark 4.4 and Kolmogorov's theorem).
We prove in the sequel that :  and it has prop 3.1.From the properties of the scalar pr erty b) in Remark oduct on N K and We verify that   has also property b) in Remark 3.1, respectively it fullfiels We verify that tension of For any vector-v we have: the required property.We prove that  is the unique extension of , both of them with the specified properties.We prove by recurrence, that for every l l and any vectorial function N N is an isomorphism, exists then an element ; that is the required assertion, in case , is true.We consider the statement satisfied in case and prove it for   that is for any 1 1 1 , there is an ele- where we have denoted with: k The same inequality is true for ; it results that For every , and any 0 l  , , it results, from above, that can find the el we , .
showing that .
we obt ger 0 l  is arbitrary choosen, ain that ; that is the extension properties with such is unique.Remark 4.6.Let , for any .
In this case, the null space is , , .
for an osition 4.7.
sitively defined and hspf associated with the ke be t lbert spaces buil :

K
he Hi t via and is stable at N Assume that the assertion is true for some and prove it for We fix an element and prove that, n find an element we ca . For th : as a nd with property b) in Remark 3.1 is the   extension w h is stable at any l k  t is hspf a hic tha We denote with , 1 .
We may c muting selfadjoint operators  on the space l K , l N  , as in Proposition 4.2.We define as in Remark 4.1 (A) the ope 1 : .
We have immediately: consequently we obtain: A recurrence argument leads to the formula: Let subject on the same conditions as in Remark 4.2.We denote with  , N K  2) Moreover, we consider in addition, that the kernel the eleme s nt , , .
, and any ). if we have , C Proof. 1) Indeed, let us show, that, Immediately, from above, we have: e time, from Cauchy-Buniakowski-Schwarz inequality, we have: .
Equatities (15) can be satisfied, in case of (16), only when In these conditions, with respect to the joint spectral measure associated to the commuting tuple A, acting on the finite dimensional Hilbert space N K , the join spectrum the spectrum of the bounded operator k A defined on N K .The set consists only of isolated, in finite number, principal values of k A ; consequently is an atomic set and the joint spectral measure A of A is an atomic one.With respect to the joint spectral measure, we have  a positive, operator-valued atomic measure and obtained the representations: and define the map To check the definition is correct, we shall use again Remark 4.4 and show that, if 1

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 nction to the null space square nctional associated sitive 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 quotient spaces of some vectorial-valued fu s with respect of some hermitian positive fu with a po the number of atoms of the obtained operator-valued atomic representing measure for the given kernel.


is also a hspf on k F , consequently it has properties 1), 2), 3) and a), b) from Remarks 3


The map , above defined, is an isomorphism of vectorial spaces onsequently: N .Obviously, i A % is a subspace in A % (the subspace generated in N K by

. Dimension Stability and Consequences in Truncated, Hausdorff, Multidimensional, Operator-Valued Moment Problems
and the required quotient Hilbert space