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 operators
is 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 TN, 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
, and
for 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 KN (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 TN) 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.