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 ![](https://www.scirp.org/html/19-7401336\c7acebc6-1b00-489c-a104-62d6963e1699.jpg)
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:
![](https://www.scirp.org/html/19-7401336\59644d5e-1c72-4595-80a2-6360dad02c63.jpg)
with
the Kronecker symbol. For
we consider the spaces of vectorial functions:
![](https://www.scirp.org/html/19-7401336\b4a3ad90-868a-4af2-8580-4f32f61c494d.jpg)
and, for each integer
, we denote with
![](https://www.scirp.org/html/19-7401336\5f22c7d1-5bae-4373-840f-cdd8dd811aaa.jpg)
with
for all multiindices
with at least one indices
, the C-vector subspaces of
For
we also use the same function,
defined by:
![](https://www.scirp.org/html/19-7401336\8366fb2a-93fb-42bd-9cea-b1572e30da00.jpg)
and, for all
![](https://www.scirp.org/html/19-7401336\ff9fe6c9-cb48-43fb-9e20-b7381c60081b.jpg)
we define the convolutions
as
![](https://www.scirp.org/html/19-7401336\f5ebfbf7-b678-4180-bcc8-9f49d38640e8.jpg)
that is ![](https://www.scirp.org/html/19-7401336\445d61b3-5d35-4b20-ab41-cc2faa4c26fa.jpg)
3. Hilbert Spaces Associated with Finite Positive Operator Valued Kernels; Algebraic Prerequisite
We consider for
an operator kernel
![](https://www.scirp.org/html/19-7401336\863ae738-db9b-48b1-9eec-9894b6c6585d.jpg)
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) ![](https://www.scirp.org/html/19-7401336\b54c3080-ab63-415a-9245-b67d146daef8.jpg)
for all sequences
in H. When
![](https://www.scirp.org/html/19-7401336\faaf341b-a737-43ef-9b99-6f9e55957fbb.jpg)
is the C-vector space of functions defined on
with vectorial values, we consider the kernel
as a double indexed, simmetric one:
![](https://www.scirp.org/html/19-7401336\27ee972e-1e14-49e3-9102-e1479a47600e.jpg)
With the aid of
, we introduce the hermitian, square positive functional
![](https://www.scirp.org/html/19-7401336\7b955304-02ac-472a-a4d8-d35fb6bc0a27.jpg)
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 ![](https://www.scirp.org/html/19-7401336\7f448863-b43e-4ae6-8fba-33f0892e973a.jpg)
3)
, for all ![](https://www.scirp.org/html/19-7401336\0dc43184-a967-4a23-a434-e5814f22bee2.jpg)
Remarks 3.1.
a)
is a hermitian, square, positive functional on
it results that
satisfies the Cauchy-Buniakovski-Schwartz inequality, respectively:
![](https://www.scirp.org/html/19-7401336\eb0ce62f-afbd-4c9f-818d-0b6a1956b097.jpg)
b) Because of the construction of the hermitian functional
and the simmetry of the kernel
satisfies the equalities:
![](https://www.scirp.org/html/19-7401336\6bd1d9f6-b35a-4334-91ee-dc18208dc6b7.jpg)
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
![](https://www.scirp.org/html/19-7401336\91f6e86c-6d10-44cc-9ac7-e0039e8d1e9c.jpg)
, ![](https://www.scirp.org/html/19-7401336\6286cb1e-d712-40cb-a58c-2c69b6b18403.jpg)
the usual Kroneker symbol,
the basis of H and V represents the linear span of these elements. When
, we consider the vector subspaces
, with
![](https://www.scirp.org/html/19-7401336\635b329e-22fe-4035-a321-e495b50b10f4.jpg)
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
![](https://www.scirp.org/html/19-7401336\41b5de92-2447-41f9-a722-37a10d3af3a3.jpg)
it results that
is a vector subspace in
and in
. We denote with
the Hilbert space
with the norm
![](https://www.scirp.org/html/19-7401336\f563a1b1-81b9-4b46-8d6f-f32620484c0d.jpg)
for all
. Because
, it results that also
and there is the natural inclusion map
![](https://www.scirp.org/html/19-7401336\021991a6-2ab2-45a4-b806-77273ad3367e.jpg)
The inclusion map
is injective one. Indeed, when
,
it results ![](https://www.scirp.org/html/19-7401336\653f7241-af6b-4b1c-b867-ca8bbd0b717a.jpg)
, 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 ![](https://www.scirp.org/html/19-7401336\c273a414-1691-44ac-b828-0c39d4e415da.jpg)
![](https://www.scirp.org/html/19-7401336\3b8d31dd-7203-4628-9d36-0b161467b831.jpg)
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
![](https://www.scirp.org/html/19-7401336\e0495e00-91d9-4654-9a96-bf7e21dc6649.jpg)
and ![](https://www.scirp.org/html/19-7401336\4f224340-11c6-4548-af0f-2a52359c882c.jpg)
the associated hspf with
. We consider the set
Because
satisfies Cauchy-Buniakovski-Schwarz inequality,
is a vector subspace in
, and also
![](https://www.scirp.org/html/19-7401336\f02beaff-1749-4d0f-9356-b30dde694f83.jpg)
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
![](https://www.scirp.org/html/19-7401336\ae419e12-f833-47fe-a05b-1a0d6bc2e36a.jpg)
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
![](https://www.scirp.org/html/19-7401336\31a175e9-4fd6-4f66-b2d1-8734c891b40b.jpg)
is obviously a subspace in
and also in
. Consequently, the Hilbert space
with respect to the norm
is defined via the hspf ![](https://www.scirp.org/html/19-7401336\a9491f5f-c955-43ac-aa33-6b8e88c5a0a2.jpg)
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 ![](https://www.scirp.org/html/19-7401336\87b6c13c-12ef-4280-8993-2d9c056fa13f.jpg)
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
, ![](https://www.scirp.org/html/19-7401336\e15fb797-4f95-49ac-96b1-775d68271a0c.jpg)
![](https://www.scirp.org/html/19-7401336\1c263a0c-e458-4c92-a0ed-eaf8505b1af3.jpg)
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:
![](https://www.scirp.org/html/19-7401336\29d4261e-db6e-48a5-a3fc-0f9d72dbd0fe.jpg)
It results, that
The maps
![](https://www.scirp.org/html/19-7401336\9dc38528-3fd1-450c-af77-5ae2c00d9a7e.jpg)
![](https://www.scirp.org/html/19-7401336\48b016c5-dd8e-4637-a471-d190619cb1d6.jpg)
are correctely defined.
d) If we consider the subspaces
![](https://www.scirp.org/html/19-7401336\d7299495-4365-43fc-952f-0d9bc81034fd.jpg)
, also
the null spaces
![](https://www.scirp.org/html/19-7401336\1806c333-55c9-40f7-8e5a-76e74176ece9.jpg)
of it. We have
![](https://www.scirp.org/html/19-7401336\1afa263e-bbc0-48b2-acc3-9c5e96c0af9e.jpg)
it results,
.
The same
![](https://www.scirp.org/html/19-7401336\41a06840-fd62-4e60-a656-96fa8c16df7a.jpg)
and the null subspace
![](https://www.scirp.org/html/19-7401336\7349d32d-7ef1-4866-ae04-3993eb3a4dbc.jpg)
of it. We have also
for the Hilbert quotient spaces we obtain
![](https://www.scirp.org/html/19-7401336\a2936ad4-fd9d-459d-8980-eec210ca5fa5.jpg)
In the same way, by recurrence, we obtain the vectorial subspaces
![](https://www.scirp.org/html/19-7401336\b0897fea-f9e4-4e5a-8007-d8dead04c9ae.jpg)
the null subspaces
![](https://www.scirp.org/html/19-7401336\28226d97-bf64-47db-a2f1-9c6be8b56549.jpg)
of it and the required quotient Hilbert space
for all
From the above construction, the inclusions of the Hilbert spaces
![](https://www.scirp.org/html/19-7401336\2a847ff5-4322-495e-bd2d-cb796f63f860.jpg)
and the naturally isometries
![](https://www.scirp.org/html/19-7401336\e6f90b08-6cf4-44e1-87f1-b99ea9b3f402.jpg)
![](https://www.scirp.org/html/19-7401336\bf9dd65d-59ab-4c74-87df-8e0fadad8f16.jpg)
are obtained, for all
Because
, respectively
are stable at
, it results
![](https://www.scirp.org/html/19-7401336\669f9b56-d88b-4872-a2ca-bd1520ff5abf.jpg)
Consequently all the isometries
are surjective one. Let as consider the operators:
,
![](https://www.scirp.org/html/19-7401336\507df9d1-bf24-40ee-a216-f43319a32b7f.jpg)
![](https://www.scirp.org/html/19-7401336\71d03823-9de9-4cdf-8326-26cad600d983.jpg)
and
![](https://www.scirp.org/html/19-7401336\7a50de95-77e3-4f02-9bce-e8481dc426f3.jpg)
by
.
![](https://www.scirp.org/html/19-7401336\f99dc9cd-84dd-4417-88f4-3d094d7cafb9.jpg)
![](https://www.scirp.org/html/19-7401336\f902300e-c7d9-48fc-bf6e-a65f0b4b822d.jpg)
and
![](https://www.scirp.org/html/19-7401336\70fb57b2-2c16-4686-9ff4-f9f36e34a095.jpg)
by
. By recurrence the operators
![](https://www.scirp.org/html/19-7401336\3ec50513-15f6-477d-b722-be104cd0d90c.jpg)
![](https://www.scirp.org/html/19-7401336\218daf75-74b7-46e2-89e7-87a49046bb5d.jpg)
![](https://www.scirp.org/html/19-7401336\a4805a85-8540-444d-96a1-000efee74090.jpg)
and
![](https://www.scirp.org/html/19-7401336\a4794ec7-4ad7-49f7-a688-8c219b1decb7.jpg)
![](https://www.scirp.org/html/19-7401336\b9eca145-37b7-4745-aadf-98049958f3b0.jpg)
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:
![](https://www.scirp.org/html/19-7401336\7476effe-ca7e-45b4-ae6d-b0428544b084.jpg)
That is the operators
are correctly defined. and extend the operators
to
We apply (6) for computing
![](https://www.scirp.org/html/19-7401336\9e117602-fdf4-456b-9a41-a6f3db04bf7d.jpg)
Consequently, it results
(7)
where we have denoted
![](https://www.scirp.org/html/19-7401336\b185e491-19e4-4df5-a95d-0cb5d69d9733.jpg)
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,
![](https://www.scirp.org/html/19-7401336\6714c6ee-9219-48c2-9f8c-45a7b5507a4f.jpg)
![](https://www.scirp.org/html/19-7401336\e60d4749-ac35-4c8c-abdb-ca09038e9a51.jpg)
From the stability of
and
at
,
is an isomorphism of vectorial spaces; it results that there exist
![](https://www.scirp.org/html/19-7401336\d846891e-ff06-4a91-9bdd-076755f1168f.jpg)
![](https://www.scirp.org/html/19-7401336\0d8ebc61-324d-47a7-8209-78a5986d5fc7.jpg)
with ![](https://www.scirp.org/html/19-7401336\ce9f1b85-6c2e-4fc9-91a4-641311c66216.jpg)
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 ![](https://www.scirp.org/html/19-7401336\867d90f4-cfec-4e72-a3c5-641b560475f3.jpg)
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:
![](https://www.scirp.org/html/19-7401336\eed3b88c-b1c9-435d-bd0d-df49cb3145ba.jpg)
Because
is stable at
,
is an isomorphism of vectorial spaces, it exists
such that:
![](https://www.scirp.org/html/19-7401336\33bb7a3d-8ce6-45c5-b5ef-cb88bae5b773.jpg)
case in which
(10)
We compute also
![](https://www.scirp.org/html/19-7401336\28274e29-dda7-4b52-bafa-e437cdae9fef.jpg)
Because
is stable at
,
is an isomorphism, it exists
such that
and
(11)
We prove that results in (10), (11) are equal; that is:
![](https://www.scirp.org/html/19-7401336\81691db9-b639-4893-8951-65648011661b.jpg)
Indeed, let us consider the element
![](https://www.scirp.org/html/19-7401336\4943ff7d-2253-45cd-9ae6-a2e7db8f1e18.jpg)
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
![](https://www.scirp.org/html/19-7401336\1395a3a0-f9f0-4f6b-a56d-c54f0e8187c8.jpg)
we can define the extension
![](https://www.scirp.org/html/19-7401336\ccfd5c24-07cc-4b32-836e-0e008225261a.jpg)
From the stability of
, respectively
is an isomorphism, it exists
with
From the same calculation as previous,
![](https://www.scirp.org/html/19-7401336\95b99ec2-7c84-4951-92af-e7be52f1b227.jpg)
that is
Consequently, we have:
(13)
From (12) and (13),
![](https://www.scirp.org/html/19-7401336\5a5ca41d-0536-4e01-9b20-08067c0865b7.jpg)
That is results in (10) and (11) are equal; commutativity occurs. ![](https://www.scirp.org/html/19-7401336\a849cb1c-17a5-4dde-a60c-f58f58518453.jpg)
Remark 4.4. In conditions of Proposition 4.3, we have
for all
,
where
stands for
, respectively
, ![](https://www.scirp.org/html/19-7401336\bb8c41d8-aa67-4e32-9ee5-6ac47bf5b7e8.jpg)
and ![](https://www.scirp.org/html/19-7401336\63ebd1db-d6b7-4b90-997d-1583eeabd56e.jpg)
Proof. Indeed, because
is stable at
, if
, from the definition of
we have
![](https://www.scirp.org/html/19-7401336\01aae1f4-d4e9-49cb-9ade-b9815bb9b9b5.jpg)
Let
with
we consider
with
and
,
or
. In this case, from Remark 4.2 d) and first assertion of above,
![](https://www.scirp.org/html/19-7401336\43738f4d-215f-4eea-ab04-1da1df1ffc19.jpg)
From the definitions of the isomorphisms
,
, it results
;
that is
![](https://www.scirp.org/html/19-7401336\be51e976-3cfe-4cc8-b61d-52448ec04e1b.jpg)
for all ![](https://www.scirp.org/html/19-7401336\16408051-5800-44c9-af6d-922fbb87f6a4.jpg)
Theorem 4.5. Let
![](https://www.scirp.org/html/19-7401336\7d8b0ffa-81a2-4489-81e1-346123c94476.jpg)
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
![](https://www.scirp.org/html/19-7401336\7fa7b569-a5af-4de6-897b-5092fee782b1.jpg)
when
Let us consider the functional
defined by ![](https://www.scirp.org/html/19-7401336\2162e471-6360-4763-ab6c-d1f49e0f8c4b.jpg)
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
![](https://www.scirp.org/html/19-7401336\17e1da30-d53d-4fc3-a032-7363d889fe89.jpg)
for all
arbitrary. Obviously, using the same properties, we have:
and
,
, ![](https://www.scirp.org/html/19-7401336\68259092-e9e6-4ff7-b6a7-c690769dff16.jpg)
arbitrary. It results that
is a hspf on
.
We verify that
is an extension of
For any vector-valued functions
![](https://www.scirp.org/html/19-7401336\b1081e09-292b-4efb-8702-d640da0f53fe.jpg)
we have:
![](https://www.scirp.org/html/19-7401336\1eaaa63f-1c2a-49e5-a8cd-52611200c09f.jpg)
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
![](https://www.scirp.org/html/19-7401336\e8004210-10fe-4851-9c5b-fd20b8f38cc5.jpg)
for all
arbitrary. Indeed,
![](https://www.scirp.org/html/19-7401336\2bc0643a-86e0-4f40-9403-0a50b1dcf2fd.jpg)
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
![](https://www.scirp.org/html/19-7401336\c1e1c70e-2447-4d33-826c-f4d7fb928f15.jpg)
We compute:
![](https://www.scirp.org/html/19-7401336\b901f1f9-ea31-4bf6-bc06-1506849168d5.jpg)
where we have denoted with:
![](https://www.scirp.org/html/19-7401336\3fe2ff2a-6959-47f6-b9e8-51988d1baec0.jpg)
The same inequality is true for
; it results that
![](https://www.scirp.org/html/19-7401336\119bf158-47ba-47c8-8ac0-ec4161e59366.jpg)
The vectorial function
, ![](https://www.scirp.org/html/19-7401336\42334931-4def-4fe3-b452-ecae7aab5ffe.jpg)
is an isomorphism (
is stable at
), there exists
such that
![](https://www.scirp.org/html/19-7401336\ca3e70f3-5b14-4686-a5e2-83c208ce0102.jpg)
that is
We have:
![](https://www.scirp.org/html/19-7401336\f4eb69b8-da94-43b6-b8e4-fcc295ef9ee8.jpg)
For every
, and any
, it results, from above, that we can find the elements
such that
Moreover,
![](https://www.scirp.org/html/19-7401336\f7649fb6-856a-4228-aac6-bc18ab90df8a.jpg)
showing that
The integer
is arbitrary choosen, we obtain that
; that is the extension with such properties is unique. ![](https://www.scirp.org/html/19-7401336\d284b5ae-87f1-4e00-bc6c-0294a822c922.jpg)
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
![](https://www.scirp.org/html/19-7401336\772e8fc7-0e22-462c-ba91-52d45aaa8bc0.jpg)
, it follows
![](https://www.scirp.org/html/19-7401336\837a961e-7e6c-49ab-84ec-ac858dc40d4f.jpg)
that is
for any
consequently,
for any ![](https://www.scirp.org/html/19-7401336\56a72a42-0df0-480a-82a9-d319b34d6ce2.jpg)
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 ![](https://www.scirp.org/html/19-7401336\5321ff77-695c-4f6b-a30f-fce9ded1196b.jpg)
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 ![](https://www.scirp.org/html/19-7401336\9779ed96-a0b0-4399-a795-689fc604d2e6.jpg)
such that
, with
the null space of
For the element
because
is an isomorphism, it exists then
such that
that is
![](https://www.scirp.org/html/19-7401336\b69f72cc-4a12-489f-861b-9480444163c5.jpg)
Using the property b) in Remark 3.1 for
, we compute:
![](https://www.scirp.org/html/19-7401336\92c81f36-9f5e-4c1a-b5cb-5608c3b91163.jpg)
where we have denoted with
![](https://www.scirp.org/html/19-7401336\396d1406-e2ea-4f2c-8e48-5ed2fe70d90a.jpg)
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 ![](https://www.scirp.org/html/19-7401336\6877ae4e-4f1d-4fa3-b576-6d2633f5a429.jpg)
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![](https://www.scirp.org/html/19-7401336\11808851-6a11-4676-95db-8bf588830d82.jpg)
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 ![](https://www.scirp.org/html/19-7401336\47689b26-6928-4361-8b1d-cb4caf4da2fb.jpg)
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
![](https://www.scirp.org/html/19-7401336\2d045254-4ed1-4310-ba32-0abf179c535f.jpg)
We have immediately:
![](https://www.scirp.org/html/19-7401336\752ea4c6-f1eb-4b7b-8b06-e593ed688a3b.jpg)
modulo
for all
It results
that imply
; respectively
![](https://www.scirp.org/html/19-7401336\e23f1e7f-720e-465d-ad80-88901a1f51eb.jpg)
Consequently
![](https://www.scirp.org/html/19-7401336\0e5cd677-c21e-4532-b747-0353683c1701.jpg)
consequently we obtain:
![](https://www.scirp.org/html/19-7401336\7182dbff-4b05-47b7-a647-c80f18a5e96a.jpg)
A recurrence argument leads to the formula:
![](https://www.scirp.org/html/19-7401336\212927a0-92a8-4022-94d3-a0702afc0f06.jpg)
Let
![](https://www.scirp.org/html/19-7401336\df9d0cbb-777b-42c5-9e48-4f85440e5e6e.jpg)
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 ![](https://www.scirp.org/html/19-7401336\19acb084-93d2-4538-b170-7287474dae2e.jpg)
2) Moreover, we consider in addition, that the kernel
is such that for all
, as in 1),
the elements
![](https://www.scirp.org/html/19-7401336\6e4b6a23-71c1-4bda-8fa8-014dff3c745c.jpg)
![](https://www.scirp.org/html/19-7401336\ba826a60-260e-42c5-8a7d-4a9cf899a237.jpg)
are C-linear independent in
and for any
,
, and any
the elements
![](https://www.scirp.org/html/19-7401336\74d17bef-ad64-4baa-aeee-424d2f6cb622.jpg)
are linear dependent in KN (in
stands for
).
Proof. 1) Indeed, let us show, that, if we have
such that (14)
![](https://www.scirp.org/html/19-7401336\633fc099-520c-4835-a463-01caf0e9e9bf.jpg)
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 ![](https://www.scirp.org/html/19-7401336\02c0635a-e622-4586-af17-266d20d363ee.jpg)
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
![](https://www.scirp.org/html/19-7401336\43188770-c444-482a-bb0a-cb138f8f0a50.jpg)
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:
![](https://www.scirp.org/html/19-7401336\dab5bd44-7fd3-4c9a-94e6-1cb2c434cabe.jpg)
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
,
![](https://www.scirp.org/html/19-7401336\4be060cb-4fdb-450f-8c7f-d56c2edcc32c.jpg)
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
![](https://www.scirp.org/html/19-7401336\685abed6-b0fb-4e8b-ac23-fed68056ddbe.jpg)
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
![](https://www.scirp.org/html/19-7401336\5acc81b5-768c-4271-8812-f308b1d745a0.jpg)
We denote with
for any
a positive, operator-valued atomic measure and obtained the representations:
![](https://www.scirp.org/html/19-7401336\10fe403e-64df-4e69-b57f-0a7cbd5d3943.jpg)
We consider the vector space
![](https://www.scirp.org/html/19-7401336\6869cef3-35c2-46b9-90c5-294debd2843c.jpg)
and define the map
when
![](https://www.scirp.org/html/19-7401336\4640ecd0-51f8-4381-86f2-411df69af1e8.jpg)
and
![](https://www.scirp.org/html/19-7401336\ac29b868-4080-4ac8-8693-88b335468dbc.jpg)
To check the definition is correct, we shall use again Remark 4.4 and show that, if
, with
, we have
. Indeed,
means that
![](https://www.scirp.org/html/19-7401336\229160cb-f0f5-4561-bf8d-ad7f4aa666ad.jpg)
That is
From the definition,
is linear; we prove that
is also injective. Let us consider
![](https://www.scirp.org/html/19-7401336\e4ed5388-b2eb-4c62-8173-e39d491329e2.jpg)
such that
that is:
![](https://www.scirp.org/html/19-7401336\a31e4239-139c-4c6f-9f09-3d00dcbcbe92.jpg)
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
![](https://www.scirp.org/html/19-7401336\8062f947-890e-457f-aadd-4984e31589c3.jpg)
The map
, above defined, is an isomorphism of vectorial spaces, consequently: ![](https://www.scirp.org/html/19-7401336\018760c7-c950-414a-8dac-46637ae0db67.jpg)
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
![](https://www.scirp.org/html/19-7401336\4bb468c3-7048-4c2e-83a3-70447a63a078.jpg)
It follows that
![](https://www.scirp.org/html/19-7401336\c3bb5f6a-feb1-4679-a741-49bd79cc5e97.jpg)
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. ![](https://www.scirp.org/html/19-7401336\2ca1a284-50ac-440a-869c-47c2d0ee7199.jpg)
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.