Positive-Definite Operator-Valued Kernels and Integral Representations

Abstract

A truncated trigonometric, operator-valued moment problem in section 3 of this note is solved. Let be a finite sequence of bounded operators, with arbitrary, acting on a finite dimensional Hilbert space H. A necessary and sufficient condition on the positivity of an operator kernel for the existence of an atomic, positive, operator-valued measure , with the property that for every with , the moment of coincides with the term of the sequence, is given. The connection between some positive definite operator-valued kernels and the Riesz-Herglotz integral representation of the analytic on the unit disc, operator-valued functions with positive real part in the class of operators in Section 4 of the note is studied.

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Lemnete-Ninulescu, L. (2012) Positive-Definite Operator-Valued Kernels and Integral Representations. Applied Mathematics, 3, 1990-1999. doi: 10.4236/am.2012.312274.

1. Introduction

About the scalar complex trigonometric moment problem we recall that: a sequence of complex numbers with is called positive semi-definite if for each

, the Toeplitz matrix is positive semi-definite. The problem of characterising the positive semi-definiteness of a sequence of complex numbers was completely solved by Carathéodory in [1], in the following theorem:

Theorem 1. The Toeplitz matrix is positive semi-definite and rank with if and only if the matrix is invertible and there exists with for and

such that

(1.1)

In the same paper [1], Charathéodory also proved that: if, then are the roots of the polynomial

which are all distinct and belong to

Another characterization of the positive semi-definiteness of a sequence of complex numbers was obtained by Herglotz in [2]. In [2], for, the moment of a finite measure on is defined by

The following characterization of the positivity of a complex moment sequence is the main result in [2].

Theorem 2. A sequence of complex numbers, is positive semi-definite if and only if there exists a positive measure on the unit circle such that

From Theorem 1 and Theorem 2, Charathéodory and Fejér in [3] deduce the following theorem:

Theorem 3. Let be given complex numbers.

Then there exists a positive measure on such that

(1.2)

if and only if the Toeplitz matrix is positive semi-definite. Moreover, if then there exists a positive measure supported on points of the unit circle which satisfies (1.2.)

Theorem 3 gives an answer to the scalar, truncated trigonometric moment problem.

Operator-valued truncated moment problems were studied in [4,5]. Regarding the truncated, trigonometric operator-valued moment problem, we recall that:

1), is called a spectral function if each is a bounded, positive operator,; it is orthogonal if each is an othogonal projection;

2) a finite sequence of bounded operators on an arbitrary Hilbert space is called a trigonometric moment sequence if, there exists a spectral function such that

for every In [4], the necessary and sufficient condition of representing a finite sequence of bounded operators on an arbitrary Hilbert space H, with as a trigonometric moment sequence is the positivity of the Toeplitz matrix

obtained with the given operators. The representing spectral function is obtained in [4] by generating an unitary operator, defined on the direct sum of copies of the Hilbert space H for obtaining an orthogonal spectral function and by applying Naimark’s dilation theorem to get the representing spectral function from it. In [5], a multidimensional operator-valued truncated moment problem is solved. That is: given a sequence of bounded operators

acting on an arbitrary Hilbert space H, with

a necessary and sufficient condition for representing any such operator

as the moment of a positive operator-valued measure is given. The necessary and sufficient condition in [5] for such a representation is again the positivity of the Toeplitz matrix

obtained with the given operators. The representing positive operator-valued measure, (spectral function), in [5] is obtained by applying Kolmogorov’s decomposition positive kernels theorem.

Concerning the complex, operator-valued moment problem on a compact semialgebraic nonvoid set, we recall that a sequence of bounded operators

acting on an arbitrary complex Hilbert spacea H, subject on the conditions, is called a moment sequence if there exists an operator-valued positive measure on such that

A sequence of bounded operators with and, acting on an arbitrary, complex, Hilbert space is called a trigonometric operator-valued moment sequence, if there exists a positive, operator-valued measure on the p-dimensional complex torus such that for all

Some of the papers devoted to operator-valued moment problems are: [6-10], to quote only few of them. The operator-valued multidimensional complex moment problem is solved in [9] in the class of commuting multioperators that admit normal extension (subnormal operators) (Theorem 1.4.8., p. 188). In [9], Corollary 1.4.10., a necessary and sufficient condition for solving a trigonometric operator-valued moment problem is given. In [10], another proof of a quite similar necessary and sufficient existence condition on a sequence of bounded operators to admit an integral representation as trigonometric moment sequence with respect to some positive operator valued measure is given. In Section 4 of this note, we prove that the two existence conditions in [9,10] are equivalent.

The present note studies in Section 3 the representation measure of the truncated operator-valued moment problem in [5], only when the given operators act on a finite dimensional Hilbert space. In Proposition 3.1, Section 3, it is shown that the representing measure, in this case, is an atomic one. In Proposition 3.2, Section 3, the necessary and sufficient existence condition in Proposition 3.1 is stated also in terms of matrices.

In Section 4 of the note, is studied the connection between the problem of representing the terms of an operator sequence

as moments of an operator valued, positive measure and the problem of Riesz-Herglotz type integral representation of some operator-valued, analytic function, with positive real part in the class of operators.

2. Preliminaries

Let arbitrary,

denote the complex, respectively the real variable in the complex, respectively real euclidian space. For

we denote

and by. The sets:

represent the torus in and the unit disc in if

and

For, we denote with the integer part of the number The addition and subtraction in, respectively in are considered on components. In the set the elements are treated in lexicographical order. If is an arbitrary complex Hilbert space and

a commuting multioperator, we denote by

for all and, as usual, is the algebra of bounded operators on; also denotes the Kronecker symbol for. Let

be a sequence of bounded operators on subject to the conditions for all

and For such a finite sequence of operators, in [5], a necessary and sufficient condition for the existance of a a positive Borel operator-valued measure on, such that the representations

(2.1.)

hold, it is given. Such a measure is called a representing measure for

In Section 3 of this note, in Proposition 3.1, we give a necessary and sufficient condition for the existence of an atomic representing measure of a truncated, operator-valued moment problem as in (2.1.) in case that the operators act on a finite dimensional Hilbert space. In Proposition 3.2 of this note, the necessary and sufficient existence condition for the representing measure in (2.1.) is reformulated in terms of matrices.

In section 4, Proposition 4.2, we establish a RieszHerglotz formula for representing an analytic, operatorvalued function on, with real positive part in the class of operators. The obtained, representation formula for such functions is the same as in the scalar case [11, 12]. In this case, the representing measure is a positive operator-valued measure. The proof of Proposition 4.1 in this note is based on the characterization on an operatorsequence to be a trigonometric, operator-valued moment sequence in [9]. The represented analytic, operator-valued function is the function which has as the Taylor’ s coefficients the operators.

3. An Operator-Valued Truncated Trigonometric Moment Problem on Finite Dimensional Spaces

Let be arbitrary and consider the set

with the lexicographical order (represents the cartesian product of the mentioned sets), H a finite dimensional Hilbert space with

and

Proposition 3.1. Let

be a sequence of bounded operators on with

for all

The following assertions are equivalent:

(i) for all sequences in

(ii) There exists the multisequence

of points and the bounded, positive operators, such that

(3.1)

for all

(iii) There exists a positive atomic operator-valued measure on such that:

Proof. On the set

we have the lexicographical order. The finite sequence of operators is considered double indexed i.e.; with this assumption, from, can be viewed as an operator-valued kernel

Let the C-vector space of functions defined on with values in the finite dimensional Hilbert space H. With the aid of, we can introduce on the non-negative hermitian product:

according to, we have the positivity condition:

The matrix associated to this kernel is a Toeplitz matrix of the form:

From Kolmogorov’s theorem, there exists the Hilbert space (essentially unique), obtained as the separate completeness of the vector space of functions with respect to the usual norm generated on the set of cosets of Cauchy sequences, (i.e.), by the nonnegative kernel, respectively the space

(when H is finite dimensional, the Hilbert space

). From the same theorem, there also exists the sequence of operators

such that for all In this particular case for, we have

where denotes the range of the operators and denotes the closed linear span of the sets,. The operators are:

with and

the Kronecker symbol. Also, from the construction of

, we have, where

denotes the range of the operators and denotes the closed linear span of the sets.

Let us consider the subsets

the subspaces in, , and the operators defined by the formula

for any with the standard basis in. From the definition of, since are linear for all, the same is true for the operators for all. For an arbitrary

we have:

for all. We extend to preserving the above definition and boundedness condition; the extensions are denoted with the same letter In case that

are C-linear independent operators with respect to the kernel, and from above, the operators are partial isometries, defined on linear closed subspaces with values in, with equal deficiency indices. In this case, admit an unitary extension on the whole space for all Let us denote the extensions of these operators to with the same letter. The adjoints of are defined by

for all Obviously, for the extended operators

In the same time, for all and all; we preserve the commuting relations for the extended operators. When is a finite dimensional Hilbert space with a basis, the same is true for the obtained Hilbert space All the vectors are C-linear independent in with respect to the kernel Indeed, if

equivalent with, this equality implies We consider that all the vectors are C-linear independent in with respect to the kernel We have then,

.

A basis in is

Let be the defined isometries, with

and

;

for

and

We have and also We consider the orthonormal algebraic complement of the space in, respectively the orthonormal complement of When

for and when; we have

Let be an orthonormal basis inrespectively an orthonormal basis in

We extend the partial isometries

to the whole spaces in the following way:

Because

and

it results that also the extensions are isometries and; that is are unitary operators for all; ( the extended operators are denoted with the same letters). The commuting relations are also preserved In the above conditions, the commuting multioperator consisting of unitary operators on admits joint spectral measurewhose joint spectrum Considering the construction of, we obtain and by induction for all

Because on the finite dimensional space, all the operators are unitary and compact one, their spectrum consists only of the principal values. The principal values are the roots of the characteristic polynomials associated with the matrix of in suitable basis in, for all The characteristic polynomials of are all complex variable polynomials of the same degree

with the roots

Let, be the family of the spectral projectors associated with the families of the principal values that is with the spectral measures of From the definition of, we have for all

and Because

we have also

Consequently, for, we have obtain:

From Kolmogorov’s decomposition theorem for, we have

with positive operators. That is:

(3.2.)

(i.e. assertion)

Let be a positive, atomic operator-valued measure on. From we have:

(i.e. assertion (iii)).

If

and is a positive operator-valued measure, we have:

that is

Proposition 3.1, in case H a finite dimensional space, statements implies also a similar, straightforward characterization, as in the scalar case [6]:

Proposition 3.2. When

operators acting on a finite dimensional space with, are as in Proposition 1, the Toeplitz matrix

is positive semidefinite if and only if it can be factorized as with

the diagonal matrix

with entries the positive operators

on the principal diagonal.

4. A Riesz-Herglotz Formula for Operator-Valued, Analytic Functions on the Unit Disk

Remark 4.1. Let be a sequence of bounded operators, acting on an arbitrary, separable, complex Hilbert space, such that for all and The following statements are equivalent:

(a) for all and all sequences of complex numbers with only finite nonzero terms.

(b) There exists a positive, operator-valued measure on such that

.

(c) The operator kernel is positive semidefinite on, that is it satisfies

for all, all sequences of vectors and all

Proof. (a) (b) was solved in [9], Corollary 1.4.10.

(b) (c) represents the sufficient condition in Proposition 1, [10].

(c) (a). Let with for an arbitrary From (c), it results

that is the operator kernel satisfies

(that is statement (a)).

Because the trigonometric polynomials are uniformly dense in the space of the continuous functions on it results that the representing measure of the operator moment sequence is unique.

For the proof of the following Proposition 4.2, we recall some observations.

A bounded monotonic sequence of positive non-negative operators converges in the strong operator topology to a non-negative operator (pp. 233, [11]). Due to this remark, if is a continuous, positive operator-valued function on the compact set, we define the Riemann integral of the function with respect to the Lebesgue measure The definition are the usual one in the class of positive operators. That is: the limits of the riemannian sums associated to the function, arbitrary divisions of and arbitrary intermediar points exists (are limits of bounded monotonic sequence of non-negative operators), and from the continuity assumption of on the compact set, are all the same. We denote the common limits, as usual with We apply this natural construction in the proof of the following result.

Proposition 4.2. Let be an analytic, vectorial function, with values in the set of bounded operators on a complex, separable Hilbert space. The following statements are equivalent:

(a)

(b) (Riesz-Herglotz formula) There exists a positive operator-valued measure on with

and an operator such that:

The proof follows quite the similar steps as the proof of the Riesz-Herglotz formula for analytic, scalar functions with real positive part ([11,12].)

Proof. (a) (b) Let

be the Taylor expansion of, with

We define for all In this case , we obtain for all,

If we consider arbitrary and

, the previous equality becomes

As a consequence of the orthogonality of the system of functions with respect to the usual scalar product defined on, from the the previous remark and s uniform convergent expansions, for all sequences and all we obtain:

We normalize this relation by dividing it with 2 and obtain, for, the following inequalities:

for all sequences and all arbitrarywith

In the above conditions from Theorem 1.4.8, [9], there exists a positive operator-valued measure on such that

For and we have

Let the homeomorphism and the positive operator-valued measure

Accordingly to this measure we obtain the representations:

and

Assured by the integral representations of the operators we have:

is analytic on, and

For the operator-valued analytic functions on we can state the same characterization theorem as in the the scalar case ( Theorem 3.3, [11],) that is:

Theorem 4.3. Let be a sequence of bounded operators acting on an arbitrary, separable, complex Hilbert space, subject to the conditions for all, The following statements are equivalent:

(a) There exists an unique, positive, operator-valued measure on such that:

(b) The Toeplitz matrix is positive semidefinite.

(c) There exists an analytic vectorial function for all and

for some with

(d) There exists a separable, Hilbert space, an operator and an unitary operator, such that and

Proof. was solved in [9], Th.1.4.8., p. 188. We sketch the proof of implication.

As in above Proposition 4.2, there exists a positive operator-valued measure

such that In this case, for the function, we have

that is is analytic on Also from (a), we have:

From the above representation, it results:

(c) (a) As the same proof in Proposition 4.2, we have

for arbitrary. From this inequality, it results that there exist the representations

with a positive operator valued measure on ([9], Th. 1.3.2), this is (a).

The equivalence,. From remark 4.1.we have ((c) from Remark 4.1.). The equivalence is the main result in [10], Proposition 1. p. 116. From [10], Proposition 1, (condition (c) in Remark 4.1.) assured the existence of a Hilbert space, an operator and an unitary operator such that, that is (d); (the Hilbert Space, the unitary operator are obtained by applyng Kolmogorov’s decomposition theorem on positive semidefinite kernels.) Conversely is immediately.

5. Conclusion

We give a necessary and sufficient condition on a finite sequence of bounded operators, acting on a finite dimensional Hilbert space, to admit an integral representation as complex moment sequence with respect to an atomic, positive, operator-valued measure. We also established a Riesz-Herglotz representation formula for operator-valued, analytic functions on the unit disc, with real positive part in the class of operators.

Conflicts of Interest

The authors declare no conflicts of interest.

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