Positive-Definite Operator-Valued Kernels and Integral Representations

A truncated trigonometric, operator-valued moment problem in section 3 of this note is solved. Let    , , , ,1 s s p n n n i i L H n Z n s i p                 be a finite sequence of bounded operators, with   1, , , 1 p p s s s N p    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 E , with the property that for every Z p n with ,1 i i n s i p    , the moment of th n E coincides with the term th n s n  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.



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.

Introduction
About the scalar complex trigonometric moment problem we recall that: a sequence of complex numbers   Z n n t  with n t t   n is called positive semi-definite if for each 0 n  , the Toeplitz matrix is positive   , 0 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: In the same paper [1], Charathéodory also proved that: if , then  which are all distinct and belong to 1 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 .The following characterization of the positivity of a complex moment sequence is the main result in [2].Theorem 3 gives an answer to the scalar, truncated trigonometric moment problem.

T
Operator-valued truncated moment problems were studied in [4,5].Regarding the truncated, trigonometric operator-valued moment problem, we recall that: of bounded operators on an arbitrary Hilbert space is called a trigonometric moment sequence if, there exists a spectral func- for every In [4], the necessary and sufficient condition of representing a finite sequence of bounded operators on an arbitrary Hilbert space H, as a trigonometric moment sequence is the positivity of the Toeplitz matrix   , 0 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 , acting on an arbitrary, complex, Hilbert space is called a trigonometric operator-valued moment sequence, if there exists a positive, operator-valued measure F  on the p-dimensional complex torus 1 p T such that for all Some of the papers devoted to operator-valued moment problems are: [6][7][8][9][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.

Let arbitrary, p N
, , , , , , denote the complex, respectively the real variable in the complex, respectively real euclidian space.For , , , , , and by 1 1 p q q q p t t t   .The sets: be a sequence of bounded operators on H subject to the conditions s s n n 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 Bor T , such that the representations 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   .) is reformulated in terms of matrices.In section 4, Proposition 4.2, we establish a Riesz-Herglotz 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  

An Operator-Valued Truncated Trigonometric Moment Problem on Finite Dimensional Spaces
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 The following assertions are equivalent: of p d points and the bounded, positive operators, p , we have the lexicographical order.The finite sequence of operators   , , , 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 C vector space of functions K F with respect to the usual norm generated on the set of cosets of Cauchy sequences, (i.e. ' ' F  ), by the nonnegative kernel s  , respectively the space From the same theorem, there also In this particular case for , m n I  .
, , , , , , the Kronecker symbol.Also, from the construction of K , we have x, where m Ranh x denotes the range of the operators m and denotes the closed linear span of the sets .
and the operators : From the definition of i A , since m are linear for all 1i h m I  , the same is true for the operators i A for all 1 i p   .For an arbitrary we have:

A y A y A h x A h x h x h x h h x x x x h h x x h x h x y
K preserving the above definition and boundedness condition; the extensions and , 1 ,1 are C-linear independent operators with respect to the kernel , and from above, the operators i  A are partial isometries, defined on linear closed subspaces

with equal deficiency indices. In this case, i
A admit an unitary extension on the whole space K for all 1 Let us denote the extensions of these operators to .i   p K with the same letter i A .The adjoints of i A are defined by   for all Obviously, for the extended operators  and all 1  we preserve the commuting relations for the extended operators.When , i j p   ; H is a nite dimensional Hilbert space with a basis   , the same is  true for the obtained Hilbert space ors .
Let be an orthonorma basis in , respectively : , to the whole spaces K in the following way: ults that also the extensions are isometries and are unitary operators for p all 1 i   same l ; ( the perators are denoted with the ers).The commuting relations i j In the above conditions, the commuting multi

 
, , operator consisting of unitary operators on K adm ectral measure, whose joint spectrum its joint sp , , .
and by , we obtain induction Because on the finite dimensional space K , all the operators The chara teristic polynomials i cof A are plex variable polynomials of the same de ee   From the definition of i j P , we have , , 1, , j q j q d    and 1 .

n m h h h A A h h A A A A h h P P P Ph
positive operators.That is: i.e. assertion ) F  is a positive operator-valued measure, we have: i implies also a similar, straightforward ch s in the scalar case [6]: 1 and with entries the positive operators on the principal diagonal.

A Riesz-Herglotz Formula for
Operator-Valued, Analytic Functions on the Unit Disk Remark 4.1.
(b) was solved in [9], ufficient (c) (a).Let Corollary 1.4.10.(b)  (c) represents the s condition in Proposition 1, [10].The proof follows quite the sim s the proof ilar steps a of the Riesz-Herglotz formula for analytic, scalar functions with real positive part ( [11,12] r a In , we obtain for all We define If we consider 0 r   arbitrary and , the previous equality becomes As a consequence of the orthogonality of the system of functions   and all we obtain: We normalize this relation by dividing it with 2 and 0 , 2 for all sequences and all arbitrary In the above conditions from Theorem 1.4.8,[9], there exists a positive operator-valued measure 1 F on such that , , and d 0 Accordingly to this measure we obtain the representations: ured by the integral representations of the operators the same characterization theorem as in the the alar case ( Theorem 3.3, [11] The following staeq (a) There exists an unique, positive, operator-valued measure tements are uivalent: on such that:   n n p q n n n p q p q p q p q p q n q p p q p q F re F re e e r e e ) nver seq sional Hilbert space, to admit an integral representation as complex m t sequence with respect to an atomic, omen positive, operator-valued measur Riesz-Herglotz representation formula for opera r-valued, analytic functions on the unit disc, with real positive part in the class of ope EFERENCES

Theorem 1 .
The Toeplitz matrix finite dimensional Hilbert space.In Proposition 3.2 of this note, the necessary and sufficient existence condition for the representing measure in (2.1.
There exists a positive, operator-valued measure on the is a continuous, positive operator-val compact set I R  , we define the Riemann integral of the function f with respect to the Lebesgue measure d .usual one in the class of the limits of the riemannian tive operators.That is: associated to the function f , arbitrary divisions  of I and arbitrary intermediar points   n n   exists (are limits of bounded monotonic sequence of non-negative operators), and from the continuity assumption of f on the compact set I , are all the same.We denote the common limits, as usual with  d .

F
is analytic on .Also from (a), we have:From the above representation, it results: 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 E  , with , ,) that is: c) There exists an analytic vectorial function , that is (d); e H ert Space K , the unitary operator U are