Positive Definite Operator Valued Kernels and Integral Representations

Positive-Definite Operator-Valued Kernels and Integral

Representations

L. Lemnete-Ninulescu

Departament of Mathematics, Politechnica University of Bucharest, Bucharest, Romania Email: luminita_lemnete@yahoo.com

Received June 5, 2012; revised October 18, 2012; accepted October 26, 2012

ABSTRACT

A truncated trigonometric, operator-valued moment problem in section 3 of this note is solved. Let

 



, , , ,1

s s p

n L H n n n Z ni si 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 condi-

tion on the positivity of an operator kernel for the existence of an atomic, positive, operator-valued measure E_, with

the property that for every Zp

n with n_i s_i,1 i p, the nth moment of E_ coincides with the nth term s_n

of the sequence, is given. The connection between some positive definite operator-valued kernels and the Riesz-Her- glotz 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.

Keywords: Unitary-Operator; Self-Adjoint Operator; Joint Spectral Measure of a Commuting Tuple of Operators; Spectral Projector; Complex Moments; Analytic Vectorial Functions

1. Introduction

About the scalar complex trigonometric moment problem we recall that: a sequence

 

tn n_Z of complex numbers with tn tn is called positive semi-definite if for each

0

n , the Toeplitz matrix

 

is positive , 0

n n i j _{i j}

T t_





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 follow- ing theorem:

Theorem 1.The Toeplitz matrix

 

, 0

n n i j _{i j}

T t_



 is po-

sitive semi-definite and rank Tn with 1 1

if and only if the matrix Tr 1

r

   r n

 is invertible and there exists  j T j1, 1, 2,,r with j k for jk

and

0, 1, ,

j j r

   

such that

1

, 0,1,

r k

k j j

j

t   for k







 , .n



(1.1)

In the same paper [1], Charathéodory also proved that: if 1 r n , then



₁, ,_r are the roots of the polynomial

 

0 1

1 0 1

1 2 1

det

1

r

r

r r

r

t t t

t t t

P z

t t t

z z

  

 

 

 

 



 

 

 

 

     

 

which are all distinct and belong to 1

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

.

T

nZ _nth

 on is defined by T1

 

 

π _int

π

1 _{e d .}

2π n

c   t  n







 

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

Theorem 2.A sequence of complex numbers

 

tn n Z ,

n n

t t_ is positive semi-definite if and only if there exists a positive measure  on the unit circle such that

1,

T

 

Z.

n

t  n for n

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

Theorem 3. Let

 

tj n_jn be given complex numbers.

that

 

j tj, j n

   , (1.2)

if and only if the Toeplitz matrix

 

is

posi-, 0

n n i j _{i j}

T t_





tive semi-definite. Moreover, if 1rank T_n  r



n1 ,



then there exists a positive measure  supported on points of the unit circle which satisfies (1.2.)

r

1

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

T

Operator-valued truncated moment problems were stu- died in [4,5]. Regarding the truncated, trigonometric op- erator-valued moment problem, we recall that:

1) E

 

 ,   π  π

 

is called a spectral function if

each

 

a E  is a bounded, positive operator,

 

b

 



E  E 



for  ; it is orthogonal if each E

 



is an othogonal projection;

2) a finite sequence



A0, , An



of bounded opera- tors on an arbitrary Hilbert space is called a trigono- metric moment sequence if, there exists a spectral func- tion E

  

 ,   π  π



such that π

 

πe d

ik k

A 



_  E 

for every In [4], the necessary and suffi-cient condition of representing a finite sequence of bounded operators on an arbitrary Hilbert space H,

kn with 0 H

0, , .

k  n

 

n k

A A_n A__n,A Id as a trigonometric

moment sequence is the positivity of the Toeplitz matrix

 

n_{, 0} n i j _{i j}

T A_



 obtained with the given operators. The representing spectral function is obtained in [4] by gen- erating 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 trun- cated moment problem is solved. That is: given a se- quence of bounded operators



n1



 

s , p, ,1

n n Z ni si i p

     ,

acting on an arbitrary Hilbert space H, with



1, , p



p, sn sn,

s s  s N  _

a necessary and sufficient condition for representing any such operator



1



, , , , ,1

s p

m m m mp Z mi si i p

      

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

th

m





21

, ,

2

i

i i i

s

l m s

l m i p

        _ _

   



obtained with the given operators. The representing posi-tive 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

K

 

, _, p

Z

    

   ,

acting on an arbitrary complex Hilbert spacea H, subject

on the conditions  ,   , ,  0,0 1H is called a moment sequence if there exists an operator-valued positive measure

K

F_ on K such that

 

, _Kz z dF z , , Zp.

     

 





A sequence of bounded operators

 

n n Z p with

, p

n n n Z

 

     and  0 IdH, acting on an arbitrary,

complex, Hilbert space is called a trigonometric opera-tor-valued moment sequence, if there exists a positive, operator-valued measure F_ on the p-dimensional com-

plex torus T1p such that

 

for all

1 d

p

T z F z



 

 



. p

Z

  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 mul- tioperators that admit normal extension (subnormal op- erators) (Theorem 1.4.8., p. 188). In [9], Corollary 1.4.10., a necessary and sufficient condition for solving a trigo- nometric operator-valued moment problem is given. In [10], another proof of a quite similar necessary and suffi- cient existence condition on a sequence of bounded op- erators to admit an integral representation as trigonomet- ric moment sequence with respect to some positive op- erator 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 representa- tion 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, Sec- tion 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 Proposi- tion 3.1 is stated also in terms of matrices.

In Section 4 of the note, is studied the connection be- tween the problem of representing the terms of an opera- tor sequence

 

n n Z, n L H

 

, n n, n Z, 0 Id 

 

          _H

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

2. Preliminaries

Let pN arbitrary,



1, ,



,

p p

s s  s N



1, ,



,



1, ,



,

p p

p p

z z  z C z z  z C



1, , p



p

t t  t R

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



1, ,



,



1, ,



p p

p

m m  m Z q q  qp N

,



, we denote

1

1 p, 0,1

m m m

p i

z z z z  i p

1

1 p

m m m

p

z z z

and by 1

1 p

q q q

p

t t t . The sets:







1 1, , , 1 for all 1

p

p i

T  z  z z   i p



represent the torus in p

C and D



zC z, 1



the

unit disc in C; if



1, ,



1

p p

z  z T

and

0, m_i mi.

i i i

m  z z

For



1, ,



p p

s s  s N , we denote with

2 i

s

      the

integer part of the number . 2

i

s

The addition and sub- traction in p

N , respectively in _Zp _{are considered on} components. In the set



nZ ,ni si,1 i p



p _the

elements are treated in lexicographical order. If H is

an arbitrary complex Hilbert space and



1, ,



 

p p

N N  N L H

a commuting multioperator, we denote by 1

1

p m m

m

p

N N N

for all p

mN and, as usual, is the algebra of

bounded operators on

 

L H

H; also _ij denotes the Kro-

necker symbol for i j, Z. Let

 

_p



 

, , 1



s s s

n _{n Z} n L H ni si i p

         

be a sequence of bounded operators on H subject to

the conditions s s

n n

 

   for all

, , 1

p

i i

nZ n s   i p

and ₀s . H

Id

  For such a finite sequence of operators,

in [5], a necessary and sufficient condition for the exis- tance of a a positive Borel operator-valued measure F_

on Bor T

 

1p , such that the representations

 





1 d , 1, ,

, 1

p ,

s n p

n _T p

i i

z F z n n n Z n s i p



    

  





(2.1.)

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

 

s _p.

n n Z



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, opera- tor-valued moment problem as in (2.1.) in case that the operators

 

p.

s n _{n Z}

 act on a finite dimensional Hilbert space. In Proposition 3.2 of this note, the necessary and sufficient existence condition for the representing meas- ure in (2.1.) is reformulated in terms of matrices.

In section 4, Proposition 4.2, we establish a Riesz- Herglotz formula for representing an analytic, operator- valued 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 operator- sequence

D

 

_{n n Z} p to be a trigonometric, operator-val-

ued moment sequence in [9]. The represented analytic, operator-valued function is the function which has as the Taylor’ s coefficients the operators

 

k k N .

3. An Operator-Valued Truncated

Trigonometric Moment Problem on Finite

Dimensional Spaces

Let



1, ,



p p

s s  s N be arbitrary and consider the

set

1

1

, 1 , ,

2 2 2

p si si si

I  _ _{ }  _ _{ } _ _  __

     

   

  

with the lexicographical order ( represents the carte- sian product of the mentioned sets), H a finite dimen-

sional Hilbert space with



dimCHr and 1



1



.

p j j

d_



_ s  _r

Proposition 3.1. Let

 

p



 

, , 1



s s s

n _{n Z} n L H ni si i p

         

be a sequence of bounded operators on H with ,

s s

n n

 

   for all nZp,ni si, 1  i p,s0IdH.

(i) ,

, s n m n m

H m n I

x x  





0 for all sequences

 

xn _{n I} in H.

(ii) There exists the multisequence















1 2



1 2

1 1

, , , _pp p, , , 1, , p

i i i T i ip d

       

of p

d points and the bounded, positive operators,

1 2

12

p p i i

F _i such that

  

   

 

1 2

1 1 2

1

1 2 12

, ,

, , 1, ,

p

1 2

p p p

p p

n

n n

s p p

n n i i i i i

i i d

F

  



 







 

 _i (3.1)

for all



1, ,



, ,1 .

p

p i i

n n  n Z n s i p

(iii) There exists a positive atomic operator-valued

measure F_ on



1



p

Bor T such that:

 





1 1

d , , ,

with ,1 .

p ,

s n p

n _T p

i i

z F z n n n Z n s i p



    

  





Proof.

   

i  ii . On the set

1

1

, 1 , ,

2 2 2

p si si si

I      _{   }  _ _ _

     

 

   

we have the lexicographical order. The finite sequence of operators

 

,

p i i

s s

n _{n Z ns}

   is considered double in- dexed i.e. s



,



s

n m n m

   ; with this assumption, from ,

 

i s _{can be viewed as an operator-valued kernel}

  



: ; , s

s s

n m

I I L H n m 

      .



Let F



f I: H the C-vector space of functions

defined on I with values in the finite dimensional Hil-

bert space H. With the aid of s, we can introduce on F the non-negative hermitian product:

     

,

, s _{i j I} , ,

H

f g _ 



_  i j f i g j ;

according to

 

i , we have the positivity condition:

     

1

,

, s _{i j I} , ,

H

f f _ 



_  i j f i f j 0.

 _{ }

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

1 2

1 2

0 0 0 1

0 1 1

0 0

.

p

p

s s s

s s s s

s

s s

s s s

   

    

 

  

   

 

 

  







  

   

 

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 non-

negative kernel _s_{, respectively the space ' '}·

KF 

(when H is finite dimensional, the Hilbert space



ˆ,



K f fF ). From the same theorem, there also

exists the sequence of operators

 

_m



,



m I

h _ L H K

such that s



,



m n

n m h h

  for all In this par-

ticular case for

,

m nI.

F, we have

1

·

' ' m I m

KF  V Ranh x

where denotes the range of the operators m and denotes the closed linear span of the sets

, m

Ranh x

m

VRanh x x

h x

m

h

Ran xH. The operators hm:HK are:

  

p_{1 k k}

m k m

h x j 



_ j x

with m



m₁, , m_p



,j



j₁, , j_p



I x, H and ij the Kronecker symbol. Also, from the construction of

K, we have K F· Vm ZpRanhm

 

  x, where Ranh xm denotes the range of the operators m and

denotes the closed linear span of the sets .

h x VRanh xm

m

h x Ran

Let us consider the subsets

1 1

1

1 1

2

1 , ,

2 2

1 , ,

2 2

1

, , ,

2 2

1 , ,

2 2

1 1, ,

2 2

1 , ,

2 2

i

i i

p p

i

i i

p p

s s

I

s s

s s

I

s s

I

s s

s s

     

  _{   }

   

 

     

  _{   }

   

 

     

 

 _{    }

    

 

     

 _{    }

   

 

     





  _{ } _{ }

   

 



  

    

  

 

 



 

 

 I,

 

 _{ }

 

 

 

the subspaces in K,

1

1i m I_i m

K V _ Ranh x,

2

2i m I_i m

K V _ Ranh x and the operators A K_i: _1i K_2i

defined by the formula

1 1

i

i i

i n n n e

n I n I

n

A h x h_ x

 

 



 









for any 1 i p with _i

 

_ik p₁

k

e   _ the standard basis

in p

C . From the definition of A_i, since _m are linear

for all _1i

h

mI , the same is true for the operators A_i for

all 1 i p. For an arbitrary

 

1 1

1, _i ,

i

i n I n n n n I

yK y



_ h x x _ H,

1 1

1 2 1

1 1

1 1

,

, ,

2

, ,

, ,

, ,

,

s

s

i i

i i i i

s s

i i i

s s

i i

s s

i i

i i i n n i m m

n I m I

n e n m e m m e n e n m

n I m I n m I

s

n m n m m n n m

n m I n m I

n n m m

n I m I

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



  _



   

  _  _

 

 _  _



  _



 

  

 



















A basis in is

for all 1 i p. We extend Ai to Ki1 preserving the

above definition and boundedness condition; the exten-sions A Ki: i1Ki2，1 i p are denoted with the same

letter Ai In case that

1 1 ₂1 1 1 1 ₂ 1

and ,

1 ,1

i i

i i p i i p

j j

s s

k k k k k k k k

h e h

j r i p

_ _  _  _{ }_ 

  



  _{  }



    e

are C-linear independent operators with respect to the

kernel , and from above, the operators Ai are partial isometries, defined on linear closed subspaces Ki1K

with values in Ki2K, with equal deficiency indices.

In this case, Ai 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 A_i. The

adjoints of Ai are defined by



i2



i2 i

i n I n n n I n e n

A



_ h x 



_ h_ x

for all 1 i p Obviously, for the extended operators i i

.

, 1 .

i i K

A A A A Id  i p

In the same time, A A x_{i j} A A x_{j i} for all xK1iK2i and all 1 we preserve the commuting relations for the extended operators. When

,

i j p

  ;

H is a nite dimen-

sional Hilbert space with a basis

 

r fi

1

j _j

e , the same is



true for the obtained Hilbert space K ors

j

espe

All the vect

k







, p, arbitrary fixed, 1, ,



r

   are

ernel .

 Indeed, if



m j

h e m I Z m

C-linear independent in K with r ct to the



 





 

, ,

0

p q m i m j m i m j

p q I   h eh e p h e h e q _H





equivalent with  2  2 2



 e ei, j _H



0, this equality implies   0. We consider that all the vectors



h e mm j, I,1 j r



are C-linear independent

in K with respect to the kernel . We have then,

dim

K



, p, 1, .



m j

B h e m I Z j r

Let A Ki: 1iK2i be the defined isometries, with 1

1i m Ii,j1,r m

K VRan _{ } h x

and

m 2

2i m Ii

K VRan _ h x;

for

1 1

, 2

p i

i

i s m

m I Z m

E VRan _{ }h x

  _{ }  



and

2

, 2

.

p i

i

i s m

m I z m

F VRan _{ }h x

  _{ }  



We have KK1iE1i and also K F2iK2i. We consider E1i the orthonormal algebr nt of the space 1i

aic compleme

K in K, respectively F_2i the

orthonor-mal complement of K2i. When





p

1, k 1 k k i

q_



_{ } s  r_

1

p and qr when p1; we have

 for



1 2

dimCEi dimCFi q.



1, ,



i i

q

u u

CK   I r d.

Let be an orthonorma basis in ,

respectively

l E1i



, i



q

v an orthonormal basis in

1,

i

v  F_2i

We extend the partial isometries A Ki: 1iK2i, 1 i p to the whole spaces K in the following way:

 

i i, 1 ,_, 1, .

i j j

A u v  j q i p

Because

, , 1 ,

i i i i i i

i j i j j j j j

K K K

A u  v v   u u ,

and

A u

, i, i i, i i, i ,

i j k j k jk j i k j k

i i

K K K K

v  v v   u A v  u u

ults that also the extensions are isometries and

A u

it res

1

i i

A A ; that is A K_i: K are unitary operators for p

all 1 i

same l

; ( the perators are denoted with the ers). The commuting relations i j j i

extended o

ett A A A A

are also preserved 1 i p In the above conditions, the

commuting multioperator



A1, , Ap



consisting of unitary operators on K adm ectral measure,

whose joint spectrum

its joint sp



1, ,



1 .

p p

A A T

   Considering

the construction of K ₀

i

i e

A h h_ and by

, we obtain induction

i n i ne

A h f all or

1

s



1, , , 1 .

2 i

n_{ }  _  i p

 

  

Because on the finite dimensional space K, all the

values

 

1,

i j _{j d}



 that is



 

i i i

j

P E _j



with Ei the

spectrum 

 

Ai T1 consists only of the A si princi-pal values pal values are the roots of the char-acteristic polynomials associated with the matrix of i

. The princi

A

in suitable basis in K, for all 1 i p. The chara

teristic polynomials i

c-of A are plex variable

polynomials of the same de ee



1



p

d s  r

all com dim  gr i 1  i



 

K

spectral measures of Ai,1 i p. From the definition of i

j

P , we have P_jiP_qi0,

 

P_ji 2P_ji for all





, , 1, ,

jq j q  d and A_i 



d_j1i_jP_ji. Because

,

i j j i

A A  A A we have also

 

with the roots

1,

j d, 1 , 1, .

i i

j j j d

   , 1 i p  i j j i_{,1 , .}

E E E E i j p

Let

 

_{Consequently, for}





1, ,

p p

m m  m Z

i

1, , 1

j _{j d}

P i

  p, be the he spectral

ctors associated w

family of t _

j

, we have ob- tain:

proje ith the families of the principal

 

₁ 1 ₁

 

1 1 .

p

p d m d m

m

m p p

p j j j j j

A _  P _  P

1

1m

A A _ _ _ _











From Kolmogorov’s decom for



position theorem , p

m n I Z , we have





         1 1 1 1

1 1 1

1 1

1 1

1 1

1

0 0 0

p p p p i i             

i i i

1 1 0

1

1 1 1

0 ₁ 0

, , 1, , , , 1, ,

1 1

0

, , 1, ,

, p p

p p p p p p p p p p p p p p m n m n

n m m n p p

m m _pn

n p

i i i s s s s

i i d s s d

n m

n m p p

i i i

i i d

n m h h h A A h h A A A A h

h P P P P h

h P P

                               







                   1 1 1 1 1 1 1

1 1 2

1

1 1

0 0 0

, , 1, ,

1 12

, , 1, ,

p p p p p p p p p p p p n m

n m p p

i i i i

i i d

n m

n m p p

i i

i i d

h h P P h

F                     





        

ith _p 0

s s   m n

1 2 1

12 1

0

p

p p

i i i i i

F  hP P h

w positive operators. That is:

  





1

1 ip 1 2 p

1

1

, , 1, ,

, , , , ,1

p p

s p p

n i i i i p i i

i i d

1n np 12 p

F n n n Z n s i p

 



 



_     

 

  (3.2.)

i.e. assertion )

 _

 

ii

Le

   

ii  iii . t (

1 2   1 2

12

, , , 1, , p

p p

p i i i

i i i d

 





 

 

p

F F be a

positive, atomic operator-valued measure on T₁ . From

  

ii 3.2. we have:

 

1

d ,

n

z F z 



p , with ,1

p

n _T  n Z ni si i

     

(i.e. assertion (iii)).

s

p

   

iii  i . If

 

1

d , , with ,1

p

s n p

n _T z  n Z ni si i p

 



    

and

F z

F is a positive operator-valued measure, we have:

 

 

 

 

1 1 , , 1 1 2 2 1 2 1 2

, d ,

d , d

d 0

p

p

n m n m _{H T} n m

H

m n I n m I

n m

p T

n I m I

s n m

H

n T

n I

x x z F z x x

z F z z F z z F z

              



 











that is

 

i .

Proposition 3.1, in case H a finite dimensional space,

statements

   

i  i

aracterization, a

i implies also a similar, straight-

forward ch s in the scalar case [6]: Proposition 3.2.When





 

1 1 and p_,

i i

p s

j n

j n Z n s

m



_ s   _{ } ,

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

dimC 

 

1 2 1 ,1 1 i i p s s m

n s i p

s s s

s s s

T                 _  

1 2 p 0 0

, 0 1 p n n Z s s

0 0 0

s       , s s                 

is positiv ed

as

s

 

     

e semidefinite if and only if it can be factoriz

m

T RDR with

   

           

   

2 1

2 1 2 1 1

1 1

1 2

2 1

1 1

2 1 2 1 1

1 1 1 1 1 2

1

,

p p p

k

s s

s s s s s s s s

p p p

k k

R

 

 

       

 

 

 

 

 

   

 

 

 

 

 



  

  

  

  

1  1

1











, p



 

, p₁



1



and

j j

RM m d C d_



_ s  r_ D the dia-

gonal matrix



12 1 1

12 11 2

12 1 1

12

0 0

0 0

, 0

0 p

p

p k

p dd d

F F

D

F

F

 

 

 

 

  

 



 

 

 

 

 

 

 

 

   

 

   

 



p, p



DM d d



12 p



F 

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

 

1

1, , 1, ,

p p i i

i i  d



 

 

_{ Z}p be a sequence of bounded on an arbitrary, separable,

operators, acting complex

Hilbert space H, such that   _ _ for all p

Z

 

and  0 IdH. The following statements are equivalent: (a)



n_{ ,} ₀_{   }   0 for all

p

n

uences of complex numbers

 

N and all

seq _{ N}p with onl

ite nonzero terms.

y fin

(b) There exists a positive, operator-valued measure F on T1p Cp such that

 

1 d

p ,

p

T z F z Z



 

 



  .

(c) The operator kernel

 

_{ Z}p is positive semidefinite on H, that is it satisfies

,

, 0

H n

x x

     



  all

n

 

for nNp, all sequences of vectors

 

x_{   n n, }H and all nNp.

(a) 

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

ufficient

(c) (a). Let

Corollary 1.4.10. (b)  (c) represents the s condition in

Proposition 1, [10].



 

f_{ } H, with f_ _x

, , 0

n

n  x x H  

       ;



that is the operator kernel satisfies

2

, , 0

0 0

n n

n

       

   

  _{ } _{ }

 

 

 

    





(that is statement (a)).

use the ls are uniformly

the sp tions on i

m

ns.

onic sequen sitive no

tors co operator topology

to a non-negative operator (pp. 233, [11]). Due to this remark, if

Beca trigonometric polynomia

dense in ace of the continuous func t

res 1

, p

T

ults that the representing measure of the operator nt se

ome quence is unique.

For the proof of the following Proposition 4.2, we re-call some observatio

A bounded monot ce of po n-nega-

tive opera nverges in the strong

   

: , 0,

f I R L H f     I

ued function 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 . The posi-

sums definition are the 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-nega- tive 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 . If  



We apply

this natural construction in the proof of the following re- sult.

Proposition 4.2. Let f D: L H

 

be an analytic,

vectorial func unded

operators on a complex, separable Hilbert space tion, with values in the set of bo

H. The following ents are equivalent:

(a)

statem

 

0,

 

0 _H.

f z z D and f Id

     

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

(

f

F on



π,π



with

 

π

πFf  d IdH

 



and an operator CL H

 

,CC such that:

 

for an

arbitrary xH. From (c), it results

 

π

i zd , .

i i π

e

e f

f z C F z D







z





 



 

The proof follows quite the similar steps as the proof

of the Riesz-Herglotz formula for analytic, scalar func-tions with real positive part ([11,12].)

Proof. (a)  (b) Let

 

 

0 i

 

0 1

n n n

f z  f  f 



_ z

be the Taylor expansion of f ,  z D with

 

0 Id , L H

 

, n Nand lim  1.

r a In , we

obtain for all

n

f

 H n n n

We define n n

 

   fo ll n1. this case

zD,

 

 

0 i

 

0 n1 n . n

z  f  f 



 z

1

f _

If we consider 0 r arbitrary and





i

e , π,π

zr    , the previous equality becomes

   

 

i i

1

i i

1

e e

2 0 nen n nen

n n

n

f r f r

f r r

 

.

 

  

 



  



 





As a consequence of the orthogonality of the system of functions

 

_eik

k Z



 with respect to the usual scalar pro-

duct defined on 2





π,π



,d



L   , from the the previous

remark and fs uniform convergent expansions, for all

sequences

 

n nC and all nN we obtain:

   

 









 













2

π _{i i i i}

0 1

π 1

1 ₂

i i i

0 1

1

i i i( )

π

1 _{1 , 0}

1 _{e e e e d}

lim 2π

e e e d

2 0 e e e

2π

n n r

k k n

k n

k n

k k k k p q

k k

r _{k p q}

f r f r

r

f r r

   

  

  

   

π _i

π

1 ₁

π

1

2 0 e

lim 2π

k k k

k

r _k

f r    

1 lim



 

  

 

 



 _{  }

 _  _{  }

 

 



 _   

 

     _

 

   















 



 _



 _    

 









 

2

0 ,

d

0 n p n p q p q 0.

q



   

 



  

We normalize this relation by dividing it with 2 and

0,

2

p p q p

f



 _ _





obtain, for  ,n ,n N



    e following ine-

2 n

n _n



  , th

qualities:



, 0 0

n

p q _{p q} p q   



for all sequences

 

n ₀ and all arbitrary

k k C

 _  nN,

with



 

 

₀

 

,

and 0 .

n n

H

L H

Id f

 

    

   

n

In the above conditions from Theorem 1.4.8, [9], there exists a positive operator-valued measure F1 on

such that 1

T

 

 

 



 

1 1

, p qd ₁ , , and d ₁ 0 0

p q p q _H

Tz z F z p q N T F z Id f



   







     .

For q0 and pN we have 

 

1

0 pd ₁ .

p

Tz F z



 



Let the homeomorphism





 

i

1

: π,π T, e

     

ued measure and the positive operator-val









 

2 1 , 2: π,π .

F F  F Bor  A H

Accordingly to this measure we obtain the representa- tions:



 

 

1

π _i

1 _π 2

d e d ,

p p

p

Tz F z F p N

 _

 

 







 

and

.

ured by the integral representations of the opera- tors

 

 

π 2

πdF  IdH f 0

   



Ass _

 we have:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

0 1

2 i

π

i π

2 i

π

1 1

0 0 0 2 2

2 2

i 0

e

0 0

e

i d , , with .

2i e

n n n n

n n

n n

n

f z f z z f f z z

f

z f f

z

C F z D C

z



 

0 1

i π

1 1

2

2

e

d i 0 d

n n

n n

n z

z F f F

 

π i π i

2

π π

0 0

e n nd e

n n

z F

 _{  }

 

 

 

 



  



   

 

 

 





   

  _    _       

 

  

 



    

















 

 

 







b





 

a

 

 









 

2 2 2

2 1

d 0,

z f z f z

F z



 _





 

2 πdF

π π _i

2 



 _e_

f is analytic on D , f

 

0 iC



_π  and

.

 

π

 

2 π

0 d H

f F  I



 



d

the same characterization theorem as in the the alar 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



For the operator-valued analytic functions on D we

can state sc

 

n n Z

  

H ,

,

subject to the conditions for all

n n

 

   nN  ₀ Id_H. The following sta-

eq

(a) There exists an unique, positive, operator-valued

measure

tements are uivalent:

on such that:

.

1

T F_

 

1 d ,

n

n _Tz F z n Z

 



 

) The Toeplitz matrix is positive semi-

definite.

(b



n m



n m_{, 0}



 



(c) There exists an analytic vectorial function

 

 

:DL H ,F z 0 for all zD and

F

 

1

i 2 n

H n

n

F z Id C z



  





for some CL

 

H

There exists a separable

with

(d) t space an op-

erator and an unitary opera

.

CC

, Hilber K,

0:

h HK tor UL K

 

,

such that n h U



0 0,

n

h n Z

    and h h_{0 0} H.



Id



Proof.

   

a  b was solved

188. We s oof of implication

in [9], Th.1.4.8., p. ketch the pr

   

a  b .

 

 

 

1

1

, 0 , 0

2

1 1 1

2 2 2

1

d

d , d 0, ,arbitrary.

p p

n m

n m n m _T

n m n m

n n n

n m n

n

T T

n m

F z z z

F z z F z F z z p N

 

  

 

 

  

 

    



 











As in above n 4.2, there exists a p

   

a  c

o

Propositio

sitive operator-valued measure F₂:



π,π



L H

 

ch that π i

 

2

e dn , .

n su

π F n N

 _

 



  In this case, for

i 2 n



the function F z

 

Id_H C



_n1_nz , we have

 

 

1

that is

1

2 ₁

2 1

sup n d , _n

n

T

x x _ z x x

 





1 1

2n _{sup 1;}

x F z

 1

sup nd

n

n n x _Tz F z

 



F is analytic on D. Also from (a), we have:

 

 

 

 

i π

2 i π

i

i d .

C z e

C F

e z

 

π _i π _i

2 i 2

π π

0

d d

n n n n

n

z

e z F e z F

e

 



0

0 1

i n n

H n n n

n n

F z Id C z z  



 



 



 





From the above representation, it results:



 

 











 

  _       _









 

 

 

π i

 

π

 

2 2

i _i 2

2

2 2 H d 2 H d 0, .

z e F

π π

z F z F z Id F Id F z D

e z



  _ 



         





 _{e z}





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

   

 

2

π _{i i i i}

1, _π 0 1

π i i

1, _π

, 0

1

lim 2 2

2π

r r R H

n n

Id

, 0 , 0

,

1

lim d

2π

d 2 d

2 0

n

r r R n

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

   

 

   

     

 

  _





 

 



 _  _{  }

 

 

 

 



  











 _

 

 _  







for arbitrary . From this inequality, it results that

there exist the representations

with

Physics, Vol. 63, 1911, pp. 501-511.

[3] C. Carathéodory und L. Fejér, “Über den Zusammenhang der Extreme von Harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz,” Rendiconti del Circolo Matematico di Palermo, Vol. 32, No. 1, 1911, pp. 218-239. doi:10.1007/BF03014796

nN

 

1 d 1 , q

q _Tz F z q Z

 



 

1

F

3.2)

a positive operator valued measure on ([9], Th. 1. , this is (a).

The equivalence, . From remark 4.1.we

ha he equi-

alen in result in [10], Proposi-

Proposition 1, (condition (c) in

Rema of a Hilbert space

and an unitary oper

position Co

5. Conclusion

We give a necessary and sufficient condition on a finite uence of bounded operators, acting on a finite dimen-

e. We also established a to

rators.

R

del C

Vol. 3 , pp. 193-207.

1

T

   

b  d

is the ma [10],

the existence

K



n



 

ogor idefinite ker diately.

[4] T. Ando, “Truncated Moment Problems for Operators,” Acta Mathematica, Vol. 31, 1970, pp. 319-334.

[5] L. Lemnete-Ninulescu, “Truncated Trigonometric and Hausdorff Moment Problems for Operators,” Proceedings of the 23th International Operator Conference, Timisoara, 29 June-4 July 2010, pp. 51-61.

[6] M. Bakonyi and V. Lopushanskaya, “Moment Problems for Real Measures on the Unit Circle,” Operator Theory Advances and Applications, Vol. 198, 2009, pp.49-60. [7] F. J. Narcowich, “R-Operators II., on the Approximation

of Certain Operator-Valued Analytic F