On the Positive Function of Bounded Type in Hilbert Space
K. El Naggar & A.A. El-Sabbagh
Department of Mathematics, Faculty of Engineering,
Benha University, Shoubra, 108 Shoubra Street, Cairo, Egypt
Fax 202 2023336
Abstract
An invariant subspace theory for contractive transformations in Hilbert space which are
due to L. de Branges and J. Roovnyak are considered. Now we are interested in using
reproducing kernel spaces methods in Hilbert space in realization problem for functions
of bounded type in the unit disk. Tlie relation between factorization and invariant
subspaces for continuous and contractive transformations in Hilbert space will be
discussed.
Key words: Reproducing Kernel Hilbert Space, Nevanlinna Function, Positive
function, invariant subspace, contractive transformation, Hilbert space.
2000 AMS SUBJECT CLASSIFICATION CODE: 47B34.
1. Preliminaries
Let H be the Hilbert space over ℂ with inner product ,  defined by  f , g    g (t ) * f (t )dt ,
a
b
f,g ∈ H , where * denotes the complex conjugate. Let H2 be the Hilbert space of all
ordered pairs {f,g}, f,g ∈ H with the inner product ,  defined by :
{ f , g},{h, k}  [ f , h]  [ g , k ],{ f , g},{h.k} ∈ H2. T is called a closed linear relation in H2 if
T is a (closed) linear manifold in H2. Such T is often considered as a graph of (closed)
linear (multi-valued) operator. To any matrix-valued Nevanlinna function there is
associated in a natural way a reproducing kernel Hilbert space. This Hilbert space
provides a model for a simple symmetric not necessarily densely defined operator. We
define the Nevanlinna class ℕn as the class of all n×n matrix functions N(  ), which are
holomorphic in ℂ0 satisfy N(  )* = N () ,  ∈ C0 and for which the kernel
K N (,  ) 
N ()  N ( ) *
,

 ,  ∈ ℂ0 ,
  ,
is nonnegative, where * denotes the adjoint , the details related these notations will be
given is section 2. The reproducing kernel Hilbert space associated with the kernel
KN(  ,λ), see [3] and [4], is denoted by L(N). For general information concerning
reproducing kernel Hilbert spaces, we refer to [5], [10], [11], [12] and [13]., we shall
give more details related to this subject in the following section. We see for each:
~
~
~
~
~
~
N(  ) ∈ N n, there exist n × n matrices A and B with A = A *, and B = B * ≥ 0,
and a non-decreasing n×n matrix function Σ on ℝ with  R (t 2  1) 1 d (t )   such that :
t 
~ ~
 1
N ( )  A  B   
 2
 d (t ),
R t  
t 

1
 ∈ ℂ0
(1.1)
With this so-called Riesz-Hergoltz representation the kernel KN ( ,  ) takes the form :
~
K N (,  )  B  
d (t )
,
R (t  )(t   )
,  ∈ ℂ0 ,
(1.2)
~
see [2], [7], [14] and [16]. We define the ℂ B~ to be the range R( B ) in ℂn×1endowed with
n
~
~
~
~
the inner product [ B c, B d] = d* B c. We note that ℂ B~ is equal to the space L( B  ). As
n
usual L2 (dΣ) is the Hilbert space of all n×1 vector functions f defined on ℝ such that
2
f    f (t ) d (t ) f (t )   . The theory of Hilbert spaces of entire functions is a detailed
R
description of eigenfunction expansions associated with formally selfadjoint differential
operators. Let H be a Hilbert space, then L[H] set of bounded linear operators from H to
H. Furthermore, when
U unitary if U -1 = U* ⇒ U unitary operator , D(U) = R(U) = H;
U, V, S, A and T are relations in H2;
V isometric if V -1 ⊂ V* ⇒ V operator ;
S symmetric if S ⊂ S* ;
A selfadjoint if A = A* ;
T closed ρ(T) = {  ∈ ℂ| (T -  )-1 ∈ L(H)}
σ(T) = ℂ \ ρ(T)
resolvent set (closed);
spectrum (open);
RT(  ) = {T -  )-1 ,  ∈ ρ(T).
When A = A*, put
H 0  Hο A(0) ;
A  {0}  A(0)  {{ f , h}  A | f  0};
A0  Aο A .
See [1] and [18].
We are interested in reviewing de Branges’ early work on Hilbert spaces of entire
functions and present applications to the theory of hermitian operators and to the spectral
theory of certain differential equations. To quote de Branges [4], the theory of Hilbert
spaces of entire functions is a detailed description of eigenfunction expansions
associated with formally selfadjoint differential operators.
Associated to θt is a reproducing kernel Hilbert space H(θt) with reproducing kernel
i  U (t ,  )iJU * (t ,  )
 2 (   * )
and a reproducing kernel Hilbert space H(At , Bt) with reproducing kernel :


 A B
 with U (t ,  )   ~ ~  (t ,  )  . (1.3)


C D


A* (t , * ) B(t ,  * )  B * (t , * ) A(t ,  * )
,
  (   * )
2
The spaces H(At, Bt) will be the de Branges spaces of entire function that we will be
mainly interested in. They form an isometrically increasing sequence of spaces and are
linked to the spectral function Δ by
 H ( A , B )  L (d) .
t
2
n
t
(1.4)
t 0
The function Δ is in fact the Weiner algebra of functions continuous on the line. More
precisely,
 ( )  I n  


e iu k (u )du
(1.5)
where k belongs to L 2nn (ℝ), see [6], [8], [9], [22] and [25].
Suppose now that we start with a spectral function Δ of the form (1.3). The associated
inverse spectral problem consists of finding the associated potential V(t), i.e., to find a
canonical equation with associated spectral function Δ. Under a suitable hypothesis,
V can be found. The theory of Hilbert spaces of entire functions plays an important role
within this hypothesis. See [26], [28] and [33]. We discuss the use of reproducing kernel
spaces method in the realization problem for functions "analytic" of bounded type in the
unit disk. All of this will be done in section 2..
2. Reproducing kernel Hilbert Space
In this section we recall the definition of Nevanlinna Function, a reproducing kernel
Hilbert space and introduce the spaces which we will study in the sequel.
Definition 2.1. The Nevanlinna pick problem consists of finding a function f(  ),   ℂ
analytic in Im  > 0, Im (  ) having a fixed sign there, to take assigned values on infinite
sequence of points with :
Im (  ) > 0.
Definition 2.2. We define the Nevanlinna class in ℕn as the class of all nxn matrix
function N(  ), which are holomorphic in ℂ\ℝ,satisfying N(  )* = N(  ),   ℂ\ℝ, and
for which the kernel :
KN ( ,  )  N ()  N ( ) * ,

 ,   ℂ\ℝ,    is nonnegative,
Definition 2.3. For each N(  )  ℕn there exist nxn matrices A and B with A = A* and
B = B*  0, and a nondecreasing nxn matrix function Σ on ℝ with  (t 2  1) 1 d(t )   ,
R
t 
 1
 2
 d(t ),
 t   t 1
such that: N ()  A  B   R 
  ℂ\ℝ .
(2.1)
with this so-called Riesz- Herglotz representation the kernel KN (  , λ) takes the form
K N ( ,  )  B  
d(t )
,
R (t  )(t   )
 ,   ℂ\ℝ.
(2.2)
the reproducing kernel Hilbert space associated with kernel KN (  ,λ) is denoted by L(N).
3
Definition 2.4. Let h be a Hilbert space of ℂ
n×1-valued
function defined on some
set Ω. h is a reproducing kernel Hilbert space with reproducing kernel the ℂn×n-valued
function K(λ,u) defined on Ω×Ω if the following conditions hold
K 0
(a) for any c in ℂn×1 , ω in Ω, the function λ  K(λ,ω)c belong to h,
(b) for any f in h, c in ℂn×1 , ω in Ω
 f,K0   = c* f (ω),
(2.3)
where ·,· denotes the inner product in h.
Definition 2.5. The function K is easily seen to be unique. It is positive onΩ: for any r
any c1 ,...,cr in ℂn×1, ω1,...,ωr in Ω, the r × r hermitian matrix with ij entry c *J K(ωi,ωj)ci
is positive. Conversely, to any function K(λ,ω) positive on some set Ω is associated a
unique reproducing kernel Hilbert space with reproducing kernel K.
Example 2.1. A typical example is the function K(λ,ω) = 1/-2πi(λ-ω.) defined for λ and
ω in the open upper half plane ℂ+. The associated reproducing kernel Hilbert space is
H2, the Hardy space of the line. More generally, if S is ℂp×q-valued, analytic, and
bounded by one in modulus in ℂ.
The functions
I p  S ( ) S * ( )
 2i(  *)
,
I q  S * ( ) S ( )
 2i(  *)
(2.4)
are positive in ℂ+ .
Notation. ρω(λ) = -2πi(λ-ω*).
Most of the reproducing Kernels to be studied and used in this paper are of the form:
K ( ,  ) 
X ( )TX * ( )
,
  ( )
(2.5)
where J is ℂm×m signature matrix, that is, J = J* = J -1, and where X is ℂk×m-valued and
meromorphic in ℂ+ .
Examples 2.2.
0
(1) If J =  I
 n

In 




0  and X = [ ,I], then K(λ,ω) = ( (λ)+ * (ω)) /ρω(λ).
0 
(2) J =  0    , and Σ signature, also X = [I , θ], then K(λ, ω) = Σ - θ (λ) Σθ*(ω) /ρω(λ).


4
0
(3) If J =  I
 n
(4) If J =
 0

  iI n
In 

0 
and X = [A,B], then K(λ,ω) = (A(λ)B* (ω)+B(λ)A* (ω)) /ρω(λ).
iI n 

0 
and X = 2 [A,B], then K(λ,ω) = (A(λ)B* (ω)-B(λ)A*(ω)) /-π(λ-ω*).
We now construct properties of J-contractive functions and of H(θ) spaces relevant
I
0 
p
to our purpose. We first suppose J = J0 =  0  I  .
q 

In this paper we use the following notations: (Consider T as a linear relation in H2)
D(T) = {x|  y (x,y) ∈ T} domain;
R(T) = {y|  x (x,y) ∈ T} range;
T(x) = {y|{x,y} ∈ T};
in particular T(0) = {y|{0,y}∈ T} multivalued part
v(T) = {x|{x,0} ∈ T} nullspace;
when S is also a linear relation in H2, then:
T+S = {{x,y + z}|{x,y} ∈ T,{x,z} ∈ S}sum;
ℝ
= Set of real numbers;
ℂ
ℂ
= Set of complex numbers;

= {  ∈ ℂ, ± Im(  ) > 0};
ℂ0
=ℂ\ℝ;
ℕ
θ
= {1, 2, …};
= is the characteristic function ;
ℂn×1
= Space of nx1 vectors with entries from C;
T–λ
= {{x,y - λ}|{x,y} ∈ T,
λ ∈ ℂ;
αT
= {{x,αy}|{x,y} ∈ T},
α ∈ ℂ;
T -1
= {{y,x}|{x,y} ∈ T} inverse;
T*
= {{u,v}|[v,x] - [u,y] = 0,  {x,y} ∈ T} adjoint;
T∔ S = {{x + f,y + g}|{x,y} ∈ T, {f,g} ∈ S} direct if T ∩ S = {{0}};
T⊕ S = T ∔ S when S⊥ S in H2 ;
T⊥
= Orthogonal complement in H2 ;
T⊖ S = T ∩ S⊥ ,
A,B
= are entire functions;
H(A B) = De Brange’s spaces of entire functions A and B;
K(λ ,ω) = is a kernel, λ , ω ∈ ℂ0;
N(  )
= is a Nevanlinna function,  ∈ ℂ0 ;
ℕn
= is the class of all n × n matrix functions N(  ).
5
3.
Analytic Functions in Unit Disk
A matrix valued function S is of bounded type in the unit disk D if its entries are quotient
of H  functions, that is of functions analytic and bounded in modulo in D. We will
suppose moreover S analytic at the origin, and the realization problem for S consists in
finding representations of S as
~
~
S(z)= ( D + z C )-1 B
(3.1)
~
~
where D = S(0) and where A, B, C are bounded operators between adequate spaces.
Such realizations always exist for functions analytic at the origin of S, some are more
useful to handle problems (even if they are not of bounded type). Among all realizations
related to S: when S is rational, of particular interest are minimal realizations; the
~
realization (3.1) is minimal if the pair (A, C ) is observable:  K   A n  {0} and the pair
0
(A,B) is controllable:  Ran A B  ℂm if A acts from ℂm into ℂm . Minimality can also be
n
n
defined for non-rational functions. In this paper we will not be interested in minimal
realizations but in realizations which are coisometric and, whenever possible,
A
B
contractive: the matrix operator  ~ ~  is a coisometry (i.e., has an adjoint which is
C D
isometric) between adequate spaces and, if possible, a contraction. Coisometric
realizations were studied first by de Branges and are useful in the study of Schur
algorithm. We first consider the case of Schur functions, i.e., of functions analytic and
bounded by one in modulus in the unit disk. See [17].
Theorem 3.1. Let S be a ℂp×q valued Schur function. The function K(z,ω) =
I p  S ( z ) S * ( )
1  z *
is positive for z and ω in D. We denote H(S) the associated reproducing kernel Hilbert space.
Theorem 3.2. The space H(S) has the following properties:
(a) if f is in H(S) so is R0 f, where R0 f(z) =
f ( z )  f (0)
, and R0 is a bounded operator
z
from H(S) into H(S),
(b) for any c in ℂp , the function z →
S ( z )  S (0)
c belongs to H(S),
z
~
(c) the map C : f → f (0) is bounded from H(S) into ℂp .
The space H(S) is contractively included in H p2 and is the closure of the range of operator
  I  S p S * | H 2 in the norm u
p
H (S )
Writing
 u, u
H 2p
.
S ( z )  S (0)  z
6
S ( z )  S (0)
z
we get a realization for S with
D = S(0) : ℂq → ℂp
Af = R0 f : H(s) → H(S)
Bc =
S ( z )  S (0)
c : ℂp → H(S)
z
(3.2)
~
C f = f (0) : H(S) → ℂp
 A B
~  is a coisometry and a contraction from H(S)⊕ ℂp into H(S) ⊕ ℂp .
C D
The matrix  ~
This realization is not minimal in general but it is observable since
 K  A
n
 {0} .
n
Recent work has shown that the space H(S) is a convenient framework to treat various
interpretation problems related to S and in particular to describe the Schur algorithm.
We now consider more general situations and suppose first that S is in a generalized
*
Schur class ρk , i.e., that the function I p  S ( z ) S* ( ) has k negative spaces. By a theorem
1  z
of Krein and Langer we can write S as S = B-1 S1 where B is a rational Blashlike product
and S0 is a ℂp×q Schur function. One can choose S1 and B such that H(B) ∩ H(S1) = {0}.
Since
B(z) = D0 + zC0 (I-zA0)-1 B0
(3.3)
with e.g.,
A0 = R0 : H(B) → H(B)
We get a realization for B -1 as
B 1 ( z )  D01  zD 01C 0 ( I  zA0* ) 1 B0 D01 .

A*
0
A computation shows that 
1
  D0 C 0
B0 D01 
 is co isometric from H(B) ⊕ ℂp onto
D01 
H(B) ⊕ ℂq , where H(B) is endowed with the negativr norm f
 A0*

  D 1C
0
0

B0 D01 

D01 
 I

 0

0

I p 
 A0*

  D 1C
0
0

*
B0 D01 

D01 

 f
 I
 0
= 
Since we know a realization for S1 ,
~
~
S1(z) = D 1 + z C1 (I – zA1) -1 B1 , where e.g. A1 = R0 : H(S1) → H .
We get the following realization for S = B -1 S1 .
7
0

I q 
H ( B1 )
~
~
S(z) = D + z C (I – zA) -1 B
~
B0 D01 C1 
 : H(B) ⊕ H(S1) → H(B) ⊕ H(S1)

A1

1 ~
B D D 
B =  0 0 1  : ℂq ⊕ ℂq → H(B) ⊕ H(S1)
B1


~
1 ~
C = (C0 , D0 C1 ): H(B) ⊕ H(S1) → ℂp ⊕ ℂq
 A0*
A = 
 0
and
D = D01 D1.
  I H ( B)

 A B
The matrix  ~ ~  is coisometric with respect to , J = 
I H ( S1 )
C D


*
1
 A0 0 B 0 D 0   I 0
0

 

 A B
 ~ ~  =  0 I
0   0 A1 B1  .
~ ~
C 0
C D
D 0   0 C1 D1 
 0


 as
I 
(3.4)
Up to now we did not use the hypothesis H(B) ∩ H(S1) = {0}, and in fact the above
argument will be valid for any function S of bounded type in D.
Under the hypothesis H(B) ∩ H(S1) = {0} we now discuss another realization for S1, for
which the scale space will be a Pontryagin space and the main operator in the state space
will be R0 , see [19] and [20]
Theorem 3.3. Let S be ρk . There exists a unique reproducing kernel Pontryagin space
K(S) with reproducing kernel
I p  S ( z ) S * ( )
1  z
. Let S = B
*
-1
Sl be a representation of S1
with H(B) ∩ H(S1) = {0}. Then K(S) = {B -1 ( f0 + f1), f0 ∈ H(B1), f1 ∈ H(S1) with norm
B 1 ( f 0  f 1 )
2
K (S )
The space K(S) is still R0 invariant and
 f1
2
H ( S1 )
 f0
2
H ( S1 )
S ( z )  S (0)
c belongs to K(S) for any c in ℂq .
z
Thus, the realization
~
~
S(z) = D + z C (I – zA) -1 B
~
(3.5)
~
where D , C , A, B are as in (3.2), but with K(S) instead of H(S), is still valid.
The above theorem still holds if S = S 01 S1 where S0 is not any more a Blaschlike finite
product, as long as H(S0) ∩ H(S1) = {0}.
More general cases, i.e., when H(S0) ∩ H(S1) ≠ {0}, or when S is merely analytic at the
8
origin are more difficult to deal with (see [7]). Spaces H(S) and K(S) are useful to
discuss factorization theorems. As a consequence of the commutant lifting theorem, one
can prove:
Theorem 3.4. Let S0 and S1 be in ρp×q and ρp×q' . Then H(S0) is contractively included
in H(S1) if and only if S1 = S0 S' where S' ∈ ρq'×q .
This suggests the problem, for a S in a class ρk , to find all S' such that K(S') is
contractively included in K (S).
ρp×g : ρp×g' valued Schur Functions, see [31] and [32].
Lemma 3.1. Let S be in ρp×q , c in ℂq and  in D. Then, the function (RωS) c belongs to
H(s) and
R S c
Proof. The function bω(z) =
 c
2
H (S )
*
I q  S * ( ) S ( )
  ( )
c.
z 
is unitary on the unit circle, and so, for u in H q2 ,
*
1  z
R S c  S
2
H 2p
 b RS c  b S
2
H 2p
 c
 S ( ) c
 S .   b  

 

By the reproducing kernel property of
2
.
H 2p
1
in H2 and since bω(ω) = 0, we have

 c
 S ( )c
S 
 b   ,

 

S * ( ) S ( )c
 c
  ( )
*
H 2p
and hence,
R S c  S
 c

S * ( ) S ( )c
 c
 S 
 b 

 

2
2
*
H 2p
S * ( ) S ( )c
c
 c

 b
  ( )

H 2p
2
*
 c*
I q  S * ( ) S ( )c
  ( )
 u
2
H q2
H q2
.
The required inequality follows then from the definition of H(S). See [23], [24] and [27].
The same argument, with more book-keeping, will show that, for ω1,...,ωN in D,
c1, … , cN in ℂq ,
 Ri Sc i
2
H (S )
  c *j
I q  S * ( j ) S ( i )
 j ( i )
i, j
9
ci .
( )
Theorem 3.5. Let S be. in ρp×q . The function K(z,ω) defined by
 I p S ( z ) S * ( )


 ( z)
K ( z,  )   * *  *
 S ( z )  S ( )

z *

S ( z )  S ( * ) 

z *
*
*
* 
I q  S ( z ) S ( )


  ( z)

(3.6)
is positive.
Proof. Let ci , i = 1,..., N, di , i = 1,..., N, ωi , i = 1,..., N be sequences respectively in
2
cj 
ℂq , ℂp , and D. We want to show that the sum: Σ   K(ωi,ωj)
d j 
 cj 
  is positive.
d 
 j
This sum is composed of four terms
I p  S ( i ) S * ( j )
*
cj .
(1) =  c i
 j ( i )
(2) =  d
I q  S * ( *i ) S * ( *j )
*
i
 j (i )
S (i ) S ( *j )
(3) =  c
*
i
i   *j
dj .
dj .
(4) = (3)*.

The term (3) can be written as  with ki ( z ) 

(3) =  R Sd i ,  k i c i
*
j
I p  S ( z ) S * (i ) 
.

 i ( z )

.
H (S )
By the Cauchy-Schwarz inequality, we get:
(3)
  R * Sd j
2
j
2
H (S )
 k i c i
2
H (S )
.
But
 k i c i
2
H (S )
  c *j k i ( i )c i  (1),
and, by inequality (  ), with  i* instead of ωi ,
 R * Sd j
j
2
 d
H (S )
*
j
I q S * ( *j ) S ( i* )
 i ( j )
d i  (2)
from which the positivity of the sum follows

c 
  i  K (i , j )
 di 
 cj 
   (1)  (2)  (3)*
d 
 j



2
(1)  (2) .
See [15] and [21].
The reproducing kernel Hilbert space with reproducing kernel (3.6) gives the de Branges
Rovnyak model for contractions. It will be denoted by D(S).
10
Theorem 3.6. Let S be in ρp×q and D(S) the associated reproducing kernel Hilbert space
with reproducing kernel (3.6). Then, D(S) is invariant under the generalized shift
 zf  Sg (0) 
f
 .
   
R
g
g
0


The adjoint of A in D(S) is
R0 f
f


   

*
*
g
 zg  S ( z ) f (0) 
and every completely nonunitary contraction T acting in a Hilbert space is unit
equivalent to the operator A* in some D(S) space. See [29], [30], [31] and [33].In these
notes we will not pursue the discussion of applications of the space D(S) to model theory
and the relationship between the de Branges Rovnyak model and the Sz. Nogy-Foias
model. Our interest in H(S) and D(S) is that they give special realizations of S.
~
Lemma 3.2. Let S(z) = S(0) + z C (I - zA) -1 where
A = R0 : H(S) → H(S)
Bc =
S ( z )  S (0)
c : ℂq → H(S)
z
~
C f = f(0): H(S) → ℂp .
B 
A
 is coisometric from H(S) ⊕ ℂq into H(S) ⊕ ℂp .
 C S (0) 
Then the operator  ~
~
Theorem 3.7. Let S(z) = S(0) + z C (I – zA)-1 B where
A* = generalized shift : D(S) → D(S)
f
B*   = g{0) : D(S) → ℂq
g
 
~f
C   = f(0) : D(S) → ℂp
g
(3.7)
 A B
~
~  is a unitary operator from H(S) ⊕ ℂq into H(S) ⊕ ℂq where D = S(0).
C D
then  ~
Proof. It is clear immediately from Theorem 3.3 and from Lemma 3.2 above.
11
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K. El Naggar & A.A. El-Sabbagh
Department of Mathematics, Faculty of Engineering,
Benha University, Shoubra, 108 Shoubra Street, Cairo, Egypt
Fax 202 2023336
E-mail alysab1@hotmail.com
13