A structure theorem for reproducing kernel Pontryagin spaces
D. Alpay
Department of Mathematics, Ben-Gurion University of the Negev, POB 653, 84105 Beer-Sheva, Israel Received 29 January 1998; received in revised form 1 July 1998
Abstract
We illustrate a relationship between reproducing kernel spaces and orthogonal polynomials via a general structure theorem. The Christofell–Darboux formula emerges as a limit case. c1998 Elsevier Science B.V. All rights reserved. Keywords:Reproducing kernel spaces; Structured kernels
1. Introduction
Finite-dimensional reproducing kernel spaces and orthogonal polynomials are two closely related topics. In this paper we review some of the links between these two subjects by proving a general structure theorem. For further relationships, in particular with the Schur algorithm and with the theory of linear systems, we refer to [1]. Recall [12] that a vector space P endowed with an (in general indenite) inner product h;iP is called a Kren space if one can write P=P++P− where
1. The space P
+ endowed with the inner product h;iP is a Hilbert space.
2. The space P− endowed with the inner product −h;iP is a Hilbert space.
3. For each p+∈P+ and p−∈P−, it holds that
hp+; p−iP= 0:
4. Each element of P admits a unique decomposition p=p++p− with p+∈P+ and p−∈P−. The space is called a Pontryagin space if dimP−¡∞. A Pontryagin space P whose elements are functions dened on a set and with values in Cp is called a reproducing kernel Pontryagin space
if there exists a Cp×p-valued function K(z; !) with the following two properties:
1. For every choice of !∈ and c∈Cp, the function
K!c:z7→K(z; !)c
belongs to P.
2. For every choice of !∈, c∈Cp and of f∈P, it holds that
hf; K!ciP=c∗f(!):
It is well to recall that the function K(z; !) has negative squares (with = dimP−) in the following sense: for every choice of integer n, of points !1; : : : ; !n∈ and vectors c1; : : : ; cn∈Cp
the n ×n hermitian matrix with ij entry equal to c∗i K(!i; !j)cj has at most strictly negative
eigenvalues, and exactly such eigenvalues for some choice ofn of points!1; : : : ; !n andc1; : : : ; cn.
There is a one-to-one correspondance between such functions and reproducing kernel Pontryagin spaces. We refer the reader to [10] for the Hilbert space case and to [22, 21, 5] for the case of Pontryagin spaces. The case of Kren spaces is more involved and will not be considered here (see [21, 2]).
In the present paper we review work on nite-dimensional reproducing kernel spaces (a partic-ular instance of which will be the case of polynomials). In particpartic-ular, following a strategy due to de Branges (and pursued later by his student Li [18]), we characterize the case where the reproducing kernel is of the form
U(z)JU(!)∗ −V(z)JV(!)∗
1−z!∗ : (1.1)
In this expression, U and V are matrix-valued functions (say Cp×p-valued) analytic in an open
subset of the open unit disk and with nonidentically vanishing determinant, and J is a signature matrix, i.e., a matrix which is both unitary and hermitian. More generally, one could consider denominators of the form
a(z)a(!)∗ −b(z)b(!)∗
where the functions a and b are analytic in a connected open subset ⊂C and such that the three sets
+={z∈; |a(z)|¿|b(z)|};
−={z∈; |a(z)|¡|b(z)|};
0={z∈; |a(z)|=|b(z)|}
are all nonempty, see [7]. This allows to treat in a unied way Toepliz and Hankel matrices.
2. Finite-dimensional reproducing kernel Hilbert spaces
When the space is nite-dimensional, the reproducing kernel is given by a simple and well-known formula:
Theorem 2.1. Let P be a nite-dimensional Pontryagin space, whose elements are functions → Cp. Let{f
1; : : : ; fN}be a basis of P. LetP be the matrix withi; jentry given byPij=hfj; fiiP. Then
P is hermitian and nonsingular and P is a reproducing kernel Pontryagin space with reproducing kernel
Proof. Indeed, let P−1= (
ij). One can write
K(z; !) =X
i; j
fi(z)ijfj(!)∗
and thus, with c∈Cp,
K(z; !)c=X
i; j
fi(z)ijfj(!)∗c
and the function z7→K(z; !)c∈P. Let f=P
llfl∈P. We have
hf; K(·; !)ciP=
X
l
lhfl; K(·; !)ciP
=X
l
X
i; j
l∗ijc∗fj(!)hfl; fiiP
=X
l; i; j
l∗ijc∗fj(!)Pil;
from which one easily concludes since ∗ij=ji and PijiPil is equal to 0 or 1, depending on j6=l
or j=l.
The matrix P is called the gramian of the basis fi; i= 1; : : : ; n ; see [20, p. 2]. It is of interest to
relate the structure of the reproducing kernel and of the Gram matrix. The celebrated Christofell– Darboux formula can be viewed as an instance of such a link, as will be made clear in the sequel. The spaces of polynomials of degree less or equal to a given integer motivate the assumption that the space P is invariant under the backward shift operators R dened by
(Rf)(z) =
f(z)−f()
z− :
In fact, a less stringent hypothesis will be made: let !∈C. Iff∈P is analytic at!andf(!) = 0, then we will require that the function
z7→f(z)1−z! ∗
z−! (2.3)
belongs to P. Such hypothesis originates with the work of de Branges and are motivated by the theory of (in general nondensely dened) relations in Hilbert space; see [13], and especially Theoreme 23, p. 59. In view of the equality
1−z!∗ z−! =
1− |!|2
z−! −!
∗
we have in particular that the function z 7→ f(z)=(z−!) belongs to P for ! o the unit circle. The invariance condition (2.3) forces the structure of the space:
Lemma 2.2. Let P be a nite-dimensional vector space of functions analytic in some open set
f(); as f runs through P; is equal to Cp. Assume furthermore that the invariance condition
(2.3) holds. Then, there is an analytic Cp×N-valued function X(z) and a matrix A∈CN×N such
that the columns of a basis of P are given by
F(z) =X(z) (I +A−zA)−1:
Proof. By hypothesis, there is a p×p matrix E whose columns are functions of P and whose determinant at some point is nonzero. (This will then happen at all points of , at the exception of a zero set, i.e., the set of zeros of an analytic function.) Let now N= dimP and F be a p×N matrix whose columns form a basis of P. The function
F(z)−E(z)E()−1F()
belongs to P and vanishes at the point . So by hypothesis the columns of
F(z)−E(z)E()−1F()
We now review some properties of reproducing kernels of the form (1.1). Most of the material can be found in [4, 5]. First let us assume that U=J=I. If the function V is rational and inner (i.e. in analytic in the open unit disk and takes unitary values on the unit circle) the reproducing kernel space with reproducing kernel (1.1) is equal to Hp2 ⊖VHp2, where Hp2 is the Hardy space for the denition of the McMillan degree. If the function V is not anymore inner but is still rational and takes unitary values on the unit circle, a theorem in [17] asserts that V=V2−1V1, where both V1
and V2 are inner and may be chosen such that
allows to conclude that P(V) consists of the functions of the form
f(z) =V2(z)−1(f1(z) +f2(z)); fi∈(Hp2 ⊖ViH2p); i= 1;2;
with the indenite inner product
kfk2=kf
1k2Hp2⊖V1Hp2 − kf2k
2
Hp2⊖V2Hp2;
see [5, Theorem 6.6, p. 133]. In particular, the space P(V) possesses the property (2) of Theorem 4.1 since the same property trivially holds for the spaces H2p⊖ViH2p associated to
in-ner functions.
Let us still assume thatU=I in (1.1) but consider an arbitrary signature matrixJ. SinceJ=J∗=
J−1, the matrix J is unitarily equivalent to
J0=
Ir 0
0 −Is
;
where r (resp. s) is the multiplicity of the eigenvalue +1 (resp. −1). Without loss of generality, we may assume that J=J0. Set
P=I +J0
2 ; Q=
I −J0
2 :
The matrix
= (PV +Q)(P+QV)−1
is well dened and is called the Potapov–Ginzburg transform of V. Because of the equality
I−(z)(!)∗
1−z!∗ = (P−V(z)Q)
−1J0−V(z)J0V(!)∗
1−z!∗ (P−V(!)Q)
−∗
(see [5, Theorem 6.8, p. 136]) one reduces the case of arbitrary J to the case J=I and shows that property (2) of Theorem 4.1 still holds. We refer to [14, 4].
We mention that an analogue of the Kren–Langer factorization theorem for the case of an arbitrary signature matrix J does not hold. The multiplicative structure ofJ-inner functions is well understood [19]. The case of J-unitary functions is mostly open, even in the rational case, see [5, 9].
4. Structure theorems
We now characterize nite-dimensional reproducing kernel Ponytryagin spaces with reproducing kernel of the form (1.1), supposing moreover that a full rank condition (explicited in the theorem) is met. First one denition: an open subset of C is said to be symmetric with respect to the unit circle if 1=z∗ ∈ for every nonzero z∈.
Theorem 4.1. Let P6={0} be a nite-dimensional reproducing kernel Pontryagin space of Cp-valued functions analytic in an open set ⊂C which is symmetric with respect to the unit
P is of the form (1.1) for some signature matrix J and Cp×p-valued rational functions U and V
with non identically vanishing determinant if and only if the following conditions hold:
1. There is a point ∈ such that the span of the vectors f(); as f runs through P; is equal
to Cp.
2. If f∈P is analytic at !∈ and f(!) = 0; the function dened by (2.3) belongs to P and has
same norm as f.
The rst condition in fact holds for all points in , with the possible exception of a zero set owing to the connectedness and analyticity hypothesis. In the scalar case and for spaces of polynomials this theorem appears in the work of Li [18]. The proof of this result for the Hilbert space case appears in [3], and is an adaptation of arguments of de Branges.
Proof of Theorem 4.1. We rst assume that the two conditions of the theorem hold, and show that the reproducing kernel is of the required form. We set K to be the reproducing kernel of P and proceed in a number of steps:
Step 1: There is 6= 0∈ such that K(; ) and K(1=∗;1=∗) are nonsingular.
Proof of step 1. Let ! be a point at which K(!; !) is singular; there exists a nonzero vector
∈Cp such that F(!)P−1F(!)∗= 0. The same argument as the one of [6, Theorem 4.2] forces
then F()P−1F()∗= 0 for all ; ∈. It follows then that for every ∈ and every f∈P,
∗f() =hf; K(·; )iP= 0;
which contradicts the full range hypothesis on the values f(); f∈P.
We note that the argument of [6] takes full advantage of the connectedness of the set of analyticity of the elements of P. For a counterexample when this hypothesis is not in force, see [6, p. 51].
Step 2: The reproducing kernel is of the form (1.1).
Proof of step 2. Let c∈Cp and let !∈C be a point where the elements of P are analytic; the
function
z7→(K(z; !)−K(z; )K(; )−1K(; !))c
belongs to P and vanishes at !. From the second assumption, the function
z7→ 1−z ∗
z− (K(z; !)−K(z; )K(; )
−1K(; !))c
still belongs to P. Let F be an element of P which vanishes at 1=∗. In view of the hypothesis on the inner product of P we have
"
F;1−z
∗
z− (K(z; !)−K(z; )K(; )
−1K(; !))c
#
P
=
"
F(z) 1−z
z−1=∗;
1−z∗ z−
1−z=
z−1=∗(K(z; !)−K(z; )K(; )
−1K(; !))c
#
=
The equality between the rst and last line of this chain of equalities is valid for every function in P which vanishes at 1=∗. Therefore,
NJ2N∗, where J1 and J2 are (a priori dierent) signature matrices and M; N are invertible matrices.
It follows that K is of the form
Hence, J1 and J2 can be taken to be equal to a common signature matrix, which we will denote
by J.
We now study the converse of the theorem. The function B=U−1V is J-unitary and rational, and
the reproducing kernel Pontryagin space P(B) with reproducing kernel (J −B(z)JB(w)∗)=(1−zw∗) is R-invariant (see [5]). Let g=Uf∈P, with f∈P(B). Assume that ! is o the unit circle, such
that g(!) = 0 and detU(!)6= 0. Then, f(!) = 0. Thus, z 7→ f(z)=(z−!)∈P(B). It follows that
z7→F(z)=(z−!) belongs to P and so does
g(z)1−z! ∗
z−! = −!
∗g(z) + (1− |!|2) g(z)
z−!:
The norm condition is veried using the properties of P(B) spaces.
Corollary 4.2. In the preceding theorem, assume that P is a Hilbert space and that J=I. Then
there is a positive measure d on the unit circle such that P is isometrically included in the
Lebesgue space Lp2(d).
Indeed, we then have P(B) =Hp
2 ⊖BH
p
2 and one can take
d(t) =U(eit)∗U(eit):
The formula
K(z; !) =K(z; )(1−z
∗)K(; )−1K(; !)(1−!∗)
(1−z!∗)(1− ||2)
−K(z;1=
∗)(z−)K(1=∗;1=∗)−1K(1=∗; !)(!∗ −∗)
(1−z!∗)(1− ||2)
really means that to compute the kernel function, it is enough to know it on two symmetric values, namely at and 1=∗.
The Toeplitz case is of special interest: let Pn be the space of polynomials of the form Pn0Ajzj,
where the Aj∈Cp×p endowed with the inner product dened by an nonsingular block-hermitian
Toeplitz matrix P. The conditions of the theorem are easily seen to be satised and in order to apply the result one looks rst for a nonzero number such that
(IpIp : : : nIp)P−1(IpIp: : : nIp)∗
and
(Ip1=Ip : : :1=nIp)P−1(Ip1=Ip : : :1=nIp)∗
are both nonsingular. The limiting process → ∞ leads to the Christofell–Darboux formula, which is another way of expressing the Gohberg–Heinig formula [15]. This formula expresses the inverse of a (say hermitian invertible) Toeplitz matrix in terms of the rst and last columns of its inverse, provided not only the matrix is invertible but also the rst main minor; this is equivalent to re-quire that the matrices (00n) and (n)
nn in the block decomposition P−1= (
(n)
Setting
and the kernel takes the form
En(z)(00(n))−1En(!)∗ −zFn(z)((nnn))−1!∗Fn(!)∗
1−z!∗ ;
i.e. we get to the Christofell–Darboux formula (see for instance [8, Theorem 4.1, p. 38]).
One recognizes in En and Fn orthogonal polynomials; a result of Kren [16] characterizes the
number of their zeros in the non positive case and the scalar case; for an extension to the matrix case, see [8].
In connection with the previous discussion, let us mention:
Problem 4.3. Can one express the kernel(1.1) in terms of two non symmetric valuesK(z; 1) and
K(z; 2)?
Acknowledgements
It is a pleasure to thank Martine Olivi (INRIA Sophia-Antipolis) for a careful reading of the manuscript.
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