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On L. Schwartz's boundedness condition for kernels

In previous works we analysed conditions for linearization of hermitian kernels. The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in the related matter of characterizing the real space generated by positive definite kernels. The aim of this note is to find more concrete expressions of the Schwartz type conditions: in the Hamburger moment problem for Hankel type kernels on the free semigroup, in dilation theory (Stinespring type dilations and Haagerup decomposability), as well as in multi-variable holomorphy. Among other things, we prove that any hermitian holomorphic kernel has a holomorphic linearization, and hence that holomorphic kernels automatically satisfy L. Schwartz's boundedness condition.


Introduction
We analysed in [7] conditions under which the linearization functor produces a Kreȋn space from a hermitian kernel, in the spirit of Kolmogorov type decompositions, and subsequently in [8] we generalized this construction to kernels invariant under the action of a semigroup with involution. We also related these constructions with the GNS representations of * -algebras, an issue of some recent interest in quantum field theory with indefinite metric ( [2], [14], [13], [19]). The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in [18] in the related matter of characterizing those hermitian kernels that are in the real linear space generated by positive definite kernels. These boundedness conditions are rather difficult to be verified, see [2], [13], and their nature is quite obscure, see [19].
The aim of the present note is to find more concrete expressions of the L. Schwartz type conditions. We first note that the invariant Kolmogorov decomposition has a countarpart in the representation theory of semigroups with involution on reproducing kernel Kreȋn spaces. Then, it is explained that the type of invariance that is considered in [8] can be viewed as a Hankel type condition and we apply this to a Hamburger moment problem for the free semigroup on N generators. This is used in order to show how the Schwartz condition is somewhat simplified when it is written for generators of a * -algebra.
In Section 4 we first show that the Stinespring dilation of hermitian linear maps fits into the general scheme of invariant Kolmogorov decompositions and we make explicit the connection with completely bounded maps and Wittstock's Theorem [20]. This opens the possibility of defining a class of non-hermitian decomposable kernels, that may successfully replace the missing class of completely bounded kernels, by using a generalization of Haagerup's decomposable linear mappings on C * -algebras, cf. [12]. An analog of Paulsen's Dilation Theorem for decomposable kernels is obtained in Theorem 4.4. In Section 5 we show that holomorphic kernels in more than one variable have Kolmogorov decompositions and hence, that hermitian holomorphic kernels automatically satisfy L. Schwartz's boundedness condition. In view of the transcription between Kolmogorov decompositions and reproducing kernel spaces, e.g. see Theorem 2.4, this result is an extension of the result of D. Alpay in [3] proved for one variable holomorphic hermitian kernels.

Preliminaries
We briefly review the structure of the Kolmogorov decomposition of invariant hermitian kernels. As it was shown in [18], the natural framework for studying hermitian kernels is given by Kreȋn spaces and for this reason we briefly discuss the necessary terminology.

Kreȋn Spaces. An indefinite inner product space (H, [·, ·] H ) is called Kreȋn space
provided that there exists a positive inner product ·, · turning (H, ·, · ) into a Hilbert space and such that [ξ, η] = Jξ, η , ξ, η ∈ H, for some symmetry J (J * = J −1 = J with respect to the Hilbert space structure) on H. Such a symmetry J is called a fundamental symmetry and we will frequently indicate by a lower index the space on which it acts. For two Kreȋn spaces H and K we denote by L(H, K) the set of linear bounded operators from H to K. For T ∈ L(H, K) we denote by T ♯ ∈ L(K, H) the adjoint of T with respect to the indefinite inner product [·, ·]. The Hilbert space adjoint of T with respect to the positive inner products ·, · is denoted by T * . It is important to note that, if J H and J K are fundamentaly symmetries on H and, respectively, K then We say that A ∈ L(H) is a selfadjoint operator if A ♯ = A. For example, in terms of fundamental symmetries, this means J H A = A * J H . Also, we say that the operator U ∈ L(H, K) is unitary if UU ♯ = I K and U ♯ U = I H , where I H denotes the identity operator on H. Equivalently, this means that U is boundedly invertible and J H U −1 = U * J K . In terms of inner products, this means that U is isometric and surjective.
A special situation occurs for a unitary operator U with domain and range the same Kreȋn space H if it commutes with some fundamental symmetry J H . Such a unitary operator is called fundamentally reducible and it can be characterized in other different ways. For instance, U is fundamentally reducible if and only if it is power bounded.
Most of the difficulties in dealing with operators on Kreȋn spaces are caused by the lack of a well-behaved factorization theory. The concept of induced space turned out to be quite useful in order to deal with this issue. Thus, let H be a Hilbert space and, for a selfadjoint operator A in L(H), we define a new inner product [·, ·] A on H by the formula A pair (K, Π) consisting of a Kreȋn space K and a bounded operator Π ∈ L(H, K) is called a Kreȋn space induced by A provided that Π has dense range and the relation holds for all ξ, η ∈ H. There are many known examples of induced spaces. A more delicate question is the uniqueness of the induced Kreȋn spaces (see [7]).

2.2.
Hermitian Kernels. We can use the concept of induced space in order to describe the Kolmogorov decomposition of a hermitian kernel. Let X be an arbitrary set. From now on we assume H is a Hilbert space with inner product denoted by ·, · . A kernel on X is a mapping K defined on X × X with values in L(H). The adjoint K * of K is defined by the formula K * (x, y) = K(y, x) * . The kernel K is called hermitian on X if K * = K. Let F 0 (X, H) denote the vector space of all functions on X with values in H which vanish except on a finite number of points. We associate to K an inner product on F 0 (X, H) by the formula: We say that the hermitian kernel L : X × X → L(H) is positive definite if the inner product [·, ·] L associated to L by the formula (2.3) is positive. One can introduce a natural partial order on the set of hermitian kernels on X with values in L(H) as follows: if A, B are hermitian kernels, then A Kolmogorov decomposition of the hermitian kernel K is a pair (V ; K) with the following properties: The next result, obtained in [7], settles the question concerning the existence of a Kolmogorov decomposition for a given hermitian kernel.
Theorem 2.1. Let K : X ×X → L(H) be a hermitian kernel. The following assertions are equivalent: (1) There exists a positive definite kernel L : X ×X → L(H) such that −L ≤ K ≤ L.
The condition in assertion (1) of the previous result appeared earlier in the work of L. Schwartz [18] concerning the structure of hermitian kernels. It is easy to see that (1) is also equivalent to the representation of K as a difference of two positive definite kernels. Thus, Theorem 2.1 says that the class of hermitian kernels admitting Kolmogorov decompostions is the same with the class of hermitian kernels in the linear span of the cone of positive definite kernels.
It is convenient for our purpose to review a construction of Kolmogorov decompositions. We assume that there exists a positive definite kernel L : X × X → L(H) such that −L ≤ K ≤ L. Let H L be the Hilbert space obtained by the completion of the quotient space F 0 (X, H)/N L with respect to [·, ·] L , where N L = {f ∈ F 0 (X, H) | [f, f ] L = 0} is the isotropic subspace of the inner product space (F 0 (X, H), [·, ·] L ). Since (1) in Theorem 2.1 is equivalent to 1/2 L for all f, g ∈ F 0 (X, H) (see Proposition 38, [18]), it follows that N L is a subset of the isotropic subspace N K of the inner product space (F 0 (X, H), [·, ·] K ). Therefore, [·, ·] K uniquely induces an inner product on H L , still denoted by [·, ·] K , such that (2.2) holds for f, g ∈ H L . By the Riesz representation theorem we obtain a selfadjoint contractive operator A L ∈ L(H L ), referred to as the Gram operator of K with respect to L, such that Let (K, Π) be a Kreȋn space induced by A L . For ξ ∈ H and x ∈ X, we define the element ξ x = δ x ξ ∈ F 0 (X, H) (here δ x is the Kronecker function delta), that is, Then we define where [ξ x ] = ξ x + N L denotes the class of ξ x in H L and it can be verified that (V ; K) is a Kolmogorov decomposition of the kernel K.
We finally review the uniqueness property of the Kolmogorov decomposition. Two Kolmogorov decompositions (V 1 , K 1 ) and (V 2 , K 2 ) of the same hermitian kernel K are unitarily equivalent if there exists a unitary operator Φ ∈ L(K 1 , K 2 ) such that for all x ∈ X we have V 2 (x) = ΦV 1 (x). The following result was obtained in [7]. We denote by ρ(T ) the resolvent set of the operator T . (1) All Kolmogorov decompositions of K are unitarily equivalent.
(2) For each positive definite kernel L such that −L ≤ K ≤ L, there exists ǫ > 0 such that either (0, ǫ) ⊂ ρ(A L ) or (−ǫ, 0) ⊂ ρ(A L ), where A L is the Gram operator of K with respect to L.

Invariant Hermitian Kernels.
We now review some results on the Kolmogorov decomposition of hermitian kernels with additional symmetries. Let φ be an action of a unital semigroup S on X. Assume that the L(H)-valued hermitian kernel K has a Kolmogorov decomposition (V, K). The action φ is linearized by the following mapping: for any a ∈ S, x ∈ X and ξ ∈ H, We notice that for a, b ∈ S, x ∈ X and ξ ∈ H we have Therefore, the family {U(a)} a∈S is a semigroup of linear operators with a common dense domain x∈X V (x)H (throughout this paper denotes the linear space generated by some set, without taking any closure). If K is a positive definite kernel then the previous construction is well-known (see, for instance, [17]). The remaining question, especially in case K is not positive definite, is: what additional conditions on the kernel K should be imposed in order to ensure the boundedness of the operators U(a), a ∈ S? We gave a possible answer in [8], by considering an additional symmetry of the kernel. Consider and this mapping can be extended by linearity to a linear mapping, also denoted by ψ a , from F 0 (X, H) into F 0 (X, H). We say that a positive definite kernel L is φ-bounded provided that for all a ∈ S, ψ a is bounded with respect to the seminorm [·, ·] 1/2 L induced by L on F 0 (X, H). We denote by B + φ (X, H) the set of positive definite φ-bounded kernels on X with values in L(H).
From now on we assume that S is a unital semigroup with involution, that is, there exists a mapping I : S → S such that I 2 = the identity on S, and I(ab) = I(b)I(a) for all a, b ∈ S. The following result was obtained in [8].
Theorem 2.3. Let φ be an action of the unital semigroup S with involution I on the set X and let K be an L(H)-valued hermitian kernel on X with the property that for all x, y ∈ X and a ∈ S. The following assertions are equivalent: for all x ∈ X, a ∈ S. In addition, U(I(a)) = U(a) ♯ for all a ∈ S.

2.4.
Reproducing kernel spaces. We now describe another construction, closely related to the Kolmogorov decomposition of a hermitian kernel. Let K be a hermitian kernel satisfying (2.7) with a Kolmogorov decomposition (V ; K) as in Theorem 2.3. Define Then R is a vector subspace of F (X, H), the class of functions defined on X with values in H. We define a map Φ : K → R by This map is linear and bijective, so that we can define on R the inner product One checks that R is a Kreȋn space with respect to this inner product. Also, Φ is a bounded operator between the Kreȋn spaces K and R, since it is closed and everywhere defined on K, hence it is unitary. Moreover, R is the closure of the linear space generated by the functions g V (y)ξ , y ∈ X and ξ ∈ H. These functions are related to the kernel K as follows: We will write g y,ξ instead of g V (y)ξ , and since g y,ξ (x) = K(x, y)ξ, these functions can be defined without using V . Therefore, the space R has the following reproducing property: We also note that property (2.7) of the kernel K is reflected into a certain symmetry of the elements of R. Thus, we define an operatorŪ (a) ∈ L(R) bȳ where U is the projective representation of S given by Theorem 2.3. We havē On the other hand, for any a, x ∈ X and f ∈ K, and we deduce that the elements of R satisfy the relation Based on this relation we obtain the following result.
Theorem 2.4. Let φ be an action of the unital semigroup S with involution I on the set X and let K be an L(H)-valued hermitian kernel on X with the property that for all x, y ∈ X and a ∈ S. The following assertions are equivalent: for all a ∈ S, x ∈ X and f ∈ R.
Proof. The implication (2) ⇒ (3) was already proved above. In order to prove (3) ⇒ (2) we define the linear mappingV (x) from H into R by the formula: The property (c) shows thatV (x) is a closed operator and by the closed graph theorem we deduce thatV (x) ∈ L(H, R). From (b) and (c) we deduce that (V ; R) is a Kolmogorov decomposition of K. Finally, This completes the proof.

Hankel type kernels
In this section we interpret the invariance property (2.7) as a Hankel condition. To see this, let S = N be the additive semigroup of natural numbers (including 0) and the action φ is given by right translation. If K satisfies (2.7), then K(n, p + m) = K(p + n, m) for m, n, p ∈ N and K is a so-called Hankel kernel. We can extend this example to a noncommutative setting as follows. Let S = F + N be the unital free semigroup on N generators g 1 , . . . , g N with lexicograhpic order ≺. The empty word is the identity element and the length of the word σ is denoted by |σ|. The length of the empty word is 0. There is a natural involution on F + N given by I(g 1 . . . g k ) = g k . . . g 1 as well as a natural action of F + N on itself by juxtaposition, φ(σ, τ ) = στ , σ, τ ∈ F + N . The condition (2.7) means in this case that for β, σ, τ ∈ F + N . It was noticed in [6] that kernels as above appear in connection with orthogonal polynomials in N indeterminates satisfyting the relations With this notation, each element P ∈ P 0 N can be uniquely written as An involution * on P 0 N can be introduced as follows: ; and, in general, if P has the representation as in (3.2) then Thus, P 0 N is a unital, associative, * -algebra over C. A linear functional Z on P 0 N is called hermitian if Z(P * ) = Z(P ) for P ∈ P 0 N . A convenient subclass of hermitian functionals, called GNS functionals, is given by those functionals admitting GNS data. A triplet (π, K, Ω) is called a GNS data associated to Z if π is a hermitian closable representation of P 0 N on a Kreȋn space K and Ω ∈ D(π), the domain of π, such that Z(P ) = [π(P )Ω, Ω] K for P ∈ P 0 N , and P ∈P 0 N π(P )Ω = D(π) (see [2], [13]). The numbers s σ = Z(Y σ ), σ ∈ F + N , are called the moments of Z.
The following is a Hamburger type description of moments.
Proof. This result is just another facet of Theorem 2.1. Assume first that the numbers s σ , σ ∈ F + N , are the moments of a GNS functional on P 0 N . Let (π, K, Ω) be a GNS data associated to Z. Define V : F + N → L(C, K) by the formula: We deduce that for σ, τ ∈ F + N and λ, µ ∈ C, Conversely, let (V, K) be a Kolmogorov decomposition of K. Define Ω = V (∅) and and we can extend π to P 0 N by linearity. Clearly, D(π) is invariant under π(P ), P ∈ P 0 N , and π(P )π(Q) = π(P Q). Also, for k, k ′ ∈ D(π), which shows that the domain of π(Y σ ) ♯ contains D(π) and π(Y σ ) ♯ |D(π) = π(Y I(σ) ) = π(Y * σ ). We can extend this argument and show that the same is true for any P ∈ P 0 N , so that (π, K, Ω) is a GNS data for Z(P ) = [π(P )Ω, Ω] K , P ∈ P 0 N . The moments of Z are As a consequence of the previous result and Theorem 2.2, we deduce a uniqueness condition for the solvability of the Hamburger moment problem for GNS functionals.

Dilations and decomposition of kernels
In this section we show that Theorem 2.3 provides a general framework for a version of the Stinespring theorem and for decompositions of hermitian linear maps. Let A be a unital * -algebra, H a Hilbert space, and let T : A → L(H) be a linear hermitian map. A Stinespring dilation of T is, by definition, a triplet (π, K, B) such that: SD1 K is a Kreȋn space with a fundamental symmetry J, and B ∈ L(H, K); SD2 π : A → L(K) is a selfadjoint (that is, π(a * ) = π(a) ♯ = Jπ(a) * J for all a ∈ A) representation, such that T (a) = B * Jπ(a)B, for all a ∈ A.
If, in addition, SD3 {π(a)BH | a ∈ A} is total in K, then the Stinespring dilation is called minimal. We consider the action φ of A on itself defined by and a hermitian kernel is associated to T by the formula It readily follows that K T is φ-invariant, that is, Since a∈A π(a)BH = x∈A V (x)H it follows that (V, K) is a Kolmogorov decomposition of T . Let us note that, by the definition of V , π(a)V (x) = π(a)π(x * )JB = π(ax * )JB = V (xa * ) a, x ∈ A, and hence, letting U = π, it follows that the Kolmogorov decomposition (V, K) is φ-invariant.
Conversely, let (V, K) be an invariant Kolmogorov decomposition of the hermitian kernel K T , that is, there exists a hermitian representation U : A → L(K) of the multiplicative semigroup A with involution * , such that U(a)V (x) = V (xa * ), a, x ∈ A.
Define π = U and B = JV (1). Since T is linear it follows easily that π is also linear, hence a selfadjoint representation of the * -algebra A on the Kreȋn space K. Then, taking into account that V (a) = U(a * )B for all a ∈ A, it follows T (a) = V (a) * JV (1) = B * U(a * ) * JB = B * JU(a)B, a ∈ A, and since a∈A π(a)BH = x∈A V (x)H we thus proved that (π, K, B) is a minimal Stinespring dilation of T . One easily check that the mapping defined in (4.4) is the inverse of the mapping associating to each invariant Kolmogorov decomposition (V, K) the minimal Stinespring dilation (π, K, B) as above.  (2) T has a minimal Stinespring dilation.
We show now that Wittstock's Decomposition Theorem [20] and Paulsen's Dilation Theorem [16] fit into the framework of invariant Kolmogorov decompositions, more precisely, the following result shows that in case A is a C * -algebra, Schwartz's boundedness condition for hermitian kernels represents an extension of the concept of completely bounded map. We use standard terminology from the theory of operator spaces, e.g. see [16] and [10]. (1) T is completely bounded.
(2) ⇒ (3) Since S is completely positive, the kernel K S is positive definite and satisfies −K S ≤ K T ≤ K S . Also, By Theorem 4.3 in [8], there exists a Kolmogorov decomposition (V, K) of K T and a fundamentally reducible representation U of U(A) on K such that One can check that for all x, y ∈ A and a ∈ U(A). For x ∈ A define T ± (x) = K ± (x, 1).
Then T ± (x) = BJπ(x * ) * J ± V (1) are linear maps on A and for x ∈ A, y ∈ U(A), we get Since K T ± and K ± are antilinear in the second variable and A is the linear span of U(A), it follows that for all x, y ∈ A. This implies that T ± are disjoint completely positive maps such that T = T + − T − . The implication (3)⇒(1) follows from Theorem 2.3.
Theorem 4.3 suggests how to extend the concept of decomposition to arbitrary kernels. In the following we repeatedly use the following observation: if L is a positive definite kernel on X and with values in L(H), and T ∈ L(H), then the kernel T * LT * is positive definite. Thus, if K is a hermitian kernel, L is a positive definite kernel, both on a set X and with entries in L(H), for some Hilbert space H, then for any x, y ∈ X  , and a contraction U on K such that K(x, y) = V (x) * UV (y) for all x, y ∈ X, and the set {V (x)h | x ∈ X, h ∈ H} is total in K.
Proof. If K(x, y) = V (x) * UV (y) for x, y ∈ X and some contraction U, then consider the positive definite kernels L 1 (x, y) = L 2 (x, y) = V (x) * V (y). We deduce that Since U is a contraction, the matrix I U U * I is positive. Next, take x 1 , . . . , x n ∈ X and after reshuffling, the matrix can be written in the form

I I I
which shows that the kernel L 1 (x, y) K(x, y) K * (x, y) L 2 (x, y) is positive definite.
Conversely, assume that K is decomposable. We consider the real and imaginary parts of K, (4.5) K 1 (x, y) = 1 2 (K(x, y) + K * (x, y)), therefore K 1 , K 2 are hermitian kernels and K = K 1 + iK 2 . Since K is decomposable, there exist positive definite kernels L 1 and L 2 such that the kernel L 1 K K * L 2 is positive definite. Therefore, is also a positive definite kernel, which implies that − 1 2 (L 1 + L 2 ) ≤ K 1 ≤ 1 2 (L 1 + L 2 ). Similarly, is a positive definite kernel, which gives that − 1 2 (L 1 + L 2 ) ≤ K 2 ≤ 1 2 (L 1 + L 2 ). Since is a positive definite kernel, we deduce from Theorem 2.1 that both K 1 and K 2 have Kolmogorov decompositions, say where V i : X → L(H, K i ) and J i are fundamental symmetries on K i , i = 1, 2. It follows that : H → K 1 ⊕ K 2 and let K be the closure in K 1 ⊕ K 2 of the linear space generated by the elements of the form V ′ (x)h, x ∈ X and h ∈ H. Finally, define V (x) = P K V ′ (x), where P K denotes the orthogonal projection of K 1 ⊕ K 2 onto K. Then the set {V (x)h | x ∈ X, h ∈ H} is total in K and K(x, y) = V (x) * P K J 1 0 0 iJ 2 P K V (y). The operator U = P K J 1 0 0 iJ 2 P K is a contraction and the proof is concluded.

Holomorphic kernels
There are many examples of hermitian kernels which are holomorphic on some domain in the complex plane, see for instance [5]. In all these cases it is known that the kernels are associated with reproducing kernel Kreȋn (Hilbert) spaces, and D. Alpay in [3] proved a general result in this direction. Our goal is to extend the result in [3] to the multi-variable case. In view of the transcription between Kolmogorov decompositions and reproducing kernel spaces, e.g. see the invariant version in Theorem 2.4, we actually show that the orginal idea of the proof in [3], which goes back to [18], can be adapted to this multi-variable setting.
We first review the well-known example of the Szegö kernel (see [4]). Let G be a Hilbert space and denote by B r (ξ) the open ball of radius r and center ξ, B r (ξ) = {η ∈ G | η − ξ < r}. We write B r instead of B r (0). For ξ, η ∈ B 1 , define and note that S is a positive definite kernel on B 1 . We now describe its Kolmogorov decomposition. Let be the Fock space associated to G, where G ⊗0 = C and G ⊗n is the n-fold tensor product of G with itself. Let (5.2) P n = (n!) −1 π∈Snπ be the orthogonal projection of G ⊗n onto its symmetric part, wherê and π is an element of the permutation group S n on n symbols. The symmetric Fock . For ξ ∈ B 1 set ξ ⊗0 = 1 and let ξ ⊗n denote the n-fold Hence n≥0 ξ ⊗n ∈ F s (G) and we can define the mapping V S from B 1 into F s (G), Lemma 5.1. The pair (V S , F s (G)) is the Kolmogorov decomposition of the kernel S.
The reproducing kernel Hilbert space associated to the Szegö kernel S, see (5.1), is given by the completion of the linear space generated by the functions s η = S(·, η), η ∈ B 1 , with respect to the inner product defined by s η , s ξ = S(ξ, η).
Note that there exists a unitary operator F from the reproducing kernel Hilbert space associated to the Szegö kernel S onto F s (G) such that F s ξ = V S (ξ), ξ ∈ B 1 .
We now explore the fact that S is a holomorphic kernel. We use the terminology and results from [15], [9] for holomorphic functions in infinite dimensions. Thus, we say that a function f defined on the open subset O of G is holomorphic if f is continuous on O and for all η ∈ O, ξ ∈ G, the mapping λ → f (η + λξ) is holomorphic on the open set {λ ∈ C | η + λξ ∈ O}. One easily check that V S is holomorphic on B 1 , therefore S(ξ, ·) is holomorphic on B 1 for each fixed ξ ∈ B 1 . We also notice that the reproducing kernel Hilbert space associated to S consists of anti-holomorphic functions on B 1 . It is somewhat more convenient to replace this space by a Hilbert space of holomorphic functions on B 1 . Thus, define the holomorphic function a ξ = S(ξ, ·) on B 1 for each ξ ∈ B 1 , and let H 2 (G) denote the completion of the linear space generated by the functions a ξ , ξ ∈ B 1 , with respect to the inner product defined by a ξ , a η = S(ξ, η), ξ, η ∈ B 1 .
We notice that H 2 (G) is an anti-unitary copy of the reproducing kernel Hilbert space of S.
We say that a hermitian kernel K, defined on an open subset O of G, is holomorphic Theorem 5.2. Let K be a holomorphic hermitian kernel on B r , with r > 0, and valued in L(G) for some Hilbert space G. Then there exists 0 < r ′ ≤ r and a Kolmogorov decomposition Proof. Let K be a holomorphic kernel on B r , r > 0. Since K is hermitian, it follows that K(·, η) is anti-holomorphic on B r for each η ∈ B r . It is convenient to reformulate this fact as follows. Let {e α } α∈A be an orthonormal basis for G. We define the mapping ξ = α∈A ξ, e α e α → α∈A ξ, e α e α = ξ * , so that the function f (ξ, η) = K(ξ * , η) is separately holomorphic on B r × B r . By Hartogs' Theorem ( [15], Theorem 36.8), f is holomorphic on B r × B r . By ( [15], Proposition 8.6), f is locally bounded. We first suppose that r > 1. Hence there exist 1 < ρ < r and C > 0 such that: uniformly on B ρ , where each p m , m ≥ 0, is an m-homogeneous continuous polynomial on G × G. That is (see [15] or [9], Chapter 1), there exists a continuous linear mapping A m on P m ((G × G) ⊗m ) such that for all ξ, η ∈ G.
This implies that P is selfadjoint and let P = P + − P − be its Jordan decomposition, where P ± are positive operators on H 2 (G). Then K can be written as the difference of two positive definite kernels. By [18] and Theorem 2.3, K has a Kolmogorov decomposition, and it is easy to see that this Kolmogorov decomposition has the required holomorphy property. In case r ≤ 1, a scalling argument as in [3] concludes the proof.
As mentioned, the above proof is based on the same idea as in [3], which goes back to [18]. An interesting aspect of this idea is that once again the Szegö kernel S has a certain universality property, that is, any holomorphic kernel on B r , r > 1, is the image of S through a bounded selfadjoint operator on H 2 (G). A different kind of universality property of S, related to the solution of the Nevanlinna-Pick interpolation problem, was established in [1].
Finally we apply Theorem 5.2 to show that non-hermitian holomorphic kernels are decomposable. Corollary 5.3. Let K be a holomorphic kernel on B r , with r > 0, and valued in L(G) for some Hilbert space G. Then there exists 0 < r ′ ≤ r, a Hilbert space K, a holomorphic mapping V : B r ′ → L(G, K), and a contraction U on K, such that K(ξ, η) = V (ξ) * UV (η) for all ξ, η ∈ B r ′ , and the set {V (ξ)h | ξ ∈ B r ′ , h ∈ G} is total in K.
Proof. We consider the real part K 1 (4.5) and, respectively, the imaginary part K 2 (4.6) of K and note that both are holomorphic hermitian kernels. Then we apply Theorem 5.2 to produce holomorphic Kolmogorov decompositions of K 1 and K 2 on a possibly smaller, but nontrivial, ball B r ′ in G and, proceeding as in the proof of Theorem 4.4, we obtain the decomposition of K as required.