Nevanlinna representations in several variables

We generalize two integral representation formulae of Nevanlinna to functions of several variables. We show that for a large class of analytic functions that have non-negative imaginary part on the upper polyhalfplane there are representation formulae in terms of densely defined self-adjoint operators on a Hilbert space. We introduce three types of structured resolvent of a self-adjoint operator and identify four different types of representation in terms of these resolvents. We relate the types of representation that a function admits to its growth at infinity.


Introduction
In a classic paper [19] of 1922 R. Nevanlinna solved the problem of the determinacy of solutions of the Stieltjes moment problem. En route he proved several other theorems that have since been influential; in particular, the following theorem, which characterizes the Cauchy transforms of positive finite measures µ on R, has had a profound impact on the development of modern analysis. Let P denote the Pick class, that is, the set of analytic functions on the upper halfplane, Π def = {z ∈ C : Im z > 0}, that have non-negative imaginary part on Π. A closely related theorem, also referred to in the literature as Nevanlinna's Representation, provides an integral representation for a general element of P.  for all z ∈ Π. Moreover, for any h ∈ P, the numbers a ∈ R, b ≥ 0 and the measure µ ≥ 0 in the representation (1.3) are uniquely determined.
What are the several-variable analogs of Nevanlinna's theorems? In this paper we shall propose four types of Nevanlinna representation for various subclasses of the n-variable Pick class P n , where P n is defined to be the set of analytic functions h on the polyhalfplane Π n such that Im h ≥ 0. In addition, we shall present necessary and sufficient conditions for a function defined on Π n to possess a representation of a given type in terms of asymptotic growth conditions at ∞.
The integral representation (1.1) of those functions in the Pick class that satisfy condition (1.2) can be written in the form where A is the operation of multiplication by the independent variable on L 2 (µ) and 1 is the constant function 1. We propose that an appropriate n-variable analog of the Cauchy transform is the formula where H is a Hilbert space, A is a densely defined self-adjoint operator on H, Y 1 , . . . , Y n are positive contractions on H summing to 1 and v is a vector in H. Theorem 1.6 below characterizes those functions on Π n that have a representation of the form (1.4). To state this theorem we require a notion based on the following classical result of Pick [21]. Theorem 1.3. A function h defined on Π belongs to P if and only if the function A defined on Π × Π by A(z, w) = h(z) − h(w) z − w is positive semidefinite, that is, for all n ≥ 1, z 1 , . . . , z n ∈ Π, c 1 , . . . , c n ∈ C, A(z j , z i )c i c j ≥ 0.
The following theorem, proved in [2], leads to a generalization of Theorem 1.3 to two variables. The Schur class of the polydisc, denoted by S n , is the set of analytic functions on the polydisc D n that are bounded by 1 in modulus.
By way of the transformations there is a one-to-one correspondence between functions in the Schur and Pick classes. Under these transformations, Theorem 1.4 becomes the following generalization of Pick's theorem to two variables.
In the light of Theorems 1.3 and 1.5 we define the Loewner class L n to be the set of analytic functions h on Π n with the property that there exist n positive semidefinite functions A 1 , . . . , A n on Π n such that (1.8) h for all z, w ∈ Π n . The Loewner class L n played a key role in [4], which gave a generalization to several variables of Loewner's characterization of the one-variable operator-monotone functions [18]. As the following theorem makes clear, L n also has a fundamental role to play in the understanding of Nevanlinna representations in several variables. In the cases when n = 1 and n = 2, Theorems 1.3 and 1.5 assert that that L n = P n , and so for n = 1, Theorem 1.6 is Nevanlinna's classical Theorem 1.1, and when n = 2, Theorem 1.6 is a straightforward generalization of that result to two variables. When there are more than two variables, it is known that the Loewner class is a proper subset of the Pick class, L n = P n [20,23]. Nevertheless, Nevanlinna's result survives as a theorem about the representation of elements of L n . Other than the work in [12] very little is known about the representation of functions in P n for three or more variables.
For a function h on Π n , we call the formula (1.4) a Nevanlinna representation of type 1. Thus, Theorem 1.6 can be rephrased as the assertion that h has a Nevanlinna representation of type 1 if and only if h ∈ L n and h satisfies condition (1.9). Somewhat more complicated representation formulae are needed to generalize Theorem 1.2. We identify three further representation formulae, of increasing generality, and show that every function in L n has a representation of one or more of the four types.
For a function h defined on Π n , we refer to a formula (1.10) h(z 1 , . . . , for some real a, some self-adjoint operator A and some vector v, where Y 1 , . . . , Y n are operators as in equation (1.4) above and z Y = z 1 Y 1 + · · · + z n Y n . Finally, Nevanlinna representations of type 4 are given by the formula where M (z) is an operator of the form acting on an orthogonal direct sum of Hilbert spaces N ⊕ M. In (1.12), v is a vector in N ⊕ M. In (1.13), A is a densely-defined self-adjoint operator acting on M and z P is the operator acting on N ⊕ M via the formula where P 1 , . . . , P n are pairwise orthogonal projections acting on N ⊕ M that sum to 1. A weaker, "generic" version of Theorem 1.9 appeared in [4, Theorem 6.9], where it was used to show that elements in L n are locally operator-monotone.
It turns out that for 1 ≤ k ≤ 4, if h is a function on Π n and h has a Nevanlinna representation of type k, then for k ≤ j ≤ 4, h also has a Nevanlinna representation of type j. Thus, it is natural to define the type of a function in L n to be the smallest k such that h has a Nevalinna representation of type k.
For h ∈ L n the type of h can be characterized in function-theoretic terms through the use of a geometric idea due to Carathéodory. A carapoint for a function ϕ in the Schur class S n is a point τ ∈ T such that lim inf Carathéodory introduced this notion in one variable in [10], along the way to refining earlier results of Julia [15]. The following was Carathéodory's main result; the notation λ nt → τ means that λ tends nontangentially to τ . Theorem 1.10. Let ϕ ∈ S 1 , τ ∈ T. If τ is a carapoint for ϕ, then ϕ is nontangentially differentiable at τ , that is, there exist values ϕ(τ ) and ϕ ′ (τ ) such that In particular, if τ is a carapoint for ϕ then there exists a unique point ϕ(τ ) ∈ T such that ϕ(λ) → ϕ(τ ) as λ nt → τ .
In several variables, carapoints have been studied in [1,14,3]. The strong conclusion of nontangential differentiability is lost in several variables; however, at a carapoint τ , there still exists a unimodular nontangential limit ϕ(τ ).
As the point χ = (1, . . . , 1) is transformed to the point ∞ = (∞, . . . , ∞) by (1.6), it is natural to say that a function h ∈ L n has a carapoint at ∞ if the associated Schur function ϕ, given by the transformation in (1.7), has a carapoint at χ, and in that case to define h(∞) by The connection between carapoints and function types is given in the following theorem. The paper is structured as follows. As is clear from the formulae used to define the various Nevanlinna representations, Nevanlinna representations are generalizations of the resolvent of a self-adjoint operator. These structured resolvents, studied in Sections 2 and 3, are analytic operator-valued functions on the polyhalfplane Π n with non-negative imaginary part, fully analogous to the familiar resolvent operator. There are also structured resolvent identities for them, studied in Section 10 of the paper.
In modern texts Nevanlinna's representation is derived from the Herglotz Representation with the aid of the Cayley transform [17,11]. In Section 4 we introduce the n-variable strong Herglotz class and then prove Theorem 1.12 by applying the Cayley transform to Theorem 1.8 of [2].
In Section 5 we derive the Nevanlinna representations of type 3, 2, and 1, we show how they arise naturally from the underlying Hilbert space geometry and we prove slight strengthenings of Theorems 1.6, 1.7 and 1.8. In Section 6 we give function-theoretic conditions for a function h ∈ L n to possess a representation of a given type.
In Section 7 we introduce the notion of carapoints for functions in the Pick class and in Section 8 we establish the criteria in Theorem 1.11 for the type of a function using the language of carapoints.
In Section 9 we give the growth estimates for functions in L n that flow from our analysis of structured resolvents, and in Section 10 we present resolvent identities for structured resolvents.
Results related to ours from a system-theoretic perspective have been obtained in ongoing work of J. A. Ball and D. Kalyuzhnyi-Verbovetzkyi [7,8]. See also [6], where Krein space methods are applied to similar problems.

Structured resolvents of operators
The resolvent operator (A − z) −1 of a densely defined self-adjoint operator A on a Hilbert space plays a prominent role in spectral theory. It has the following properties.
(1) It is an analytic bounded operator-valued function of z in the upper halfplane Π; (2) it satisfies the growth estimate (A − z) −1 ≤ 1/ Im z for z ∈ Π; (3) (A − z) −1 has non-negative imaginary part for all z ∈ Π; (4) it satisfies the "resolvent identity". Here we are interested in several-variable analogs of the resolvent. These will again be operator-valued analytic functions with non-negative imaginary part, but now on the polyhalfplane Π n . Because of the additional complexities in several variables we encounter three different types of resolvent; all of them have the four listed properties, with very slight modifications, and therefore deserve the name structured resolvent.
For any Hilbert space H, a positive decomposition of H will mean an n-tuple Y = (Y 1 , . . . , Y n ) of positive contractions on H that sum to the identity operator. For any z = (z 1 , . . . , z n ) ∈ C n and any n-tuple T = (T 1 , . . . , T n ) of bounded operators we denote by z T the operator j z j T j . Here each T j is a bounded operator from H 1 to H 2 , for some Hilbert spaces H 1 , H 2 , so that z T is also a bounded operator from H 1 to H 2 .
Definition 2.1. Let A be a closed densely defined self-adjoint operator on a Hilbert space H and let Y be a positive decomposition of H. The structured resolvent of A of type 2 corresponding to Y is the operator-valued function The following observation is essentially [4, Lemma 6.25].
Proposition 2.2. For A and Y as in Definition 2.1 the structured resolvent (A − z Y ) −1 is well defined on Π n and satisfies, for all z ∈ Π n , The range of the bounded operator (A − z Y ) −1 is of course D(A), the domain of A.
Proof. For any vector ξ in the domain of A, Thus A − z Y has lower bound min j Im z j > 0, and so has a bounded left inverse. A similar argument with z replaced byz shows that (A − z Y ) * also has a bounded left inverse, and so A − z Y has a bounded inverse and the inequality (2.1) holds.
Resolvents of type 2 are the simplest several-variable analogues of the familiar onevariable resolvent but they are not sufficient for the analysis of the several-variable Pick class. To this end we introduce two further generalizations. Let us first recall some basic facts about closed unbounded operators. Lemma 2.3. Let T be a closed densely defined operator on a Hilbert space H, with domain D(T ). The operator 1 + T * T is a bijection from D(T * T ) to H, and the operators are everywhere defined and contractive on H. Moreover B is self-adjoint and positive, and ran C ⊂ D(T * ).
Proof. All these statements are proved in [22,Sections 118,119], although the final statement about ran C is not explicitly stated. We must show that for all v ∈ H there exists y ∈ H such that, for all h ∈ H, T h, Cv = h, y .
It is straightforward to check that this relation holds for y = v − Bv, and so ran C ⊂ D(T * ).
Definition 2.4. Let A be a closed densely defined self-adjoint operator on a Hilbert space H and let Y be a positive decomposition of H. The structured resolvent of A of type 3 corresponding to Y is the operator-valued function M : Π n → L(H) given by We denote the ℓ 1 norm on C n by · 1 . Note that z Y ≤ z 1 for all z ∈ C n and all positive decompositions Y . M (z) ≤ (1 + 2 z 1 ) 1 + 1 + z 1 min j Im z j .
The following alternative formula for the structured resolvent of type 3, valid on the dense subspace D(A) of H, allows us to show that Im M (z) ≥ 0.
Moreover, for every v ∈ D(A), and therefore

This establishes equation (2.7).
The expression (2.8) follows from equation (2.7) since By equation (2.8) we have, for any z ∈ Π n and v ∈ D(A), and hence, by equation (2.2),  In the case of bounded A there is yet another expression for the structured resolvent of type 3.
Remark 2.9. In the case of unbounded A the expression (2.11) for M (z) is valid wherever it is defined, but it is not to be expected that this will be a dense subspace of H in general.
Here are two examples of structured resolvents of type 3, one on C 2 and one on an infinite-dimensional space. Then Example 2.11. Let H = L 2 (R), let A be the operation of multiplication by the independent variable t and let Y = (P, Q) where P, Q are the orthogonal projection operators onto the subspaces of even and odd functions respectively in L 2 . Thus It follows that z Y and z Y ′ commute with A 2 , and it may be checked that A straightforward calculation now shows that the structured resolvent M (z) of A corresponding to Y is given by for all z ∈ Π 2 , f ∈ L 2 (R) and t ∈ R. In particular, we note for future use that if f is an even function,

The matricial resolvent
The third and last form of structured resolvent that we consider has a 2 × 2 matricial form. As will become clear, this extra complication is needed for the description of the most general type of function in the several-variable Loewner class.
By an orthogonal decomposition of a Hilbert space H we shall mean an n-tuple P = (P 1 , . . . , P n ) of orthogonal projection operators with pairwise orthogonal ranges such that n j=1 P j is the identity operator.
is a bounded operator defined on all of H, and For any z ∈ C n , Consider the third and fourth factors in the product on the right hand side of equation (3.1); the product of these two factors is well defined as an operator on H since (1 − iA) −1 maps M to D(A). It is even a bounded operator, since, by virtue of equation (3.4), we can immediately see that the operator (3.5) is bounded. We can get an estimate of the norm of the operator matrix (3.5) if we replace each of the four operator entries by an upper bound for its norm. We find that Now consider the second factor in the definition (3.1) of M (z). We find that which maps H into N ⊕ D(A). Hence the product of the first two factors in the product on the right hand side of equation (3.1) is On combining the estimates (3.9) and (3.6) we obtain the bound (3.2) for M (z) .
Notice in particular that the (2, 2) entry (that is, the compression of M (z) to M) is the structured resolvent of A of type 3 corresponding to Y , the compression of P to M, as in equation (2.3). We shall also refer to M (z) as the matricial resolvent of A with respect to P . The important property that Im M (z) ≥ 0 is not at once apparent from the formula (3.1); as with structured resolvents of type 3, there are alternative formulae from which this property is more easily shown. Once again the alternatives suffer the minor drawback that they give M (z) only on a dense subspace of H.
for all z ∈ Π n . Moreover, for all z, w ∈ Π n , Proof. By Lemma 2.3 the operators (1 + A 2 ) −1 and We claim that, as operators on N ⊕ D(A), (3.14) We have This is an identity between operators on H, in both cases a composition H → N ⊕D(A) → H, and moreover the first factor on the left hand side and the second factor on the right hand side are invertible, from N ⊕ D(A) to H and from H to N ⊕ D(A) respectively. We may pre-and post-multiply appropriately to obtain equation (3.14), but note that the equation is then only valid as an identity between operators on N ⊕ D(A).
On combining equations (3.1) and (3.14) we deduce that we deduce further that which proves equation (3.10). It is straightforward to verify that and so on suitably pre-and post-multiplying equation (3.16), we obtain equation (3.11).
To prove equation (3.12), check first that We have seen that S(z) is invertible for any z ∈ Π n , so that W (z) is a bounded operator on H. Clearly Here The next result shows that the matricial resolvent belongs not just to the operator Pick class, but to the smaller operator Loewner class. Proof. The identity (3.13) shows that such a relation holds on N ⊕D(A); we must extend it to all of H. Write P j as an operator matrix with respect to the decomposition H = N ⊕M, as in equation (3.3). Then z P has the matricial expression (3.4). For z ∈ Π n let Then F ♯ (z) is an operator from N ⊕ D(A) to H, and we find that  Proof. In the notation of Proposition 3.5, on choosing w = z in equation (3.20) and dividing by 2i we obtain the relation Here is a concrete example of a matricial resolvent.
Example 3.7. The function is the matricial resolvent corresponding to

Nevanlinna representations of type 4
In this section we derive a multivariable analog of the most general form of Nevanlinna representation for functions in the one-variable Pick class (Theorem 1.2). We start with a multivariable Herglotz theorem [2, Theorem 1.8]. We shall say that an analytic operatorvalued function F on D n is a Herglotz function if Re F (λ) ≥ 0 for all λ ∈ D n . For present purposes we need the following modification of the notion. In [2] these functions were called F n -Herglotz functions. The class of strong Herglotz functions has also been called the Herglotz-Agler class (for example [16,8]). It is clear that every strong Herglotz function is a Herglotz function, and in the cases n = 1 and 2 the converse is also true [2].
Conversely, every function F : D n → L(K) expressible in the form (4.1) for some H, P, V and U with the stated properties is a strong Herglotz function and satisfies F (0) = 1.
Note that λ P = j λ j P j has operator norm at most λ ∞ < 1 for λ ∈ D n , and hence equation (4.1) does define F as an analytic operator-valued function on D n .
On specialising to scalar-valued functions in the n-variable Herglotz class we obtain the following consequence.
Conversely, for any H, L, P, a and v with the properties described, equation (4.2) defines f as an n-variable strong Herglotz function.
Again, the right hand side of equation (4.2) is an analytic function of λ ∈ D n since is a bounded operator and is analytic in λ.
for all z ∈ Π n , where M (z) is the structured resolvent of A of type 4 corresponding to P (given by the formula (3.1)).
We wish to convert Corollary 4.3 to a representation theorem for suitable analytic functions on Π n . The fact that the corollary only applies to strong Herglotz functions results in representation theorems for a subclass of the Pick class P n . Recall from the introduction: Definition 4.5. The Loewner class L n is the set of analytic functions h on Π n with the property that there exist n positive semi-definite functions A 1 , . . . , A n on Π n , analytic in the first argument, such that We can now prove Theorem 1.9 from the introduction: a function h defined on Π n has a Nevanlinna representation of type 4 if and only if h ∈ L n .
Proof. Let h ∈ L n . Define an n-variable Herglotz function f : D n → C by When λ ∈ D n the point z belongs to Π n , and so f (λ) is well defined, and since Im h(z) ≥ 0 we have Re f (λ) ≥ 0, so that f is indeed a Herglotz function. In fact f is even a strong Herglotz function: since h ∈ L n , the function ϕ ∈ S n corresponding to h lies in the Schur-Agler class of the polydisc, and so f = (1 + ϕ)/(1 − ϕ) is a strong Herglotz function. By Corollary 4.3 there exist a real number a, a Hilbert space H, a vector v ∈ H, a unitary operator L on H and an orthogonal decomposition P on H such that, for all z ∈ Π n , Here and in the rest of this section z, λ are identified with the operators z P , λ P on H, and in consequence the relation Since L is unitary on H and λ ∈ D n , the operator M (z) is bounded on H for every z ∈ Π n and, by equation (4.6), we have for all z ∈ Π 2 . Theorem 1.9 will follow provided we can show that M (z) is given by equation (3.1) for a suitable self-adjoint operator A.
Observe that Formally we may now write but whereas equation (4.10) is a relation between bounded operators defined on all of H, equation (4.11) involves unbounded, partially defined operators and we must verify that the product of operators on the right hand side is meaningful. Let Since   .7)). We may therefore take inverses in the equation to obtain as operators on N ⊕ D(A). Similar reasoning applies to the equation On combining equations (4.10), (4.14) and (4.15) we obtain Premultiply this equation by 2 and postmultiply by 1 2 to deduce that M (z) is indeed the structured resolvent of A of type 4 corresponding to P , as defined in equation (3.1). Thus the formula (4.8) is a Nevanlinna representation of h of type 4.
Conversely, let h ∈ L n have a type 4 representation (4.3). By Proposition 3.5 there exists an analytic operator-valued function F : Π n → L(H) such that, for all z, w ∈ Π n , The A j are clearly positive semidefinite on Π n , and hence h belongs to the Loewner class L n .

Nevanlinna representations of types 3, 2 and 1
Nevanlinna representations of type 4 have the virtue of being general for functions in L n , but they are undeniably cumbersome. In this section we shall show that there are three simpler representation formulae, corresponding to increasingly stringent growth conditions on h ∈ L n .
In Nevanlinna's one-variable representation formula of Theorem 1.2, it may be the case for a particular h ∈ P that the bz term is absent. The analogous situation in two variables is that the space N in a type 4 representation may be zero. Equivalently, in the corresponding Herglotz representation, the unitary operator L does not have 1 as an eigenvalue. This suggests the following notion.
Thus h has a type 3 representation if h(z) = a+ M (z)v, v where M (z) is the structured resolvent of A of type 3 corresponding to Y , as given by equation (2.3).
In [5] the authors derived a somewhat simpler representation which can also be regarded as an analog of the case b = 0 of Nevanlinna's one-variable formula (5.1).
Definition 5.2. A Nevanlinna representation of type 2 of a function h on Π n consists of a Hilbert space H, a self-adjoint densely defined operator A on H, a positive decomposition Y of H, a real number a and a vector α ∈ H such that, for all z ∈ Π n This means of course that, for all z ∈ Π n , where M (z) is the structured resolvent of A of type 2 corresponding to Y (compare equation (2.1)). We wish to understand the relationship between type 3 and type 2 representations.
Proposition 5.3. If h ∈ P n has a type 2 representation then h has a type 3 representation. Conversely, if h ∈ P n has a type 3 representation as in equation (5.2) with the additional property that v ∈ D(A) then h has a type 2 representation.
Proof. Suppose that h ∈ P n has the type 2 representation for some a 0 ∈ R, positive decomposition Y and α ∈ H. We must show that h has a representation of the form (5.2) for some a ∈ R and v ∈ H. By Proposition 2.6, it suffices to find a ∈ R and v ∈ D(A) such that To this end, let C = A(1 + A 2 ) −1 and let (5.4) a = a 0 + Cα, α .
Since 1 + iA is invertible on H and ran(1 + iA) −1 ⊂ D(A) we may define Then as required. Thus h has a type 3 representation. Conversely, let h have a type 3 representation (5.2) such that v ∈ D(A), that is where a ∈ R and M is the structured resolvent of A of type 3 corresponding to Y , as in equation (2.3). Since v ∈ D(A) we may define the vector α def = (1 + iA)v ∈ H, and furthermore, by Proposition 2.6, where a 0 ∈ R is given by equation (5.4). Thus h has a representation of type 2.
A special case of a type 2 representation occurs when the constant term a in equation (5.3) is 0. In one variable, this corresponds to Nevanlinna's characterization of the Cauchy transforms of positive finite measures on R. Accordingly we define a type 1 representation of h ∈ L n to be the special case of a type 2 representation of h in which a = 0 in (5.3).
Definition 5.4. An analytic function h on Π n has a Nevanlinna representation of type 1 if there exist a Hilbert space H, a densely defined self-adjoint operator A on H, a positive decomposition Y of H and a vector α ∈ H such that, for all z ∈ Π n , A representation of type 1 is obviously a representation of type 2. The following proposition is an immediate corollary of Proposition 5.3.
Proposition 5.5. A function h ∈ L n has a type 1 representation if and only if h has a type 3 representation as in equation (5.2) with the additional properties that v ∈ D(A) and For consistency with our earlier terminology for structured resolvents and representations we should have to define a structured resolvent of type 1 to be the same as a structured resolvent of type 2. We refrain from making such a confusing definition.
We conclude this section by giving examples of the four types of Nevanlinna representation in two variables.
(2) Likewise is a representation of type 2.
(3) Let where we take the branch of the square root that is analytic in C \ [0, ∞) with range Π. We claim that h ∈ P 2 and that h has the type 3 representation where M (z) is the structured resolvent of type 3 given in Example 2.11 and v(t) = 1/ π(1 + t 2 ). To see this, let h be temporarily defined by equation (5.8).
Since v is an even function in L 2 (R), equation (2.12) tells us that Since the denominator is an even function of t, the integrals of all the odd powers of t in the numerator vanish, and we have, provided z 1 z 2 = −1, and so we find that h is indeed given by equation (5.7) in the case that z 1 z 2 = −1. When . Thus equation (5.8) is a type 3 representation of the function h given by equation (5.7). This function is constant and equal to i on the diagonal z 1 = z 2 .
(4) The function clearly belongs to P 2 . It has the representation of type 4 where M (z) is the matricial resolvent given in Example 3.7 and v = 1 √ 2 1 0 .
We claim that each of the above representations is of the simplest available type for the function in question; for example, the function h in part (4) does not have a Nevanlinna representation of type 3. To prove this claim (which we shall do in Example 8.2 below) we need characterizations of the types of functions -the subject of the next two sections.

Asymptotic behavior and types of representations
In this section we shall give function-theoretic conditions for a function in L n to have a representation of a given type. These conditions will be in terms of the asymptotic behavior of the function at ∞.
Every function in L n has a type 4 representation, by Theorem 1.9. Let us characterize the functions that possess a type 3 representation. We denote by χ the vector (1, . . . , 1) of ones in C n . The following statement contains Theorem 1.8.
Theorem 6.1. The following three conditions are equivalent for a function h ∈ L n .
(1) The function h has a Nevanlinna representation of type 3; (2) (3) Proof. (1)⇒(3) Suppose that h has a Nevanlinna representation of type 3: Let ν be the scalar spectral measure for A corresponding to the vector v ∈ H. By the Spectral Theorem Since Im h(isχ) = 1 + t 2 s 2 + t 2 dν(t). The integrand decreases monotonically to 0 as s → ∞ and so, by the Monotone Convergence Theorem, equation (6.2) holds.
Thus, for s > 0, since once again (isχ) P = is, Let the projections of v onto N , N ⊥ be v 1 , v 2 respectively. Then It follows that v 1 = 0. Let the compression of the projection P j to N ⊥ be Y j : then Y = (Y 1 , . . . , Y n ) is a positive decomposition of N ⊥ , and the compression of z P to N ⊥ is z Y . By Remark 3.2 the (2,2) block M 22 (z) in M (z) is Since v 1 = 0 it follows that which is the desired type 3 representation of h. Hence (2)⇒(1).
In [8] it is shown that condition (3) in the above theorem is also a necessary and sufficient condition that −ih have a Π n -impedance-conservative realization.
Type 2 representations were characterized by the following theorem in [5] in the case of two variables. The following result, which contains Theorem 1.7, shows that the result holds generally.
Theorem 6.2. The following three conditions are equivalent for a function h ∈ L n .

Proof. (1)⇒(3) Suppose that h has the type 2 representation
for a suitable real a, self-adjoint A, positive decomposition Y and vector v. Let ν be the scalar spectral measure for A corresponding to the vector v. Then, for s > 0, A− (isχ) Y = A − is and so The integrand is positive and increases monotonically to 1 as s → ∞. Hence, by the Dominated Convergence Theorem Im h(isχ) = 0.
By Theorem 6.1 h has a type 3 representation (6.3) for suitable a ∈ R, H, A, Y and v ∈ H. Let ν be the scalar spectral measure for A corresponding to the vector v. Then for s > 0 As s → ∞ the integrand increases monotonically to 1 + t 2 . Condition (2) now implies that It follows that v ∈ D(A). Hence, by Proposition 5.3, h has a representation of type 2.
In [5] we proved Theorem 6.2 for n = 2 using a different approach from the present one. From this theorem the characterization of type 1 representations follows just as in the one-variable case. We obtain a strengthening of Theorem 1.6. Theorem 6.3. The following three conditions are equivalent for a function h ∈ L n .
(1)⇒(3) Suppose that h has a type 1 representation as in equation (5.6) for some H, A, Y and v. Then
Let ν be the scalar spectral measure for A corresponding to the vector α ∈ H. Then Re sh(isχ) = st t 2 + s 2 dν(t), Im sh(isχ) = The integrand in the first integral tends pointwise in t to 0 as s → ∞, and by the inequality of the means it is no greater than 1 2 ; thus the integral tends to 0 as s → ∞ by the Dominated Convergence Theorem. The integrand in the second integral increases monotonically to 1 as s → ∞. Thus Re sh(isχ) → 0, Im sh(isχ) → α 2 as s → ∞.
(2)⇒(1) Suppose that h satisfies condition (6.5) of Theorem 6.2. Therefore h has a representation of type 2, say It remains to show that a = 0. The inequality (6.8) implies that there exists a sequence s n tending to ∞ such that h(is n χ) → 0. But Re h(is n χ) = a + A(A 2 + s 2 n ) −1 α, α → a. Hence a = 0 and h has a type 1 representation. This establishes (2)⇒(1).

Carapoints at infinity
How can we recognise from function-theoretic properties whether a given function in the n-variable Loewner class admits a Nevanlinna representation of a given type? In the preceding section it was shown that it depends on growth along a single ray through the origin. In this section we describe the notion of carapoints at infinity for a function in the Pick class, and in the next section we shall give succinct criteria for the four types in the language of carapoints.
Carapoints (though not with this nomenclature) were first introduced by Carathéodory in 1929 [10] for a function ϕ on the unit disc, as a hypothesis in the "Julia-Carathéodory Lemma". For any τ ∈ T, a function ϕ in the Schur class satisfies the Carathéodory condition at τ if The notion has been generalized to other domains by many authors. Consider domains U ⊂ C n and V ⊂ C m and an analytic function ϕ from U to the closure of V . The function ϕ is said to satisfy Carathéodory's condition at τ ∈ ∂U if lim inf λ→τ dist(ϕ(λ), ∂V ) dist(λ, ∂U ) < ∞.
Thus, for example, when U = Π n , V = Π, a function h ∈ P n satisfies Carathéodory's condition at the point x ∈ R n if This definition works well for finite points in ∂U , but for our present purpose we need to consider points at infinity in the boundaries of Π n and Π. We shall introduce a variant of Carathéodory's condition for the class P n with the aid of the Cayley transform which furnishes a conformal map between D and Π, and hence a biholomorphic map between D n and Π n by co-ordinatewise action. We obtain a one-to-one correspondence between S n \ {1} and P n via the formulae where 1 is the constant function equal to 1 and λ, z are related by equations (7.3). For ϕ ∈ S n we define τ ∈ T n to be a carapoint of ϕ if We can now extend the notion of carapoints to points at infinity. The point (∞, . . . , ∞) in the boundary of Π n corresponds to the point χ in the closed unit disc; as in the last section, χ denotes the point (1, . . . , 1) ∈ C n .
Definition 7.1. Let h be a function in the Pick class P n with associated function ϕ in the Schur class S n given by equation (7.4). Let τ ∈ T n , x ∈ (R ∪ ∞) n be related by (7.6) x j = i 1 + τ j 1 − τ j for j = 1, . . . , n.
We say that x is a carapoint for h if τ is a carapoint for ϕ. We say that h has a carapoint at ∞ if h has a carapoint at (∞, . . . , ∞), that is, if ϕ has a carapoint at χ.
Note that, for a point x ∈ R n , to say that x is a carapoint of h is not the same as saying that h satisfies the Carathéodory condition (7.2) at x. Consider the function h(z) = −1/z 1 in P n . Clearly h does not satisfy Carathéodory's condition at 0 ∈ R n . However, the function ϕ in S n corresponding to h is ϕ(λ) = −λ 1 , which does have a carapoint at −χ, the point in T n corresponding to 0 ∈ R n . Hence h has a carapoint at 0.
We shall be mainly concerned with carapoints at 0 and ∞. The following observation will help us identify them. For any h ∈ P n we define h ♭ ∈ P n by For ϕ ∈ S n we define ϕ ♭ (λ) = ϕ(−λ).
If h and ϕ are corresponding functions, as in equations (7.4), then so are h ♭ and ϕ ♭ .
Proposition 7.2. The following conditions are equivalent for a function h ∈ P n .
(1) ∞ is a carapoint for h; (2) 0 is a carapoint for h ♭ ; Proof. (1)⇔(2) Since −χ ∈ T n corresponds under the Cayley transform to 0 ∈ R n , we have (2)⇔(3) A consequence of the n-variable Julia-Carathéodory Theorem [14,1], is that τ ∈ T n is a carapoint of ϕ ∈ S n if and only if It follows that Let iy ∈ Π be the Cayley transform of −r ∈ (−1, 0), so that y → 0+ as r → 1−. In view of the identity  3. If f ∈ P n satisfies Carathéodory's condition (7.8) lim inf z→x Im f (z) Im z < ∞ at x ∈ R n then x is a carapoint for f . If

Proof.
Let h = f ♭ ∈ P n . Clearly |h ♭ (z) + i| ≥ 1 for all z ∈ Π n . If the condition (7.8) holds for x = 0 then Im h ♭ (z) min j Im z j < ∞ and hence, by (2)⇔(3) of Proposition 7.2, 0 is a carapoint for h ♭ = f . The case of a general x ∈ R n follows by translation.
If h ∈ P n has a carapoint at x ∈ (R ∪ ∞) n then it has a value at x in a natural sense. If ϕ ∈ S n has a carapoint at τ ∈ T n , then by [14] there exists a unimodular constant ϕ(τ ) such that Here λ nt → τ means that λ tends nontangentially to τ in D n .
Definition 7.4. If h ∈ P n has a carapoint at x ∈ (R ∪ ∞) n then we define where τ ∈ T n corresponds to x as in equation (7.6).
Although the value of h(∞) is defined in terms of the Schur class function ϕ, it can be expressed more directly in terms of h. Proposition 7.5. If ∞ is a carapoint of h then Here we say that z nt → ∞ if z → (∞, ..., ∞) in the set {z ∈ Π n : (−1/z 1 , . . . , −1/z n ) ∈ S} for some set S ⊂ Π n that approaches 0 nontangentially, or equivalently, if z → (∞, . . . , ∞) in a set on which z ∞ / min j Im z j is bounded.

Types of functions in the Loewner class
In this section we shall show that the type of a function h ∈ L n is entirely determined by whether or not ∞ is a carapoint of h and by the value of h(∞). Let us make precise the notion of the type of a function in L n . Definition 8.1. A function h ∈ L n is of type 1 if it has a Nevanlinna representation of type 1. For n = 2, 3 or 4 we say that h is of type n if h has a Nevanlinna representation of type n but has no representation of type n − 1.
Clearly every function in L n is of exactly one of the types 1 to 4. We shall now prove Theorem 1.11. Recall that it states the following, for any function h ∈ L n . On combining these two limits we find that lim inf y→∞ y Im h(iyχ) < ∞, and so, by Theorem 6.2, h has a representation of type 2. Since h(∞) = 0 it is clear that h does not have a representation of type 1. Thus (2) holds.
A trivial modification of the above argument proves that (1) is also true.
(4) Let h be of type 4. Then h has no type 3 representation, and so, by Theorem 6.1, there exists δ > 0 and a sequence (s n ) of positive numbers tending to ∞ such that 1 s n Im h(is n χ) ≥ δ > 0.
Since |h ♭ (z) + i| > Im h ♭ (z) for all z, we have Hence (0, 0) is a carapoint of h ♭ , and so ∞ is a carapoint of h.
Conversely, suppose that ∞ is a carapoint of h and that h(∞) = ∞. We shall show that and it will follow from Theorem 6.1 that h does not have a representation of type 3, that is, h is of type 4. Let ϕ ∈ S n correspond to h and let r ∈ (0, 1) correspond to is ∈ Π. Then By hypothesis, χ is a carapoint for ϕ and ϕ(χ) = 1. By definition of carapoint, The n-variable Julia-Carathéodory Lemma (see [14,1]) now tells us that α > 0 and for all r ∈ (0, 1).
(2) It is immediate that the function 1 + h, with h as in (1), is of type 2, and that ∞ is a carapoint of 1 + h with value 1.
(3) We have seen that the function has a representation of type 3. To show that h is indeed of type 3 we must prove that ∞ is not a carapoint of h. (4) The function h(z) = z 1 z 2 z 1 + z 2 = −1 − 1 z 1 − 1 z 2 is clearly in P 2 . We gave a type 4 representation of h in Example 5.6. We claim that ∞ is a carapoint of h. We have h(iy, iy) = 1 2 iy, and thus lim inf y→∞ y Im h(iy, iy) |h(iy, iy) + i| 2 = lim inf y→∞ 1 2 y 2 | 1 2 iy + i| 2 = 2. Hence ∞ is a carapoint for h. Furthermore h(iy, iy) = 1 2 iy → ∞ as y → ∞, and so h(∞) = ∞. Thus h is of type 4.
Another example of a function of type 4 is h(z) = √ z 1 z 2 .

Rates of growth in the Loewner class
The Nevanlinna representation formulae give rise to growth estimates for functions in the n-variable Loewner class. It turns out that growth is mild, both at infinity and close to the real axis. Even though the type of a function is determined by its growth on the single ray {iyχ : y > 0}, in turn the growth of the function on the entire polyhalfplane is constrained by its type.
Consider first the one-variable case. If h is the Cauchy transform of a finite positive measure µ then |h(z)| ≤ dµ(t) |t − z| ≤ dµ(t) Im z = C Im z for some C > 0 and for all z ∈ Π. For a general function h in the Pick class, by Nevanlinna's representation (Theorem 1.2) there exist a ∈ R, b ≥ 0 and a finite positive measure µ on R such that, for all z ∈ Π, h(z) = a + bz + 1 + tz t − z dµ(t) = a + bz + 1 + z 2 t − z + z dµ(t) and therefore |h(z)| ≤ |a| + b|z| + 1 + |z| 2 Im z + |z| µ(R) ≤ C 1 + |z| + 1 + |z| 2 Im z for some C > 0. Similar estimates hold for the Loewner class.
Proposition 9.1. For any function h ∈ L n there exists a non-negative number C such that, for all z ∈ Π n , (9.1) |h(z)| ≤ C 1 + z 1 + 1 + z 2 1 min j Im z j .
For any function h ∈ L n of type 2 there exists a non-negative number C such that, for all z ∈ Π n , (9.2) |h(z)| ≤ C 1 + 1 min j Im z j .
For any function h ∈ L n of type 1 there exists a non-negative number C such that, for all z ∈ Π n , Since 1 + z 1 + z 2 1 ≤ 3 2 (1 + z 1 2 ), we have |h(z)| ≤ C 1 + z 1 + 1 + z 2 1 min j Im z j for some choice of C > 0 and for all z ∈ Π n . Thus the estimate (9.1) holds. Similarly, the estimates (9.2) and (9.3) follow easily from the simple resolvent estimate (2.1).

Structured resolvent identities
To conclude the paper we point out that there are structured analogs of the classical resolvent identity for any z, w in the resolvent set of an operator A.