The hyperbolic geometry of the symmetrized bidisc

J. Agler  N. J. Young

Abstract

We solve the Caratheodory and Kobayashi extremal problems for the open symmetrized bidisc

G =^{def} {(z_{1} + z_{2}, z_{1}z_{2}) : |z_{1}| < 1, |z_{2}| < 1} ⊂ ℂ^{2}.

We prove the equality of the Caratheodory and Kobayashi distances on G and describe the extremal functions for the two problems; they are rational of degree 1 or 2. G is the first example of a non-convexifiable domain for which the two distances coincide.

J. Agler’s work is supported by an NSF grant in Modern Analysis. This work was also supported by EPSRC Visiting Fellowship GR/N 19250. MSC 2000 classification: 32F45,93B50,30E05,93B36 Keywords: Kobayashi extremal problem, Caratheodory extremal problem, interpolation, spectral Nevanlinna-Pick problem.

1 Introduction

Motivation for the study of the function theory and complex geometry of a non-classical domain closely related to the bidisc comes from a problem in control engineering known as robust stabilization. One approach to this engineering problem requires the construction of analytic matrix-valued functions on the open unit disc 𝔻 subject to interpolation conditions and to a certain boundedness condition. In its full generality this function-theoretic problem, called the μ-synthesis problem, still lacks a satisfactory analytic or numerical solution. Several authors have accordingly studied special cases in an attempt to throw light on the analytic obstacles to be overcome [10]. The simplest unsolved special case of the μ-synthesis problem (known as the spectral Nevanlinna-Pick problem for 2 by 2 matrix functions) led us to study the complex geometry of the domain G ⊂ ℂ^{2} defined by

G = {(z_{1} + z_{2}, z_{1}z_{2}) : z_{1}, z_{2} ∈ 𝔻^{2}},

which we shall call the symmetrized bidisc. Although G can be identified with the orbit space of the bidisc 𝔻^{2} under an obvious ℤ_{2} action, its geometry is very different from that of the bidisc. The symmetrized bidisc is not convex (though it is polynomially convex, hypoconvex and starlike). It is inhomogeneous and has distinguished boundary the symmetrized torus, which is topologically a Möbius band. It has a rich geometry, much of which can be described explicitly, as we shall demonstrate in this paper.

Although the genesis of the present investigation was an engineering application, there are several purely mathematical reasons to study this domain. One can think of G as the coefficient body of the set of quadratic polynomials whose zeros lie in 𝔻, a classical object of study in mathematics and engineering. Functions on G are essentially the same as symmetric functions on 𝔻^{2}. Special properties of G have consequences for operator theory (the development of an analogue of the Sz.-Nagy model theory for certain pairs of commuting operators) and operator algebras (a new example of an algebra for which every bounded representation is completely bounded [4]). However, in this paper we focus on the intrinsic complex geometry of G. We shall solve the Caratheodory and Kobayashi extremal problems on G. This immediately yields a full solution of the spectral Nevanlinna-Pick problem in the case of two interpolation points and two-by-two matrices; see [5] for the relationship between the two problems.

We denote by 𝕋 the unit circle, by Δ the closure of 𝔻 and by Γ the closure of G, so that Γ = π(Δ^{2}) where π denotes the “symmetrization map”:

π(z_{1}, z_{2}) = (z_{1} + z_{2}, z_{1}z_{2}).

We write d for the pseudohyperbolic distance on 𝔻:

d(z, w) = |\frac{z − w}{1 − \overline{w}z} | , z, w ∈ 𝔻.

The Schur class is the class of functions analytic and bounded in modulus by 1 on 𝔻.

The Caratheodory extremal problem for G and for a given pair of points z_{1}, z_{2} ∈ G is to find the quantity

C_{G}(z_{1}, z_{2}) =^{def} sup{d(F (z_{1}), F (z_{2})) : F maps G analytically into 𝔻}.

Any F for which the supremum on the right hand side is attained is called a Caratheodory extremal function for the domain G and the points z_{1}, z_{2}, and C_{G} is called the Caratheodory distance on G. In fact it is customary to define the Caratheodory distance on a general domain Ω to be tanh^{−1} C_{Ω}, with C_{Ω} as above, and the Caratheodory distance is then a Finsler metric (e.g. [12]), but for the purposes of this paper we prefer to omit the tanh^{−1}.

To describe the solution of the Caratheodory extremal problem for G we shall need the functions Φ_{α}, α ∈ Δ, defined by

Φ_{α}(s, p) = \frac{2αp − s}{2 − αs}.  (1.1)

Theorem 1.1 For z_{1}, z_{2} ∈ G,

C_{G}(z_{1}, z_{2}) = max_{ω∈𝕋}d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})).

It follows from the theorem that for each pair of points z_{1}, z_{2} ∈ G, there is an ω_{0} ∈ 𝕋 such that Φ_{ω_{0}} is a Caratheodory extremal for z_{1}, z_{2}. Generically there is a unique such ω_{0}, but it can happen that there are two possible values of ω_{0}, or even that Φ_{ω} is an extremal for all ω ∈ 𝕋. It is also true that the Φ_{ω} solve the infinitesimal Caratheodory extremal problem for G (Corollary 4.3).

Our second main result is the solution of the Kobayashi extremal problem for a pair of points z_{1}, z_{2} ∈ G. This is the problem: find the quantity

δ_{G}(z_{1}, z_{2}) = inf d(λ_{1}, λ_{2})

over all pairs λ_{1}, λ_{2} ∈ 𝔻 such that there exists an analytic function h : 𝔻 → G such that h(λ_{1}) = z_{1}, h(λ_{2}) = z_{2}. Any such function h for which the infimum is attained is called a Kobayashi extremal function for the domain G and the points z_{1}, z_{2}. The Kobayashi distance on G is defined to be the largest pseudodistance on G dominated by δ_{B} (e.g. [12]). It is standard that

C_{Ω} ≤ K_{Ω} ≤ δ_{Ω}

for any domain Ω [12]. A significant result in this paper is that the Caratheodory and Kobayashi distances coincide on G. In Theorem 5.6 we shall give a concrete description of the solution to the Kobayashi extremal problem for any pair of points in G; here is a simplified version of that description.

Theorem 1.2 Let z_{1}, z_{2} be distinct points in G and let ω_{0} ∈ 𝕋 be such that Φ_{ω_{0}} is a Caratheodory extremal function for z_{1}, z_{2} and G. There exists an analytic function h = (s, p) : 𝔻 → G such that z_{1}, z_{2} ∈ h(𝔻), p is a Blaschke product of degree at most 2 and Φ_{ω_{0}} ∘ h is the identity on 𝔻.

The fact that h in the theorem has an analytic left inverse (namely Φ_{ω_{0}}) implies that h is a solution to the Kobayashi extremal problem for z_{1}, z_{2} and G (see Theorem 5.6), and also shows that

C_{G}(z_{1}, z_{2}) = K_{G}(z_{1}, z_{2}) = δ_{G}(z_{1}, z_{2});

see Corollary 5.7. The importance of this conclusion lies in the fact that whereas the equation K = C is known for domains that are convexifiable, that is, biholomorphic to a convex domain, G is not convexifiable [11, 8].

The paper is arranged as follows. In Section 2 we gather together elementary facts about G for the convenience of the reader; these are mostly variants of facts which have appeared before in scattered references. In Section 3 we prove Theorem 1.1, giving the solution to the Caratheodory extremal problem on G with the aid of a little operator theory, applied to operators on 2-dimensional Hilbert spaces. In Section 4 we solve the infinitesimal version of the Caratheodory extremal problem for G. In Section 5 we solve the Kobayashi extremal problem for G, proving Theorem 1.2 and showing that K_{G} = C_{G}.

2 Elementary properties of the symmetrized bidisc

In this section we shall give a variety of characterizations of points of G and shall establish some geometric properties of G, to wit, that G is not convex but is polynomially convex, hypoconvex and starlike. We shall also describe the distinguished boundary of Γ and exhibit some of the properties of the important functions Φ_{α} defined in equation (1.1).

The general point in G has co-ordinates s = z_{1} + z_{2}, p = z_{1}z_{2} with z_{1}, z_{2} ∈ 𝔻. For some purposes the co-ordinates z_{1}, z_{2} are convenient, but for others the (s, p) co-ordinates are better. It is therefore useful to have a characterization of G in terms of the (s, p) co-ordinates. There is a remarkable diversity of such characterizations.

Theorem 2.1 Let s, p ∈ ℂ. The following statements are equivalent:

(1) (s, p) ∈ G;

(2) the roots of the equation z^{2} − sz + p = 0 lie in 𝔻;

(3) |s − \overline{s}p| < 1 − |p|^{2};

(4) |s| < 2 and, for all z ∈ Δ,

|\frac{2zp − s}{2 − zs}| < 1;

(5) |p| < 1 and there exists β ∈ 𝔻 such that s = βp + \overline{β};

(6) 2|s − \overline{s}p| + |s^{2} − 4p| + |s|^{2} < 4.

Proof. The equivalence of (1) and (2) is trivial. The fact that (2) and (3) are equivalent is a classical theorem of Schur [16] specialized to quadratic polynomials. We shall show that (1) ⇒ (4) ⇒ (3). The identity

|2 − w_{1} − w_{2}|^{2} − |2w_{1}w_{2} − w_{1} − w_{2}|^{2} =
2(1 − |w_{1}|^{2})|1 − w_{2}|^{2} + 2(1 − |w_{2}|^{2})|1 − w_{1}|^{2},

is valid for any w_{1}, w_{2} ∈ ℂ. Consider any point (s, p) ∈ G and write

s = ζ_{1} + ζ_{2}, p = ζ_{1}ζ_{2}

with ζ_{2}, ζ_{2} ∈ 𝔻. Certainly |s| < 2, and on setting w_{1} = zζ_{1}, w_{2} = zζ_{2} we obtain, for any z ∈ Δ,

|2 − zs|^{2} − |2z^{2}p − zs|^{2}
= 2(1 − |zζ_{1}|^{2})|1 − zζ_{2}|^{2} + 2(1 − |zζ_{2}|^{2})|1 − zζ_{1}|^{2}
> 0,

and hence

|\frac{2z^{2}p − zs}{2 − zs}| < 1.

When |z| = 1 we have

|\frac{2zp − s}{2 − zs}| < 1,

and so, by the Maximum Principle, the inequality continues to hold for all z ∈ Δ. Thus (1) ⇒ (4). Conversely, suppose that (4) holds. Then, for all z ∈ 𝕋,

|2zp − s|^{2} < |2 − zs|^{2},

which expands to give

4Re\{z(s − \overline{s}p)\} < 4(1 − |p|^{2}).

Hence

|s − \overline{s}p| < 1 − |p|^{2}.

Thus (1) to (4) are all equivalent.

We now show that (5) and (3) are equivalent. Suppose (5) holds; then |p| < 1 and

s − \overline{s}p = βp + \overline{β} − (\overline{β}\overline{p} + β)p = \overline{β}(1 − |p|^{2})

and so

|s − \overline{s}p| < 1 − |p|^{2}.

Thus (5) ⇒ (3). Conversely, suppose (3) holds and put

β = \frac{\overline{s} − s\overline{p}}{1 − |p|^{2}} .

Then |β| < 1 and

βp + \overline{β} = \frac{(\overline{s} − s\overline{p})p + s − \overline{s}p}{1 − |p|^{2}} = s.

Thus (3) ⇔ (5).

We have shown that conditions (1) to (5) are equivalent, and it only remains to show that these conditions are all equivalent to (6). Suppose (4) holds. Then, since |s| < 2, the mapping

z ↦ \frac{2zp − s}{2 − zs}

is analytic on Δ, and maps Δ to the closed disc with

centre 2\frac{\overline{s}p − s}{4 − |s|^{2}} and radius \frac{|s^{2} − 4p|}{4 − |s|^{2}}.

By hypothesis this disc is contained in 𝔻, so that

\frac{2|s − \overline{s}p| + |s^{2} − 4p|}{4 − |s|^{2}} < 1.

Thus (4) ⇒ (6). The steps of this argument are reversible, so (4) ⇔ (6). □

Analogous criteria for a point to belong to Γ are given in [7, Theorem 1.1].

Corollary 2.2 For s, p ∈ ℂ,

(s, p) ∈ G ⟺ |s| < 2 and, for all ω ∈ 𝕋, |Φ_{ω}(s, p)| < 1.

Proof. Necessity is immediate from the equivalence of (1) and (4). To see sufficiency observe that if |s| < 2 then Φ_{z}(s, p) is an analytic function of z on Δ. By the Maximum Principle, the assumption that |Φ_{z}(s, p)| < 1 for all z ∈ 𝕋 implies that the same inequality holds for all z ∈ 𝔻, and hence, by the implication (4) ⇒ (1), we have (s, p) ∈ G. □

Theorem 2.3 G is hypoconvex, polynomially convex and starlike about (0, 0), but not convex.

Proof. The closure Γ of G is not convex, for

(2, 1) = (1 + 1, 1.1) ∈ Γ, (2i, −1) = (i + i, i.i) ∈ Γ,

but the mid-point of these two points is

(1 + i, 0) = π(1 + i, 0) ∉ Γ.

Thus G is not convex.

To prove hypoconvexity, consider a point z = (z_{1} +z_{2}, z_{1}z_{2}) ∉ G. We may suppose z_{1} ∉ 𝔻, and we seek a complex hyperplane in ℂ^{2} which contains z but is disjoint from G. We may simply take the complex hyperplane

{(z_{1} + w, z_{1}w) : w ∈ ℂ} = (z_{1}, 0) + ℂ(1, z_{1}),

which clearly has the required properties.

A domain Ω is said to be polynomially convex provided that, for each compact subset K of Ω, the polynomial hull K̂ of K is contained in Ω. Let us first show that the closure Γ of G is polynomially convex. Consider any point z = (z_{1} + z_{2}, z_{1}z_{2}) in the complement of Γ. At least one of |z_{1}|, |z_{2}| is greater than 1. If both |z_{1}| ≥ 1 and |z_{2}| ≥ 1 then the function f(w_{1}, w_{2}) = w_{2} has the required properties |f(z)| > 1 and |f| ≤ 1 on Γ. If |z_{1}| > 1 but |z_{2}| < 1 then the polynomial

f_{p}(w_{1} + w_{2}, w_{1}w_{2}) = \frac{1}{2}(w_{1}^{p} + w_{2}^{p}),

for a sufficiently large positive integer p, satisfies |f_{p}(z)| > 1 and |f_{p}| ≤ 1 on Γ. Thus Γ is polynomially convex.

Now consider any compact subset K of G. For some r ∈ (0, 1) we have

K ⊂ Γ_{r} =^{def} {(z_{1} + z_{2}, z_{1}z_{2}) : |z_{1}| ≤ r, |z_{2}| ≤ r} ⊂ G.

Since Γ_{r} is polynomially convex we have

K̂ ⊂ Γ̂_{r} = Γ_{r} ⊂ G,

and so G is polynomially convex.

We shall show that G is starlike about (0, 0): if (s, p) ∈ G and 0 ≤ r < 1 then (rs, rp) ∈ G. Fix (s, p) ∈ G and r ∈ [0, 1). By Theorem 2.1 we have, for all z ∈ Δ,

|2 − zs|^{2} − |2zp − s|^{2} > 0.

One may verify the identity, valid for all z ∈ 𝕋,

|2 − zrs|^{2} − |2zrp − rs|^{2} =
r^{2}{|2 − zs|^{2} − |2zp − s|^{2}} + 4(1 − r)(1 + r − rRe(zs)).

The right hand side is strictly positive for all z ∈ 𝕋, and hence

|\frac{2zrp − rs}{2 − zrs}| < 1

for all z ∈ 𝕋, and consequently also for all z ∈ Δ. By Theorem 2.1, (rs, rp) ∈ G. Thus G is starlike about the point (0, 0). □

We denote by bΓ the symmetrized torus

bΓ = {(z_{1} + z_{2}, z_{1}z_{2}) : z_{1}, z_{2} ∈ 𝕋}.

Recall that if K is a compact subset of ℂ^{n} then the distinguished boundary of K is defined to be the Šilov boundary of the algebra A(K) of functions continuous on K and analytic on the interior of K, that is, the smallest closed subset of K on which every function in A(K) attains its maximum modulus.

Theorem 2.4 The distinguished boundary of Γ is bΓ. Topologically bΓ is a Möbius band.

Proof. The first assertion follows easily from the fact that the distinguished boundary of the bidisc is the 2-torus 𝕋^{2}. To see that bΓ is a Möbius band write z_{1}z_{2} = e^{iθ}: then

z_{1} + z_{2} = z_{1} + \overline{z}_{1}e^{iθ} = e^{iθ/2} 2 Re(z_{1}e^{−iθ/2}),

and we may parametrize bΓ by

bΓ = {(2xe^{iθ/2}, e^{iθ}) : −1 ≤ x ≤ 1, 0 ≤ θ ≤ 2π}.

Thus bΓ is homeomorphic to the set

{(x, θ) : −1 ≤ x ≤ 1, 0 ≤ θ ≤ 2π}

with the points (x, 0) and (−x, 2π) identified. Thus bΓ is homeomorphic to a the quotient space of a rectangle obtained by identifying one pair of opposite sides with a twist, hence is a Möbius band. □

In fact the symmetrised n-torus, for n ≥ 2, is a higher dimensional Möbius band [15].

It is easy to see that an element (s, p) ∈ ℂ^{2} lies in bΓ if and only if

|s| ≤ 2 and |p| = 1 and s = s̄p.  (2.1)

A complex geodesic in G is an analytic mapping h : 𝔻 → G which is an isometry with respect to the pseudohyperbolic and Caratheodory distances. Equivalently, a complex geodesic is an analytic embedding of 𝔻 in G that has an analytic left inverse. Two particular classes of complex geodesics will play an important role. We define the royal variety to be the set

V = {(2z, z^{2}) : z ∈ ℂ} ⊂ ℂ^{2}.

Then V ∩ 𝔻 (or more exactly, the embedding z ↦ (2z, z^{2})) is a complex geodesic, having the left inverse (s, p) ↦ \frac{1}{2}s. By Theorem 2.1, for any β ∈ 𝔻 the mapping z ↦ (βz + β̄, z) maps 𝔻 to G. It has an obvious analytic left inverse; we call this embedding the flat geodesic with parameter β.

By a Möbius function we mean an automorphism of 𝔻, that is, a mapping of the form cB_{α} for some c ∈ 𝕋 and α ∈ 𝔻, where B_{α} is the Blaschke factor vanishing at α:

B_{α}(z) = \frac{z − α}{1 − ᾱz} , z ∈ 𝔻.

In equation (1.1) we met the family of linear fractional mappings Φ_{ω} from Γ to 𝔻; it comprises a universal set for the Caratheodory extremal problem for G. We conclude this section by recording some of the properties of the Φ_{ω}. Recall that for α ∈ Δ and s, p ∈ ℂ, s ≠ 2/α,

Φ_{α}(s, p) = \frac{2αp − s}{2 − αs}.

Theorem 2.5 For any ω ∈ 𝕋 and s, p ∈ ℂ with s ≠ 2ω̄,

|Φ_{ω}(s, p)| = 1

if and only if

Re {ω(s − s̄p)} = 1 − |p|^{2}.

Furthermore, if (s, p) ∈ Γ \ {(2ω̄, ω̄^{2})},

|Φ_{ω}(s, p)| = 1

if and only if

ω(s − s̄p) = 1 − |p|^{2}.

Proof. Since s ≠ 2ω̄, the ratio Φ_{ω}(s, p) is defined. Now

|Φ_{ω}(s, p)| = 1 ⟺ |2ωp − s|^{2} = |2 − ωs|^{2}
⟺ 4|p|^{2} − 4Re (ωs̄p) + |s|^{2} = 4 − 4Re (ωs) + |s|^{2}
⟺ Re {ω(s − s̄p)} = 1 − |p|^{2}.

Now suppose (s, p) ∈ Γ. We have

|s − s̄p| ≤ 1 − |p|^{2}

by Theorem 2.1 – or strictly, by its analogue for Γ, which we may derive by considering (rs, r^{2}p) ∈ G and letting r → 1−. It follows that

Re {ω(s − s̄p)} = 1 − |p|^{2} ⟺ ω(s − s̄p) = 1 − |p|^{2}.

Hence

|Φ_{ω}(s, p)| = 1 ⟺ Re {ω(s − s̄p)} = 1 − |p|^{2}
⟺ ω(s − s̄p) = 1 − |p|^{2}.
□

For ω ∈ 𝕋, Φ_{ω} is continuous on G but it is impossible to extend Φ_{ω} to a continuous function on Γ. The two paths

(2rω̄, ω̄^{2}) and (2rω̄, r^{2}ω̄^{2}), 0 ≤ r < 1,

in G both tend to (2ω̄, ω̄^{2}) in Γ as r → 1, and one may check that the limits of Φ_{ω} along these two paths exist and differ (being ±ω̄).

3 The Caratheodory distance on G

In this section we shall prove the formula given in Theorem 1.1, which identifies the Caratheodory extremal functions on G and implies that the Caratheodory distance is given by the maximum modulus of a fractional quadratic function on the unit circle (Corollary 3.5). The derivation is based on an interpretation in terms of operators on 2-dimensional Hilbert spaces [1]. Recall that a domain Ω ⊂ ℂ^{n} is said to be a spectral domain for an n-tuple (T_{1}, T_{2}, …, T_{n}) of commuting operators if the joint spectrum σ(T_{1}, T_{2}, …, T_{n}) is contained in Ω and, for any bounded analytic function f on Ω,

||f(T_{1}, T_{2}, …, T_{n}))|| ≤ sup_{Ω} |f|.

It is shown in [1] that the calculation of the Caratheodory distance on any domain Ω in ℂ^{n} can be closely related to the characterization of commuting ntuples for which Ω is a spectral domain. We shall state the result only for the case Ω = G. Corresponding to any pair of points z_{1} = (s_{1}, p_{1}), z_{2} = (s_{2}, p_{2}) ∈ G and any basis u = (u_{1}, u_{2}) of unit vectors in a 2-dimensional space H we define a commuting pair (S_{u}, P_{u}) of operators on H, comprising the operators whose matrices with respect to the basis u are diag {s_{1}, s_{2}} and diag {p_{1}, p_{2}} respectively, so that

S_{u}u_{i} = s_{i}u_{i}, P_{u}u_{i} = p_{i}u_{i}, i = 1, 2.  (3.1)

A basis consisting of unit vectors will be called a normalized basis. Theorem 2.14 of [1] contains the following result.

Theorem 3.1 For z_{1} = (s_{1}, p_{1}), z_{2} = (s_{2}, p_{2}) ∈ G,

C_{G}(z_{1}, z_{2})^{2} = 1 − sup | < u_{1}, u_{2} > |^{2}  (3.2)

where the supremum is taken over all normalized bases u_{1}, u_{2} of a twodimensional Hilbert space such that G is a spectral domain for the pair (S_{u}, P_{u}) given by equations (3.1).

To apply this result to the computation of the Caratheodory distance on G we need a criterion for G to be a spectral domain for a pair of operators. The following is a modification of a characterization given in [4, Theorem 1.5].

Theorem 3.2 Let (S, P) be a pair of commuting operators on a Hilbert space and suppose that the spectral radius of S is less than 2. Then G is a spectral domain for (S, P) if and only if, for all ω ∈ 𝕋,

||Φ_{ω}(S, P)|| ≤ 1.  (3.3)

Proof. By [4, Theorem 1.5], Γ is a spectral set for (S, P) if and only if, for all α ∈ 𝔻,

2(1 − ᾱ^{2}P^{∗}α^{2}P) − (αS − ᾱS^{∗}α^{2}P) − (ᾱS^{∗} − ᾱ^{2}P^{∗}αS) ≥ 0.

This inequality can be written

(2 − αS)^{∗}(2 − αS) − (2α^{2}P − αS)^{∗}(2α^{2}P − αS) ≥ 0,

and so, since the spectral radius of αS is less than 2, is equivalent to

||(2α^{2}P − αS)(2 − αS)^{−1}|| ≤ 1.  (3.4)

Suppose G is a spectral domain for (S, P). Trivially Γ is a spectral set for (S, P). The inequality (3.4) therefore holds for all α ∈ 𝔻, and so also for α = ω ∈ 𝕋. Hence inequality (3.3) holds for all ω ∈ 𝕋.

Conversely, suppose that (3.3) holds for all ω ∈ 𝕋. The mapping

α ↦ αΦ_{α}(S, P)

is analytic on Δ; since it is contractive on 𝕋, by the Maximum Principle it is bounded by 1 on Δ. Hence, for α ∈ 𝔻,

1 − ᾱΦ_{α}(S, P)^{∗}αΦ_{α}(S, P) ≥ 0,

or equivalently,

(2 − αS)^{∗}(2 − αS) − (2α^{2}P − αS)^{∗}(2α^{2}P − αS) ≥ 0.

By [4, Theorem 1.5], Γ is a spectral set for (S, P). It follows simply that G is a spectral domain for (S, P). □

Lemma 3.3 For ω ∈ 𝕋 and any normalised basis u = (u_{1}, u_{2}),

||Φ_{ω}(S_{u}, P_{u})|| ≤ 1 ⟺ |⟨u_{1}, u_{2}⟩|^{2} ≤ 1 − d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2}))^{2}.

Proof. Φ_{ω}(S_{u}, P_{u}) is a contraction if and only if

[⟨(1 − Φ_{ω}(S_{u}, P_{u})^{∗}Φ_{ω}(S_{u}, P_{u}))u_{i}, u_{j}⟩]_{i,j=1}^{2} ≥ 0.  (3.5)

Since

Φ_{ω}(S_{u}, P_{u})u_{i} = Φ_{ω}(z_{i})u_{i},

the inequality (3.5) is equivalent to

[(1 − Φ̄_{ω}(z_{j})Φ_{ω}(z_{i}))⟨u_{i}, u_{j}⟩]_{i,j=1}^{2} ≥ 0

and so, since the diagonal entries of this matrix are positive, to

(1 − |Φ_{ω}(z_{1})|^{2})(1 − |Φ_{ω}(z_{2})|^{2}) ≥ |1 − Φ̄_{ω}(z_{1})Φ_{ω}(z_{2})|^{2}|⟨u_{i}, u_{j}⟩|^{2}.

Since

\frac{(1 − |Φ_{ω}(z_{1})|^{2})(1 − |Φ_{ω}(z_{2})|^{2})}{|1 − Φ̄_{ω}(z_{1})Φ_{ω}(z_{2})|^{2}} = 1 − d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2}))^{2},

the lemma follows. □

Proof of Theorem 1.1. Let z_{1} = (s_{1}, p_{1}), z_{2} = (s_{2}, p_{2}). By Theorem 3.1

C_{G}(z_{1}, z_{2})^{2} = 1 − sup |⟨u_{1}, u_{2}⟩|^{2}

over all normalised bases u_{1}, u_{2} such that G is a spectral domain for the pair (S_{u}, P_{u}) defined by equations (3.1). Note that the spectral radius of S_{u} is max(|s_{1}|, |s_{2}|), which is less than 2. In view of Theorem 3.2 and Lemma 3.3 we have

C_{G}(z_{1}, z_{2})^{2} = 1 − sup{|⟨u_{1}, u_{2}⟩|^{2} : ∀ω ∈ 𝕋 ||Φ_{ω}(S_{u}, P_{u}) ≤ 1}
= 1 − sup{|⟨u_{1}, u_{2}⟩|^{2} : ∀ω ∈ 𝕋 |⟨u_{1}, u_{2}⟩|^{2} ≤ 1 − d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2}))^{2}}
= sup d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2}))^{2}.

The supremum is clearly a maximum, and so

C_{G}(z_{1}, z_{2}) = max_{ω∈𝕋}d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})).  (3.6)
□

Corollary 3.4 For every pair of points z_{1}, z_{2} ∈ G there exists ω_{0} ∈ 𝕋 such that Φ_{ω_{0}} is a Caratheodory extremal function for z_{1}, z_{2}.

For if the maximum in equation (3.6) is attained at ω_{0} ∈ 𝕋 then Φ_{ω_{0}} is an analytic function from G to 𝔻 such that sup d(F (z_{1}), F (z_{2})), over all such analytic functions F, is attained at F = Φ_{ω_{0}}. □

If we insert the formula for Φ_{ω} into equation (3.6) we obtain the following expression for C_{G}(z_{1}, z_{2}).

Corollary 3.5 For any z_{1}, z_{2} ∈ G,

C_{G}(z_{1}, z_{2}) = sup_{ω∈𝕋} |\frac{(s_{2}p_{1} − s_{1}p_{2})ω^{2} + 2(p_{2} − p_{1})ω + s_{1} − s_{2}}{(s_{1} − s̄_{2}p_{1})ω^{2} − 2(1 − p_{1}p̄_{2})ω + s̄_{2} − s_{1}p̄_{2}} |

where z_{1} = (s_{1}, p_{1}), z_{2} = (s_{2}, p_{2}).

In some cases we can obtain still simpler and more explicit expressions. For example [6], if z_{1} = (0, 0), z_{2} = (s, p) ∈ G then

C_{G}(z_{1}, z_{2}) = \frac{2|s − s̄p| + |s^{2} − 4p|}{4 − |s|^{2}},

an expression which gives further insight into condition (6) of Theorem 2.1. Another example is

C_{G}((s, 0), (−s, 0)) = |s|.

4 The infinitesimal Caratheodory metric

The infinitesimal form of the Caratheodory distance studied in Section 3 is a non-negative function c_{G} defined on the tangent bundle T G of G [13, Chapter 4.2]. If u ∈ T G then

c_{G}(u) = sup_{f} ∥f_{∗}u∥

where ∥f_{∗}u∥ is the length of the tangent vector f_{∗}u of 𝔻 with respect to the Poincaré metric on 𝔻, and the supremum is taken over all analytic functions f : G → 𝔻. Concretely, if u is the tangent vector

v = v_{1} \frac{∂}{∂s} + v_{2} \frac{∂}{∂p} ∈ T G_{z}

for some z ∈ G, then

∥f_{∗}u∥ = \frac{|v_{1} \frac{∂f}{∂s} (z) + v_{2} \frac{∂f}{∂p}(z)|}{1 − |f(z)|^{2}}.

We may thus express the above definition of c_{G} as

c_{G}(z; v) = sup \frac{|f′(z)v|}{1 − |f(z)|^{2}}  (4.1)

where the supremum is over all analytic functions f : G → 𝔻. Any such f for which the supremum is attained will be called a Caratheodory extremal function for z along v. In this section we shall derive a formula for c_{G} and will show that for any point of T G there is a Caratheodory extremal function of the form Φ_{ω} for some ω ∈ 𝕋. As in Section 3 the tool we use is two-dimensional operator theory, and in particular the theory of spectral domains. The following statement is a specialization to G of [1, Theorem 3.10].

Theorem 4.1 Let z = (s_{0}, p_{0}) ∈ G and let

v = v_{1} \frac{∂}{∂s} + v_{2} \frac{∂}{∂p} ∈ T G_{z}.

For κ ≥ 0 let S_{κ}, P_{κ} be the operators on the Hilbert space ℂ^{2} (with its standard inner product) given by

S_{κ} = [ \begin{matrix}s_{0} & κv_{1} \\ 0 & s_{0} \end{matrix} ] ,  P_{κ} = [ \begin{matrix}p_{0} & κv_{2} \\ 0 & p_{0} \end{matrix} ] .  (4.2)

Then

c_{G}(z; v) = inf{1/κ : κ > 0 and G is a spectral domain for (S_{κ}, P_{κ})}.

(4.3)

Furthermore, an analytic function f : G → 𝔻 is a Caratheodory extremal function for z along v if and only if

∥f(S_{κ}, P_{κ})∥ = 1

for some κ > 0 such that G is a spectral domain for (S_{κ}, P_{κ}).

The proof was left to the reader in [1], so we sketch it here. Observe that, for any analytic function f : G → ℂ,

f(S_{κ}, P_{κ}) = [ \begin{matrix}f(z) & κf′(z)v \\ 0 & f(z) \end{matrix} ] ,  (4.4)

a formula which follows easily from the functional calculus for pairs of commuting matrices. We shall show that, for any κ > 0,

c_{G}(z; v) ≤ \frac{1}{κ} ⇔ G is a spectral domain for (S_{κ}, P_{κ}).  (4.5)

Suppose G is a spectral domain for (S_{κ}, P_{κ}) and consider any analytic function f : G → 𝔻 such that f(z) = 0. By definition of spectral domain f(S_{κ}, P_{κ}) is a contraction, while from equation (4.4),

f(S_{κ}, P_{κ}) = [ \begin{matrix}0 & κf′(z)v \\ 0 & 0 \end{matrix} ] .

Hence

|κf′(z)v| ≤ 1.

Taking suprema over f we obtain

c_{G}(z; v) = sup_{f} |f′(z)v| ≤ \frac{1}{κ}.

Conversely, suppose c_{G}(z; v) ≤ \frac{1}{κ} and let f : G → ℂ be analytic and bounded by 1. By definition of c_{G}

\frac{|f′(z)v|}{1 − |f(z)|^{2}} ≤ \frac{1}{κ}.

From equation (4.4)

B_{f(z)} ∘ f(S_{κ}, P_{κ}) = B_{f(z)} [ \begin{matrix}f(z) & κf′(z)v \\ 0 & f(z) \end{matrix} ]

= [ \begin{matrix}0 & \frac{κf′(z)}{1−|f(z)|^{2}} \\ 0 & 0 \end{matrix} ] ,  (4.6)

which is a contraction. Hence

f(S_{κ}, P_{κ}) = B_{−f(z)} [ \begin{matrix}0 & \frac{κf′(z)}{1−|f(z)|^{2}} \\ 0 & 0 \end{matrix} ]

is a contraction. Thus G is a spectral domain for (S_{κ}, P_{κ}).

We have proved the equivalence (4.5), and the formula (4.3) for c_{G} follows at once.

The assertion about Caratheodory extremal functions is trivially true if v = 0, so we suppose that v ≠ 0. It follows that c_{G}(z; v) > 0, and the set

K = {κ ≥ 0 : G is a spectral domain for (S_{κ}, P_{κ})}

is closed and bounded.

Suppose that f is a Caratheodory extremal for z_{0} along v, so that f : G → 𝔻 is analytic and

\frac{|f′(z)v|}{1 − |f(z)|^{2}} = \frac{1}{κ}

where κ = 1/c_{G}(z; v). Then G is a spectral domain for (S_{κ}, P_{κ}), by relation (4.5), and it is clear from equation (4.6) that

∥B_{f(z)} ∘ f(S_{κ}, P_{κ})∥ = 1.

Hence also

∥f(S_{κ}, P_{κ})∥ = 1,

and the necessity of the condition for extremals is proved. Conversely, suppose that f : G → 𝔻 is analytic and satisfies

∥f(S_{κ}, P_{κ})∥ = 1

for some κ > 0 such that G is a spectral domain for (S_{κ}, P_{κ}). Again from equation (4.6),

\frac{|κf′(z)v|}{1 − |f(z)|^{2}} = 1.

In view of relation (4.5) we have

c_{G}(z; v) ≥ \frac{|f′(z)v|}{1 − |f(z)|^{2}} = \frac{1}{κ} ≥ c_{G}(z; v).

Hence

\frac{|f′(z)v|}{1 − |f(z)|^{2}} = c_{G}(z; v),

so that f is a Caratheodory extremal, as asserted. □

On combining Theorem 4.1 with our criterion for G to be a spectral domain (Theorem 3.2) we obtain the following.

Corollary 4.2

c_{G}(z_{0}; v) = inf \frac{1}{κ}

where the infimum is taken over all κ > 0 such that

∥(2ωP_{κ} − S_{κ})(2 − ωS_{κ})^{−1}∥ ≤ 1

for all ω ∈ 𝕋.

We can infer that the functions Φ_{ω}, ω ∈ 𝕋, also provide extremals for the infinitesimal Caratheodory distance problem.

Corollary 4.3 For every z ∈ G and v ∈ ℂ^{2} there exists ω_{0} ∈ 𝕋 such that Φ_{ω_{0}} is a Caratheodory extremal function for z along v.

Proof. We may suppose v ≠ 0 (and therefore c_{G}(z; v) > 0). For κ ≥ 0 let

g(κ) = sup_{ω∈𝕋}∥Φ_{ω}(S_{κ}, P_{κ})∥.

g is a continuous function on ℝ^{+} which tends to infinity as κ → ∞. Let

κ_{0} = sup{κ ≥ 0 : g(κ) ≤ 1}.

Then g(κ_{0}) = 1 and hence there exists ω_{0} ∈ 𝕋 such that

∥Φ_{ω_{0}}(S_{κ}, P_{κ})∥ = 1.

Since further

∥Φ_{ω}(S_{κ}, P_{κ})∥ ≤ 1

for all ω ∈ 𝕋, it follows that G is a spectral domain for (S_{κ}, P_{κ}). Hence, by Theorem 4.1, Φ_{ω_{0}} is a Caratheodory extremal for z along v. □

Corollary 4.4 If z = (s, p) ∈ G and v = v_{1} \frac{∂}{∂s} + v_{2} \frac{∂}{∂p} ∈ T G_{z} then

c_{G}(z; v) = sup_{ω∈𝕋} |\frac{v_{1}(1 − ω^{2}p) − v_{2}(2 − ωs)}{(s − \overline{s}p)ω^{2} − 2(1 − |p|^{2})ω + \overline{s} − \overline{p}s}| .  (4.7)

and a Caratheodory extremal function for z along v is any Φ_{ω_{0}}, ω_{0} ∈ 𝕋, such that the supremum in equation (4.7) is attained at ω_{0}.

Proof. Since each Φ_{ω}, ω ∈ 𝕋, is an analytic function from G into 𝔻 and some Φ_{ω} is a Caratheodory extremal for z along v, it follows from the definition of c_{G} that

c_{G}(z; v) = sup_{ω∈𝕋} \frac{|Φ′_{ω}(z)v|}{1 − |Φ_{ω}(z)|^{2}} .

We have

Φ′_{ω}(z)v = v_{1} \frac{∂Φ_{ω}}{∂s} (z) + v_{2} \frac{∂Φ_{ω}}{∂p} (z)

= \frac{−2v_{1}(1 − ω^{2}p) + 2v_{2}ω(2 − ωs)}{(2 − ωs)^{2}}.

Hence

\frac{|Φ′_{ω}(z)v|}{1 − |Φ_{ω}(z)|^{2}} = \frac{2|v_{1}(1 − ω^{2}p) − v_{2}(2 − ωs)|}{|2 − ωs|^{2} − |2ωp − s|^{2}}

= \frac{|v_{1}(1 − ω^{2}p) − v_{2}(2 − ωs)|}{2(1 − |p|^{2}) − ω(s − \overline{s}p) − \overline{ω}(\overline{s} − \overline{p}s)}

from which the Corollary follows. □

5 The Kobayashi distance on G

In this section we shall solve the Kobayashi extremal problem for G: we shall prove the equality K_{G} = C_{G} and describe the construction of a Kobayashi extremal function for any pair of points of G. A celebrated theorem of Lempert [14] asserts the equality of the Caratheodory and Kobayashi distances on any convex domain. G is not convex, but the conclusion K_{G} = C_{G} holds notwithstanding.

To prove that K_{G} = C_{G} we must show that for z_{1}, z_{2} ∈ G there exist λ_{1}, λ_{2} ∈ 𝔻 such that d(λ_{1}, λ_{2}) = C_{G}(z_{1}, z_{2}) and an analytic function (s, p) : 𝔻 → G such that (s, p)(λ_{j}) = z_{j} for j = 1, 2. Here is a simple necessary condition for the solvability of an interpolation problem for analytic functions from 𝔻 to G.

Theorem 5.1 Let λ_{1}, λ_{2}, …, λ_{n} be distinct points in 𝔻 and let z_{j} = (s_{j}, p_{j}) ∈ G for j = 1, …, n. If there exists an analytic function h : 𝔻 → G such that h(λ_{j}) = z_{j} for j = 1, …, n then, for all ω ∈ 𝕋,

[\frac{2(1 − p_{i}p̄_{j}) − ω(s_{i} − p_{i}s̄_{j}) − ω̄(s̄_{j} − s_{i}p̄_{j})}{1 − λ_{i}λ̄_{j}}]_{i,j=1}^{n} ≥ 0.  (5.1)

This is [3, Theorem 5.2]; here is a short proof. Suppose h as described does exist. For any ω ∈ 𝕋, since Φ_{ω} maps G analytically into 𝔻, the composition Φ_{ω} ∘h maps 𝔻 analytically into 𝔻, and Φ_{ω} ∘h(λ_{j}) = Φ_{ω}(z_{j}). Hence, by Pick’s Theorem,

[\frac{1 − Φ_{ω}(z_{i})\overline{Φ_{ω}(z_{j})}}{1 − λ_{i}λ̄_{j}}]_{i,j=1}^{n} ≥ 0.

That is,

[\frac{1 − (2 − ωs_{i})^{−1}(2ωp_{i} − s_{i})(2ω̄p̄_{j} − s̄_{j})(2 − ω̄s̄_{j})^{−1}}{1 − λ_{i}λ̄_{j}} ]_{i,j=1}^{n} ≥ 0.

This inequality is equivalent to the relation

[\frac{(2 − ωs_{i})(2 − ω̄s̄_{j}) − (2ωp_{i} − s_{i})(2ω̄p̄_{j} − s̄_{j})}{1 − λ_{i}λ̄_{j}}]_{i,j=1}^{n} ≥ 0,

which multiplies out to give the inequality (5.1). □

We shall show in a future paper that the converse of Theorem 5.1 fails to hold in general. However, to solve the Kobayashi extremal problem we only need a sufficient condition for interpolation in the case of two interpolation points, and the burden of this section is to show that the converse of Theorem 5.1 does hold if n = 2. First we deal with a special case of the n-point interpolation problem in which the 1-parameter family of Pick conditions (5.1) does suffice for the existence of an interpolating function.

Theorem 5.2 Let λ_{1}, …, λ_{n} be distinct points in 𝔻 and let (s_{j}, p_{j}) ∈ G, 1 ≤ j ≤ n. Suppose that the matrix

[\frac{1 − p_{i}p̄_{j}}{1 − λ_{i}\overline{λ}_{j}} ]_{i,j=1}^{n}  (5.2)

is singular. There exists an analytic function h : 𝔻 → Γ such that h(λ_{j}) = (s_{j}, p_{j}), 1 ≤ j ≤ n, if and only if, for all ω ∈ 𝕋,

[\frac{2(1 − p_{i}p̄_{j}) − ω(s_{i} − p_{i}s̄_{j}) − ω̄(s̄_{j} − s_{i}p̄_{j})}{1 − λ_{i}\overline{λ}_{j}}]_{i,j=1}^{n} ≥ 0.  (5.3)

Furthermore, if the inequality (5.3) is satisifed and [x_{1} … x_{n}]^{T} is a non-zero null vector for the Pick matrix (5.2), then the required interpolating function h : 𝔻 → G is given by h = (s, p) where the rational functions s, p are defined by

s = \frac{∑ x_{j}s̄_{j}k_{λ_{j}}}{∑ x_{j}p̄_{j}k_{λ_{j}}} , p = \frac{∑ x_{j}k_{λ_{j}}}{∑ x_{j}p̄_{j}k_{λ_{j}}}  (5.4)

and

k_{λ}(z) = (1 − λ̄z)^{−1}  (5.5)

for λ, z ∈ 𝔻.

For the proof of Theorem 5.2 and later results we shall need some notation. Corresponding to interpolation data λ_{j} ∈ 𝔻, (s_{j}, p_{j}) ∈ G, 1 ≤ j ≤ n, we introduce the Hilbert space

M = span {k_{λ_{1}}, …, k_{λ_{n}}} ⊂ H^{2},

where k is the Szegő kernel (5.5), and operators S, P on M given by

Sk_{λ_{j}} = s̄_{j}k_{λ_{j}}, P k_{λ_{j}} = p̄_{j}k_{λ_{j}}, 1 ≤ j ≤ n.  (5.6)

Proof of Theorem 5.2. Necessity is Theorem 5.1.

Suppose that the inequality (5.3) holds for every ω ∈ 𝕋. We may integrate (5.3) with respect to Lebesgue measure on 𝕋 to show that the Pick matrix (5.2) is positive semi-definite, and hence that the Nevanlinna-Pick problem with data λ_{j} ↦ p_{j} is solvable. Since the matrix (5.2) is singular, this Nevanlinna-Pick problem is extremally solvable. It therefore has a unique solution p, which is a Blaschke product of degree at most n − 1 [17].

The inequality (5.3) can be written

[⟨(1 − P^{∗}P − \frac{1}{2}ω(S^{∗} − P^{∗}S) − \frac{1}{2} \overline{ω}(S − S^{∗}P)) k_{λ_{j}}, k_{λ_{i}}⟩]_{i,j=1}^{n} ≥ 0

and so is equivalent to

1 − P^{∗}P − \frac{1}{2}ω(S^{∗} − P^{∗}S) − \frac{1}{2} ω̄(S − S^{∗}P) ≥ 0.  (5.7)

Since the matrix (5.2) is singular, 1 − P^{∗}P is a singular operator on M, and hence ∥P ∥ = 1. The space M is invariant under T_{p}^{∗}, the adjoint of the Toeplitz operator with symbol p, and

P = T_{p}^{∗}|M.

Let f be a unit maximizing vector of P in M. We have (1 − P^{∗}P)f = 0 and

1 = ∥P f∥ = ∥T_{p}^{∗}f∥ = ∥P_{+}(p̄f)∥ ≤ ∥p̄f∥ = ∥f∥ = 1,

where P_{+} : L^{2} → H^{2} is orthogonal projection. Hence

p̄f = P_{+}(p̄f) = P f ∈ M,

and so (since pp̄ = 1)

p = f/P f.  (5.8)

By the inequality (5.7), for any ω ∈ 𝕋 we have

⟨(1 − P^{∗}P)f − \frac{1}{2}ω(S^{∗} − P^{∗}S)f − \frac{1}{2} \overline{ω}(S − S^{∗}P)f, f⟩ ≥ 0.

The integral with respect to ω over 𝕋 of the left hand side is

⟨(1 − P^{∗}P)f, f⟩,

hence is zero, and so, for almost all ω ∈ 𝕋,

(1 − P^{∗}P − \frac{1}{2}ω(S^{∗} − P^{∗}S) − \frac{1}{2} \overline{ω}(S − S^{∗})) f = 0.  (5.9)

By continuity, this relation holds for all ω ∈ 𝕋. Equation (5.9) can be written

(2 − \overline{ω}S^{∗})(1 − Φ_{ω}(S, P)^{∗}Φ_{ω}(S, P))(2 − ωS)f = 0

while the inequality (5.7) is equivalent to

1 − Φ_{ω}(S, P)^{∗}Φ_{ω}(S, P) ≥ 0.

Thus ∥Φ_{ω}(S, P)∥ = 1 for all ω, and a maximizing vector for Φ_{ω}(S, P) is (2 − ωS)f. Observe that

Φ_{ω}(S, P)k_{λ_{j}} = (2ωP − S)(2 − ωS)^{−1}k_{λ_{j}}
= (2ωp̄_{j} − s̄_{j})(2 − ωs̄_{j})^{−1}k_{λ_{j}}
= Φ\overline{ω}(s_{j}, p_{j})^{−}k_{λ_{j}},

It follows that the Nevanlinna-Pick data

λ_{j} ↦ Φ\overline{ω}(s_{j}, p_{j}), 1 ≤ j ≤ n,

are extremally solvable, having a unique solution q_{ω}, which is a Blaschke product of degree at most n − 1, and is given explicitly by

q_{ω} = \frac{(2 − ωS)f}{Φ_{ω}(S, P)(2 − ωS)f} = \frac{2f − ωSf}{2ωP f − Sf} .

The interpolation conditions satisfied by q_{ω} are:

q_{ω}(λ_{j}) = Φω̄(s_{j}, p_{j}), 1 ≤ j ≤ n.  (5.10)

In view of equation (5.8) we have

q_{ω} = \frac{2p − ωSf/P f}{2ω − Sf/P f} = Φ\overline{ω} (\frac{Sf}{P f} , p) .

Define the rational function s by

s(λ) = \frac{Sf(λ)}{P f(λ)}.  (5.11)

Then for each ω ∈ 𝕋, Φ\overline{ω} ∘ (s, p) is the Blaschke product q_{ω}. We claim that (i) s is analytic in 𝔻, and (ii) |s| < 2 everywhere on 𝔻. To see (i), suppose that λ ∈ 𝔻 is a pole of s. Then

q_{ω}(λ) = \frac{2p − ωs}{2ω − s}(λ) = ω,

whence q_{ω} is constant on 𝔻. Thus, for all ω ∈ 𝕋,

ω = q_{ω}(λ_{1}) = Φ\overline{ω}(s_{1}, p_{1}) = \frac{2p_{1} − ωs_{1}}{2ω − s_{1}} ,

which is absurd. Hence (i) holds.

Now suppose |s| = 2 somewhere in 𝔻: say s(λ) = 2ω, for some λ ∈ 𝔻 and ω ∈ 𝕋. Since q_{ω} is analytic at λ and

q_{ω} = \frac{2p − ωs}{2ω − s}

we must have p(λ) = ω^{2}. Since p is a Blaschke product, p must be constant and so

|p_{1}| = |p(λ_{1})| = |ω^{2}| = 1,

contrary to hypothesis. Thus |s| cannot take the value 2 on 𝔻. We shall shortly see that

|s(λ_{1})| = |s_{1}| < 2,

and hence |s| < 2 throughout 𝔻 by the Intermediate Value Theorem. Equation (5.10) yields, for 1 ≤ j ≤ n and all ω ∈ 𝕋,

Φ\overline{ω}(s_{j}, p_{j}) = q_{ω}(λ_{j}) = Φ\overline{ω} ∘ (s, p)(λ_{j}) = Φ\overline{ω}(s(λ_{j}), p_{j}),

from which it follows easily that

s(λ_{j}) = s_{j}, 1 ≤ j ≤ n.

Define h to be the function (s, p). Then h is analytic from 𝔻 to ℂ^{2} and

h(λ_{j}) = (s_{j}, p_{j}), 1 ≤ j ≤ n.

In fact h maps 𝔻 into Γ. Consider any λ ∈ 𝔻. We have |s(λ)| < 2, and hence

Φ(z, s(λ), p(λ)) = \frac{2zp(λ) − s(λ)}{2 − zs(λ)}

is an analytic function of z on Δ. When z = ω ∈ 𝕋,

|Φ_{ω}(s(λ), p(λ))| = |q\overline{ω}(λ)| ≤ 1,

and so, by the Maximum Principle,

|Φ_{z}(s(λ), p(λ))| ≤ 1

for all z ∈ 𝔻. By Theorem 2.1, (s(λ), p(λ)) ∈ Γ. Thus

h = (s, p) : 𝔻 → Γ

is the desired interpolating function.

Suppose that the inequality (5.3) is satisfied and that [x_{1} … x_{n}]^{T} is a non-zero null vector for the Pick matrix (5.2). This means that

f = ∑_{j=1}^{n} x_{j}k_{λ_{j}} ∈ M

is a maximizing vector for P, and we have

Sf = ∑_{j=1}^{n} x_{j}s̄_{j}k_{λ_{j}}, P f = ∑_{j=1}^{n} x_{j}p̄_{j}k_{λ_{j}}.

On comparing with the definitions (5.8) and (5.11) we establish the formulae (5.4). □

The following observation is elementary.

Lemma 5.3 Let λ_{1}, λ_{2}, μ_{1}, μ_{2} ∈ 𝔻. The following conditions are equivalent:

(1) d(μ_{1}, μ_{2}) ≤ d(λ_{1}, λ_{2});

(2) |B_{μ_{2}}(μ_{1})| ≤ |B_{λ_{2}}(λ_{1})|;

(3) det [\frac{1 − μ_{i}μ̄_{j}}{1 − λ_{i}λ̄_{j}} ]_{i,j=1}^{2} ≥ 0.

Moreover, the equivalences remain valid when the inequality signs are replaced by strict inequalities or equalities.

The next statement is simply proved by the standard technique of Schur reduction.

Lemma 5.4 Let λ_{1}, λ_{2} be distinct points in 𝔻 and let p_{1}, p_{2} ∈ 𝔻 be such

d(λ_{1}, λ_{2}) > d(p_{1}, p_{2}).

For any ω_{1}, ω_{2} ∈ 𝕋 there exists a unique Blaschke product p of degree 2 such that

p(λ_{1}) = p_{1}, p(λ_{2}) = p_{2} and p(ω_{1}) = ω_{2}.

The technical result which follows plays an important role in the proof of the main theorem of this section. It will also be needed for the identification of the automorphisms of G in a future paper.

Theorem 5.5 Let z_{1} and z_{2} be distinct points in G. The following are equivalent:

(1) Φ_{ω} is a Caratheodory extremal for z_{1}, z_{2} for every ω ∈ 𝕋;

(2) z_{1} and z_{2} lie either on V or on a flat geodesic.

Proof. (2) ⇒ (1) Suppose (2) holds. It is readily checked that z ∈ V if and only if Φ_{ω}(z) is independent of ω. Hence if z_{1} and z_{2} lie on V then d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) is constant in ω.

If z_{1} = (s_{1}, p_{1}) and z_{2} = (s_{2}, p_{2}) lie on a flat geodesic then

s_{1} = βp_{1} + β̄ and s_{2} = βp_{2} + β̄  (5.12)

for some β ∈ 𝔻. Now Φ_{ω}(βp + β̄, p) = υ(p) for a suitable Möbius function υ independent of p. Hence

d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) = d(υ(p_{1}), υ(p_{2})) = d(p_{1}, p_{2}).

In either case, d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) is constant in ω, so that all Φ_{ω} are extremal for z_{1}, z_{2}.

(1) ⇒ (2) Suppose (1), that is, d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) is constant in ω on 𝕋. Let

β_{j} = \frac{s̄_{j} − s_{j}p̄_{j}}{1 − |p_{j}|^{2}} ∈ 𝔻, j = 1, 2,

so that

s_{j} = β_{j}p_{j} + β̄_{j}

(compare Theorem 2.1). The points z_{1}, z_{2} lie on the same flat geodesic if and only if β_{1} = β_{2}. Let

f_{j}(z) = \frac{2zp_{j} − s_{j}}{2 − zs_{j}} j = 1, 2,

and

g_{2}(z) = \frac{2p̄_{2} − s̄_{2}z}{2z − s̄_{2}} ,

so that

d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) = |F (ω)|

where

F (z) = \frac{f_{1}(z) − f_{2}(z)}{1 − g_{2}(z)f_{1}(z)}.  (5.13)

F is a rational function of degree at most 2 having constant modulus on 𝕋. There are four cases. Case 1: f_{1} and f_{2} are constant functions. This implies that z_{1}, z_{2} ∈ V. Case 2: exactly one of f_{1}, f_{2} is non-constant – say f_{2}. Then z_{1} = (2ζ, ζ^{2}) for some ζ ∈ 𝔻,

β_{1} = \frac{2ζ̄ − 2ζζ̄^{2}}{1 − |ζ|^{4}} = \frac{2ζ̄}{1 + |ζ|^{2}}

and Φ_{ω}(z_{1}) = −ζ. We have, for all ω ∈ 𝕋,

C_{G}(z_{1}, z_{2}) = d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) = d(−ζ, Φ_{ω}(z_{2}))
= |B_{−ζ} ∘ Φ_{ω}(z_{2})|.

Thus the mapping

α ↦ B_{−ζ} ∘ Φ_{α}(z_{2}) = \frac{(2p_{2} − ζs_{2})α − s_{2} + 2ζ}{(2p_{2}ζ̄ − s_{2})α − s_{2}ζ̄ + 2}

maps 𝕋 to the circle centred at 0 with radius C_{G}(z_{1}, z_{2}). It is easily seen that if the linear fractional transformation z ↦ (az + b)/(cz + d) maps 𝕋 to a circle centred at 0 then ac̄ = bd̄. Hence we have

(2p_{2} − ζs_{2})(2p̄_{2}ζ − s̄_{2}) = (s_{2} − 2ζ)(s̄_{2}ζ − 2),

which may be re-arranged to give

(s̄_{2} − s_{2}p̄_{2})ζ^{2} − 2(1 − |p_{2}|^{2})ζ + s_{2} − s̄_{2}p_{2} = 0

or

β_{2}ζ^{2} + β̄_{2} = 2ζ.

This equation may be solved to give

β_{2} = \frac{2ζ̄}{1 + |ζ|^{2}} = β_{1}.

Hence z_{1}, z_{2} lie on a flat geodesic. Case 3: f_{1} and f_{2} are non-constant and there is cancellation on the right hand side of equation (5.13). That is, there exists α ∈ ℂ such that

f_{1}(α) = f_{2}(α) and f_{1}(α)g_{2}(α) = 1.

Then f_{2}(α)g_{2}(α) = 1, and so

(2p_{2}α − s_{2})(2p̄_{2} − s̄_{2}α) = (2 − s_{2}α)(2α − s̄_{2}),

which simplifies to

α^{2}(s_{2} − s̄_{2}p_{2}) − 2α(1 − |p_{2}|^{2}) + s̄_{2} − s_{2}p̄_{2} = 0.

We have

β̄_{2}α^{2} + β_{2} = 2α,

which we may solve for β_{2} to give

β_{2} = \frac{2α}{1 + |α|^{2}} .

Hence

f_{2}(α) = \frac{2p_{2}α − β_{2}p_{2} − β̄_{2}}{2 − α(β_{2}p_{2} + β̄_{2})} = −ᾱ,

and therefore

f_{1}(α) = f_{2}(α) = −ᾱ.

Thus

2p_{1}α − s_{1} = −ᾱ(2 − αs_{1})

which yields

s_{1} = \frac{2α}{1 + |α|^{2}} p_{1} + \frac{2ᾱ}{1 + |α|^{2}}
= β_{2}p_{1} + β̄_{2}.

Hence z_{1}, z_{2} both lie on the flat geodesic s = β_{2}p + β̄_{2}.

Case 4: f_{1} and f_{2} are non-constant and there is no cancellation on the right hand side of equation (5.13). On expanding this equation we obtain

F (z) = \frac{(s_{1}p_{2} − s_{2}p_{1})z^{2} + 2(p_{1} − p_{2})z + s_{2} − s_{1}}{−(s_{1} − s̄_{2}p_{1})z^{2} + 2(1 − p̄_{2}p_{1})z − s̄_{2} + p̄_{2}s_{1}} .

Since f_{1}, f_{2} are of degree exactly 1 and there is no cancellation in equation (5.13), F has 2 zeros (counting multiplicities) in ℂ. The relation

F (z)\overline{F (\frac{1}{z̄})} = C_{G}(z_{1}, z_{2})^{2}

holds on 𝕋, and hence wherever it is defined on ℂ. Thus, if α is a zero of F then 1/ᾱ is a pole of F, with the same multiplicity. It follows that there exists c ∈ ℂ such that |c| = C_{G}(z_{1}, z_{2}) and

s_{2}p_{1} − s_{1}p_{2} = c(s_{2} − s̄_{1}p_{2}),  (5.14)
2(p_{2} − p_{1}) = −c2(1 − p̄_{1}p_{2}),  (5.15)
s_{1} − s_{2} = c(s̄_{1} − s_{2}p̄_{1}).  (5.16)

By equation (5.15)

c = −B_{p_{1}}(p_{2})

and so

p_{1} − c = \frac{p_{2}(1 − |p_{1}|^{2})}{1 − p̄_{1}p_{2}}.

By equation (5.14),

s_{2}(p_{1} − c) = (s_{1} − cs̄_{1})p_{2},

and so

\frac{s_{2}p_{2}(1 − |p_{1}|^{2})}{1 − p̄_{1}p_{2}} = (s_{1} − cs̄_{1})p_{2}.  (5.17)

If p_{2} = 0 then β_{2} = s̄_{2}, c = p_{1} and (from equation (5.16))

s_{1} − s̄_{1}p_{1} = s_{2}(1 − |p_{1}|^{2}),

whence β_{1} = s̄_{2} = β_{2}. That is, z_{1}, z_{2} lie on a flat geodesic. On the other hand, if p_{2} ≠ 0 we have, from equation (5.17),

s_{2} = \frac{(1 − p̄_{1}p_{2})(s_{1} − cs̄_{1})}{1 − |p_{1}|^{2}}
= \frac{(1 − p̄_{1}p_{2})s_{1} + (p_{2} − p_{1})s̄_{1}}{1 − |p_{1}|^{2}}
= \frac{s̄_{1} − p̄_{1}s_{1}}{1 − |p_{1}|^{2}} p_{2} + \frac{s_{1} − s̄_{1}p_{1}}{1 − |p_{1}|^{2}}
= β_{1}p_{2} + β̄_{1}.

Thus z_{1}, z_{2} lie on a flat geodesic.

We have shown that in all four cases, either z_{1}, z_{2} ∈ V or z_{1}, z_{2} lie on a flat geodesic. Hence (1) ⟺ (2). □

We shall now prove the main result of this section, which contains the equality of the Caratheodory and Kobayashi distances on G and is a refinement of Theorem 1.2.

Theorem 5.6 For distinct points z_{1} = (s_{1}, p_{1}) and z_{2} = (s_{2}, p_{2}) in G let ω_{0} ∈ 𝕋 be such that Φ_{ω_{0}} is a Caratheodory extremal function for z_{1}, z_{2} and let λ_{j} = Φ_{ω_{0}}(z_{j}) ∈ 𝔻 for j = 1, 2. There exists a unique Blaschke product p of degree at most 2 such that

p(λ_{1}) = p_{1}, p(λ_{2}) = p_{2} and p(\overline{ω}_{0}) = \overline{ω}_{0}^{2}.

Furthermore if s is defined by

s(λ) = 2 \frac{ω_{0}p(λ) − λ}{1 − ω_{0}λ}, λ ∈ 𝔻,

and h = (s, p) : 𝔻 → ℂ^{2} then h maps 𝔻 to G, the composition Φ_{ω_{0}} ∘ h is the identity on 𝔻 and h is a Kobayashi extremal function for the points z_{1}, z_{2}.

Proof. Clearly λ_{1} and λ_{2} are distinct. We have

d(λ_{1}, λ_{2}) = C_{G}(z_{1}, z_{2}) ≥ d(p_{1}, p_{2}),

this inequality holding because f : G → 𝔻 : (s, p) ↦ p is analytic and f(z_{j}) = p_{j}.

We consider two cases. Case 1: d(λ_{1}, λ_{2}) = d(p_{1}, p_{2}). Then

[ \frac{1 − p̄_{i}p_{j}}{1 − λ̄_{i}λ_{j}} ]_{i,j=1}^{2} ≥ 0.  (5.18)

and the left hand side is a singular matrix. By Theorem 5.2 there exists an analytic function h = (s, p) : 𝔻 → G such that h(λ_{1}) = (s_{1}, p_{1}), h(λ_{2}) = (s_{2}, p_{2}). According to the Schwarz-Pick Lemma, there is a unique function f in the Schur class mapping λ_{1}, λ_{2} to p_{1}, p_{2} respectively, and f is a Möbius function. Since p is a Schur function and maps λ_{1}, λ_{2} to p_{1}, p_{2}, it must be that p = f and so p is a Möbius function. Moreover, for j = 1, 2,

Φ_{ω_{0}} ∘ h(λ_{j}) = Φ_{ω_{0}}(z_{j}) = λ_{j}.

Thus Φ_{ω_{0}} ∘ h is an analytic self-map of 𝔻 which preserves a pair of distinct points, hence is the identity. Therefore, for all λ ∈ 𝔻,

\frac{2ω_{0}p(λ) − s(λ)}{2 − ω_{0}s(λ)} = λ,

which equation can be re-arranged to give

s(λ) = 2\frac{ω_{0}p(λ) − λ}{1 − ω_{0}λ}.

Since s is rational and is bounded on 𝔻, we have

p(ω̄_{0}) = ω̄_{0}^{2}.

Hence all the conclusions of the Theorem hold in Case 1.

Case 2: d(λ_{1}, λ_{2}) > d(p_{1}, p_{2}). By Corollary 3.5, d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) is the modulus of a rational function of ω, and so either it is constant in ω or it attains its maximum on 𝕋 only finitely often. If it is constant then by Theorem 5.5 either z_{1}, z_{2} lie on the royal variety or there exists β ∈ 𝔻 such that s_{j} = βp_{j} + β̄. In the latter case we find, for any choice of ω ∈ 𝕋,

λ_{j} = Φ_{ω}(βp_{j} + β̄, p_{j})
= cB_{γ}(p_{j})

where

c = \frac{2ω − β}{2 − ωβ̄} ∈ 𝕋 and γ = \frac{β̄}{2ω − β} ∈ 𝔻.

It follows that d(λ_{1}, λ_{2}) = d(p_{1}, p_{2}) and we are in Case 1.

Hence z_{1}, z_{2} lie on V, say z_{j} = (2ζ_{j}, ζ_{j}^{2}) for some ζ_{j} ∈ 𝔻. Then

λ_{j} = Φ_{ω}(2ζ_{j}, ζ_{j}^{2}) = −ζ_{j},

and so we may define h : 𝔻 → G satisfying h(λ_{j}) = z_{j} by

h(λ) = (−2λ, λ^{2}).

It then follows exactly as in Case 1 that all the assertions of the theorem hold.

Alternatively, there are only finitely many ω ∈ 𝕋 such that Φ_{ω} is a Caratheodory extremal for z_{1} and z_{2}. Let E be the set of ω ∈ 𝕋 such that Φ_{ω} is not a Caratheodory extremal for z_{1} and z_{2}. E has finite complement in 𝕋.

By Lemma 5.4, there exists a unique Blaschke product p of degree 2 such that

p(λ_{1}) = p_{1}, p(λ_{2}) = p_{2} and p(ω̄_{0}) = ω̄_{0}^{2}.

Let s be the rational function of degree at most 2 defined by

s(λ) = 2 \frac{ω_{0}p(λ) − λ}{1 − ω_{0}λ}.

Since p(ω̄_{0}) = ω̄_{0}^{2} the singularity at λ = ω̄_{0} is removable, and so s is analytic on Δ and has the same denominator as p. Let h = (s, p). It is clear that h is a rational function and is analytic on Δ. We have, for j = 1, 2,

s(λ_{j}) = 2\frac{ω_{0}p(λ_{j}) − λ_{j}}{1 − ω_{0}λ_{j}} = 2\frac{ω_{0}p_{j} − Φ_{ω_{0}}(s_{j}, p_{j})}{1 − ω_{0}Φ_{ω_{0}}(s_{j}, p_{j})}
= s_{j},

and hence h(λ_{j}) = z_{j}.

We shall show that h maps 𝔻 into G. For ω ∈ 𝕋 and λ ∈ Δ let

φ_{ω}(λ) = Φ_{ω} ∘ h(λ) = \frac{2ωp(λ) − s(λ)}{2 − ωs(λ)}

wherever this ratio is defined, that is, for λ such that s(λ) ≠ 2ω̄. This definition makes φ_{ω} rational of degree at most 2. For λ ∈ 𝕋 \ {ω̄_{0}} we have, since |p(λ)| = 1,

s(λ) − s̄(λ)p(λ) = 2\frac{ω_{0}p(λ) − λ}{1 − ω_{0}λ} − 2\frac{ω̄_{0}p̄(λ) − λ̄}{1 − ω̄_{0}λ̄}p(λ)\frac{ω_{0}λ}{ω_{0}λ}
= 2\frac{ω_{0}p(λ) − λ}{1 − ω_{0}λ} − 2\frac{λ − ω_{0}p(λ)}{ω_{0}λ − 1}
= 0,

and thus

Re {ω (s(λ) − s̄(λ)p(λ))} = 0 = 1 − |p(λ)|^{2}.

Hence, by Theorem 2.5, for ω ∈ 𝕋,

|φ_{ω}(λ)| = |Φ_{ω}(s(λ), p(λ))| = 1

for all λ ∈ 𝕋 except for those λ such that s(λ) = 2ω̄ (of which there are at most 3). Thus φ_{ω} is a rational function of degree at most 2 which is unimodular on 𝕋. It follows that φ_{ω} is expressible in one of the forms

φ_{ω} = cB_{α_{1}}B_{α_{2}} or φ_{ω} = cB_{α} or φ_{ω} = c  (5.19)

for some c ∈ 𝕋 and α, α_{1}, α_{2} ∈ ℂ \ 𝕋, where

B_{α}(λ) =^{def} \frac{λ − α}{1 − ᾱλ};

note that |α| can be either greater than or less than one.

Our next goal is to show that in fact φ_{ω} is inner. For ω ∈ 𝕋 consider the kernel

K_{ω}(λ, μ) = \frac{1 − φ̄_{ω}(μ)φ_{ω}(λ)}{1 − μ̄λ},

defined on 𝔻, with the possible exception of at most two points (the poles of φ_{ω}). In particular, K_{ω} is defined at (λ_{i}, λ_{j}) for i, j = 1, 2, and

K_{ω}(λ_{i}, λ_{j}) = \frac{1 − φ̄_{ω}(λ_{j})φ_{ω}(λ_{i})}{1 − λ̄_{j}λ_{i}} = \frac{1 − Φ_{ω}(z_{i})Φ̄_{ω}(z_{j})}{1 − λ_{i}λ̄_{j}}.

For ω ∈ E,

d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) < d(λ_{1}, λ_{2})

and hence, by Lemma 5.2,

[\frac{1 − Φ_{ω}(z_{i})Φ̄_{ω}(z_{j})}{1 − λ_{i}λ̄_{j}}]_{i,j=1}^{2} > 0.

That is, the two-by-two kernel [K_{ω}(λ_{i}, λ_{j})] is positive definite. Thus the kernel K_{ω} has rank at least 2 for every ω ∈ E.

We now show that, for ω ∈ E, φ_{ω} is of the form cB_{α_{1}}B_{α_{2}} for some c ∈ 𝕋 and α_{1}, α_{2} ∈ 𝔻. The relation

1 − B̄_{α}(μ)B_{α}(λ) = \frac{(1 − |α|^{2})(1 − μ̄λ)}{(1 − αμ̄)(1 − ᾱλ)}

is valid where defined. Hence, if φ_{ω} in the expressions (5.19) has degree 0 or 1 then K_{ω} either is identically zero or is the rank one kernel

K_{ω}(λ, μ) = \frac{1 − |α|^{2}}{(1 − αμ̄)(1 − ᾱλ)},

in either case contradicting the fact that the rank of K_{ω} is at least 2. Hence φ_{ω} is of the form cB_{α_{1}}B_{α_{2}}, and so, for ω ∈ E,

K_{ω}(λ, μ) = \frac{1 − B̄_{α_{1}}(μ)B_{α_{1}}(λ)}{1 − μ̄λ} + B̄_{α_{1}}(μ)\frac{1 − B̄_{α_{2}}(μ)B_{α_{2}}(λ)}{1 − μ̄λ}B_{α_{1}}(λ)
= (1 − |α_{1}|^{2})f̄_{1}(μ)f_{1}(λ) + (1 − |α_{2}|^{2})f̄_{2}(μ)f_{2}(λ)

where

f_{1}(λ) = \frac{1}{1 − ᾱ_{1}λ}, f_{2}(λ) = \frac{B_{α_{1}}(λ)}{1 − ᾱ_{2}λ}.

Thus

[K_{ω}(λ_{i}, λ_{j})]_{i,j=1}^{2} = (1 − |α_{1}|^{2})M_{1} + (1 − |α_{2}|^{2})M_{2}

for some positive rank one matrices M_{1}, M_{2}. Since the left hand side is positive definite, it follows that

1 − |α_{1}|^{2} > 0, 1 − |α_{2}|^{2} > 0.

Thus α_{1}, α_{2} ∈ 𝔻, and φ_{ω} is a Blaschke product for ω ∈ E.

We claim that |s(λ)| < 2 for all λ ∈ 𝔻. For suppose that s(λ_{0}) = 2ω̄_{1} for some ω_{1} ∈ 𝕋 and λ_{0} ∈ 𝔻. For ω ∈ E, ω ≠ ω_{1}, we have

φ_{ω}(λ_{0}) = \frac{2ωp(λ_{0}) − s(λ_{0})}{2 − ωs(λ_{0})} = \frac{2ωp(λ_{0}) − 2ω̄_{1}}{2 − 2ωω̄_{1}} .

We know that |φ_{ω}(λ_{0})| < 1, and on letting ω → ω_{1}, ω ∈ E, we deduce that p(λ_{0}) = ω̄_{1}^{2} contrary to the fact that p is a non-constant Blaschke product. Thus |s(λ)| never takes the value 2 on 𝔻. However,

|s(λ_{1})| = |s_{1}| < 2,

and so, by the Intermediate Value Theorem,

|s(λ)| < 2 for all λ ∈ 𝔻.

It follows that φ_{ω}(= Φ_{ω} ∘ h) is well defined and analytic on 𝔻 for all ω ∈ 𝕋. For ω ∈ E, φ_{ω} is a Blaschke product, and hence by continuity φ_{ω} is inner for all ω ∈ 𝕋. Now φ_{ω} cannot be constant, else we have

1 = |φ_{ω}(λ_{1})| = |Φ_{ω}(z_{1})|,

and hence, by Theorem 2.5

ω(s_{1} − s̄_{1}p_{1}) = 1 − |p_{1}|^{2},

whereas in fact, by Theorem 2.1

|s_{1} − s̄_{1}p_{1}| < 1 − |p_{1}|^{2}.

We deduce that, for any λ ∈ 𝔻, |s(λ)| < 2 and

|Φ_{ω}(s(λ), p(λ))| = |φ_{ω}(λ)| < 1

for all ω ∈ 𝕋. Hence, by Theorem 2.1, (s(λ), p(λ)) ∈ G. Thus h is an analytic function from 𝔻 to G.

To conclude we must show that h is a Kobayashi extremal for the points z_{1} = h(λ_{1}), z_{2} = h(λ_{2}). It suffices to observe that Φ_{ω_{0}} ∘ h is the identity on 𝔻:

Φ_{ω_{0}} ∘ h(λ) = \frac{2ω_{0}p(λ) − s(λ)}{2 − ω_{0}s(λ)}
= \frac{2ω_{0}p(λ)(1 − ω_{0}λ) − 2(ω_{0}p(λ) − λ)}{2(1 − ω_{0}λ) − ω_{0}2(ω_{0}p(λ) − λ)} = λ.

This concludes the proof of Case 2 and so of the Theorem. □

Corollary 5.7

C_{G} = K_{G} = δ_{G}.

Proof. Let z_{1}, z_{2} be distinct points in G. We always have [12],

C_{G}(z_{1}, z_{2}) ≤ K_{G}(z_{1}, z_{2}) ≤ δ_{G}(z_{1}, z_{2}).

By Theorem 1.1 there exists ω_{0} ∈ 𝕋 such that Φ_{ω_{0}} is a Caratheodory extremal for z_{1} and z_{2}, that is,

C_{G}(z_{1}, z_{2}) = d(Φ_{ω_{0}}(z_{1}), Φ_{ω_{0}}(z_{2})).

Let λ_{1} = Φ_{ω_{0}}(z_{1}), λ_{2} = Φ_{ω_{0}}(z_{2}). Then λ_{1}, λ_{2} ∈ 𝔻, and by Theorem 5.6 there exists an analytic function h : 𝔻 → G such that h(λ_{1}) = z_{1}, h(λ_{2}) = z_{2} . By definition of δ_{G},

δ_{G}(z_{1}, z_{2}) ≤ d(λ_{1}, λ_{2}) = C_{G}(z_{1}, z_{2}).

Hence

C_{G}(z_{1}, z_{2}) = K_{G}(z_{1}, z_{2}) = δ_{G}(z_{1}, z_{2}).
□

Corollary 5.8 Through any pair of distinct points of G there is at least one complex geodesic.

Proof. Let z_{1}, z_{2} be distinct in G. By the defininition of δ_{G} and a normal families argument we may pick an analytic function h : 𝔻 → G such that

d(λ_{1}, λ_{2}) = δ_{G}(z_{1}, z_{2})

for some pair of points λ_{1}, λ_{2} ∈ 𝔻 such that h(λ_{1}) = z_{1} and h(λ_{2}) = z_{2}. Let Φ be a Caratheodory extremal function for z_{1}, z_{2}. Since δ_{G} = C_{G} we have

d(λ_{1}, λ_{2}) = C_{G}(z_{1}, z_{2}) = d(Φ ∘ h(λ_{1}), Φ ∘ h(λ_{2})),

and since Φ ∘ h is an analytic self map of 𝔻, it follows that Φ ∘ h is a Möbius function. Thus, for any λ, μ ∈ 𝔻,

C_{G}(h(λ), h(μ)) ≥ d(Φ ∘ h(λ), Φ ∘ h(μ)) = d(λ, μ).

On the other hand, we have C_{G}(h(λ), h(μ)) ≤ d(λ, μ) by Schwarz’ Lemma, so that h is an isometry with respect to d and C_{G}, that is, h is a complex geodesic of G. □

In a subsequent paper we shall prove that there is in fact exactly one complex geodesic through any pair of distinct points of G [9] and shall derive a concrete formula for the general complex geodesic of G. This formula was given in [2], where also a realization formula for the general complex geodesic of G is derived.

We conclude the section by showing that the converse of Theorem 5.1 holds when n = 2: the Pick condition (5.1) is sufficient for the existence of an interpolating function in the case n = 2. Let λ_{1}, λ_{2} ∈ 𝔻, let z_{1} = (s_{1}, p_{1}), z_{2} = (s_{2}, p_{2}) ∈ G and suppose that

[\frac{2(1 − p_{i}p̄_{j}) − ω(s_{i} − p_{i}s̄_{j}) − ω̄(s̄_{j} − s_{i}p̄_{j})}{1 − λ_{i}λ̄_{j}}]_{i,j=1}^{2} ≥ 0

for all ω ∈ 𝕋. Then

[\frac{1 − Φ_{ω}(z_{i})Φ̄_{ω}(z_{j})}{1 − λ_{i}λ̄_{j}}]_{i,j=1}^{2} ≥ 0,

and so, by Theorem 2.5, for all ω ∈ 𝕋

d(Φ_{ω}(z_{1}), Φ_{ω}(z_{2})) ≤ d(λ_{1}, λ_{2}).

On taking the supremum over 𝕋 we infer from Theorem 1.1 that

C_{G}(z_{1}, z_{2}) ≤ d(λ_{1}, λ_{2}),

and hence, by Corollary 5.7,

δ_{G}(z_{1}, z_{2}) ≤ d(λ_{1}, λ_{2}).

Thus there exist μ_{1}, μ_{2} ∈ 𝔻 and an analytic function f : 𝔻 → G such that f(μ_{1}) = z_{1}, f(μ_{2}) = z_{2} and

d(μ_{1}, μ_{2}) = δ_{G}(z_{1}, z_{2}) ≤ d(λ_{1}, λ_{2}).

By the Schwarz-Pick Lemma there exists an analytic self-map g of 𝔻 such that g(λ_{1}) = μ_{1} and g(λ_{2}) = μ_{2}. Then h =^{def} f ∘ g : 𝔻 → G is analytic and satisfies h(λ_{1}) = z_{1}, h(λ_{2}) = z_{2}, as required. □

6 Some more geometry

In forthcoming papers we plan to prove further results about the geometry of G. We shall identify the automorphisms of G, and this will enable us to show that G is inhomogeneous. We shall give a concrete description of the complex geodesics of G [9], and shall calculate the second order indicatrices of G, thereby proving that G is not convexifiable [8].

References

[1] J. Agler, Operator theory and the Caratheodory metric, Inventiones Math. 101 (1990) 483-500.

[2] J. Agler, F. B. Yeh and N. J. Young, Realization of functions into the symmetrised bidisc, in D. Alpay(Ed.), Reproducing kernel spaces and applications. Operator Theory: Advances and Applications 143(2003) 1-37, Birkhaüser Verlag.

[3] J. Agler and N. J. Young, A commutant lifting theorem for a domain in ℂ^{2} and spectral interpolation, J. Functional Analysis 161 (1999) 452– 477.

[4] J. Agler and N. J. Young, Operators having the symmetrized bidisc as a spectral set, Proc Edinburgh Math. Soc. 43 (2000) 195-210.

[5] J. Agler and N.J. Young, The two-point spectral Nevanlinna-Pick problem, Integral Equations Operator Theory 37 (2000) 375–385.

[6] J. Agler and N.J. Young, A Schwarz Lemma for the symmetrized bidisc, Bull. London Math. Soc. 33 (2001) 175-186.

[7] J. Agler and N. J. Young, A model theory for Γ-contractions, J. Operator Theory 49 (2003) 45-60.

[8] J. Agler and N.J. Young, Nonconvexity of the symmetrized bidisc, in preparation.

[9] J. Agler and N.J. Young, The complex geodesics of the symmetrised bidisc, in preparation.

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[13] S. Kobayashi, Hyperbolic Complex Spaces, Springer Verlag, Berlin, 1998.

[14] L. Lempert, Complex geometry in convex domains. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 759–765, Amer. Math. Soc., Providence, RI, 1987.

[15] H. R. Morton, Symmetric products of the circle, Proc. Cambridge Philos. Soc. 63 (1967) 349–352.

[16] I. Schur, Uber Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. für Math. 147(1917), 205-232 and 148 (1918), 122-145.

[17] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, AMS Colloquium Publications Vol. XX, New York, 1935.

Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA.

Department of Mathematics, University of Newcastle upon Tyne, Merz Court, Newcastle upon Tyne NE1 7RU, England.