Realization formulae for bounded holomorphic functions on certain - - PowerPoint PPT Presentation

Realization formulae for bounded holomorphic functions on certain domains and an application to the Carath´ eodory extremal problem

Nicholas Young

Leeds and Newcastle Universities Joint work with Jim Agler, UCSD and Zina Lykova, Newcastle CIRM, Trento, October 2018

Themes The idea of a Hilbert space model of a bounded analytic function (related to realization formulae) The use of models in complex geometry

The Carath´ eodory extremal problem Let Ω be a domain in Cd and let (λ, v) be a point in the complex tangent bundle TΩ. Thus λ ∈ Ω and v ∈ TλΩ ∼ Cd. Define |λ, v|car

def

= sup

F∈D(Ω)

|F∗(λ, v)| = sup

F∈D(Ω)

|DvF(λ)| 1 − |F(λ)|2. Here D(Ω) is the set of holomorphic maps from Ω to D and F∗ : TΩ → TD is the pushforward by F. Given (λ, v), find |λ, v|car and all extremal F : Ω → D for which it is attained, the Carath´ eodory extremal functions.

Non-uniqueness of Carath´ eodory extremals Let (λ, v) ∈ TΩ. If F : Ω → D is a Carath´ eodory extremal for (λ, v) and m ∈ Aut D, then m ◦ F is also an extremal (because of the invariance of the Poincar´ e metric on D under automorphisms). We say that an extremal function F for (λ, v) is essentially unique if the only Carath´ eodory extremal functions for (λ, v) are the functions m ◦ F, m ∈ Aut D. Extremal functions need not be essentially unique. When Ω is the bidisc D2 and λ = (0, 0), v = (1, 1) then z1 and z2 are inequivalent Carath´ eodory extremal functions. There is a large class of pairwise inequivalent Carath´ eodory extremals for (λ, v), indexed by [0, 1] × ¯ D(D2).

Models for ¯ D(D) A model for a function ϕ ∈ ¯ D(D) is a pair (M, u) where M is a Hilbert space and u is a map from D to M such that, for all λ, µ ∈ D, 1 − ϕ(µ)ϕ(λ) = (1 − ¯ µλ) u(λ), u(µ)M . Theorem Every function in ¯ D(D) has a model. A standard argument via “lurking isometries” then shows: every function ϕ ∈ ¯ D(D2) is expressible by a realization for- mula of the form ϕ(λ) = A + Bλ(1 − Dλ)−1C where

• A

B C D

• is a contractive operator on C ⊕ M.

Models for ¯ D(D2) A model for a function ϕ ∈ ¯ D(D2) is a pair (M1 ⊕ M2, u) where M1, M2 are Hilbert spaces and u = (u1, u2) is a map from D2 to M1 ⊕ M2 such that, for all λ = (λ1, λ2), µ = (µ1, µ2) ∈ D2, 1−ϕ(µ)ϕ(λ) = (1 − ¯ µ1λ1) u1(λ), u1(µ)M1 + (1 − ¯ µ2λ2) u2(λ), u2(µ)M2 . Theorem (Agler, 1990) Every function in ¯ D(D2) has a model.

The symmetrized bidisc G

This is the set G def = {(z + w, zw) : z, w ∈ D}.

s (−2,1) (0,−1) (2,1) p

G ∩ R2 in the (s, p)-plane

The Carath´ eodory problem in G

G = {(z+w, zw) : |z| < 1, |w| < 1} = {(s, p) : |s−¯ sp| < 1−|p|2}.

• Theorem. Let (λ, v) ∈ TG. There exists ω ∈ T such that

the rational function Φω(s, p) = 2ωp − s 2 − ωs is a Carath´ eodory extremal function for the tangent (λ, v). Thus, if λ = (s, p), v = (v1, v2), |λ, v|car = sup

ω∈T

|(Φω)∗(λ, v)| = sup

ω∈T

• v1(1 − ω2p) − v2ω(2 − ωs)

(s − ¯ sp)ω2 − 2(1 − |p|2)ω + ¯ s − ¯ ps

• .

Models for ¯ D(G) A model for a function ϕ : G → C is a triple (M, T, u) where M is a separable Hilbert space, T is a unitary operator on M and u : G → M is a map such that, for all λ, µ ∈ G, 1 − ϕ(µ)ϕ(λ) =

1 − ΦT(µ)∗ΦT(λ) u(λ), u(µ)M .

Here, if λ = (s, p) ∈ G, ΦT(λ) def = (2pT − s)(2 − sT)−1. Theorem A function ϕ : G → C has a model if and only if ϕ ∈ ¯ D(G). Proof is by symmetrization of the model for the bidisc.

Theorem Generically, Carath´ eodory extremals are unique in G. That is: Let λ ∈ G. For a generic direction Cv ∈ CP 2, there is an essentially unique Carath´ eodory extremal function for the tangent (λ, v) ∈ TG. In fact there is a smooth curve C and a finite set E in CP 2 such that essential uniqueness holds for all Cv / ∈ C ∪ E.

Theorem (Kosi´ nski-Zwonek 2016) If (λ, v) ∈ TG and there is a unique ω0 ∈ T such that Φω0 is a Carath´ eodory extremal function for (λ, v), then every Carath´ eodory extremal function for (λ, v) has the form m ◦ Φω0 for some automorphism m of D. Proof. Let ϕ be an extremal for (λ, v). Then ϕ has a model (M, T, u). Let the spectral decomposition of T be

• T ω dE(ω). Then, for any λ, µ ∈ G,

1 − ϕ(µ)ϕ(λ) = (1 − ΦT(µ)∗ΦT(λ))u(λ), u(µ)M =

• T
• 1 − Φω(µ)Φω(λ)
• dE(ω)u(λ), u(µ)M .

Proof continued ϕ has a holomorphic right inverse k : D → G. Put λ = k(z), µ = k(w). For all z, w ∈ D, 1 − ¯ wz =

• T
• 1 − Φω ◦ k(w)Φω ◦ k(z)
• dE(ω)u ◦ k(z), u ◦ k(w)M .

Hence 1 = I1 + I2 where I1(z, w) = 1 − Φω0 ◦ k(w)Φω0 ◦ k(z) 1 − ¯ wz E({ω0})u ◦ k(z), u ◦ k(w)M , I2(z, w) =

• T\{ω0}

1 − Φω ◦ k(w)Φω ◦ k(z) 1 − ¯ wz dE(ω)u ◦ k(z), u ◦ k(w) . The integrand is a positive kernel of rank 1 if ω = ω0, and

• f rank 2 if ω = ω0. Since the left hand side 1 is positive of

rank 1, it follows that I2 ≡ 0 and E(ω0)u ◦ k is a constant x.

Proof – end Recall the identity 1 − ϕ(µ)ϕ(λ) =

• T
• 1 − Φω(µ)Φω(λ)
• dE(ω)u(λ), u(µ)M .

Put µ = k(w) again, but let λ be a general point of G. We

• btain

1 − ¯ wϕ(λ) = (1 − ¯ wΦω0(λ)) u(λ), x from which it follows that u(λ), x = 1 and ϕ = Φω0.

Essentially non-unique extremals Consider a tangent of the form λ = (2z, z2), v = 2c(1, z) for some z ∈ D and c = 0 (a royal tangent). Φω is an extremal for (λ, v) for all ω ∈ T. The Carath´ eodory extremal functions for (λ, v) are the func- tions ϕ(s, p) = m

 1

2s + 1 4(s2 − 4p)

Ψ(s, p) 1 − 1

2sΨ(s, p)

 

where m ∈ Aut D and Ψ ∈ ¯ D(G).

Purely balanced tangents These are the tangents in TG for which Φω is a Carath´ eodory extremal for precisely 2 values of ω ∈ T. A variant of the above argument via models gives some information about the extremal functions for such models, but not yet a full description. Suppose (λ, v) is a purely balanced tangent and Φζ, Φη are the two extremal Φωs for (λ, v). Then (Φζ, Φη) is an iso- morphism of G with an open subset of D2. Using this fact we can write down a large class of Carath´ eodory extremals indexed by [0, 1]× ¯ D(D2), though we do not claim that these are all extremals.

Flat tangents These are the tangents (λ, v) ∈ TG where λ = (s, p), v = c(¯ s − ¯ ps, 1 − |p|2) for some c = 0. Φω is extremal for every flat tangent and every ω ∈ T. We construct a class of inequivalent Carath´ eodory extremal functions for flat (λ, v), parametrized by ¯ D(D).

References

• L. Kosi´

nski and W. Zwonek, Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock, J. Geom. Analysis 26 (2016) 1863– 1890.

• J. Agler and N. J. Young, Realization of functions on the

symmetrized bidisc, J. Math. Anal. Applic. 453 (2017) 227–240.

• J. Agler, Z. A. Lykova and N. J. Young, Carath´

eodory ex- tremal functions on the symmetrized bidisc, arXiv:1802.09067

The end