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 C d and let ( λ, v ) be a point in the complex tangent bundle T Ω. Thus λ ∈ Ω and v ∈ T λ Ω ∼ C d . Define def | λ, v | car = sup | F ∗ ( λ, v ) | F ∈ D (Ω) | D v F ( λ ) | = sup 1 − | F ( λ ) | 2 . F ∈ D (Ω) Here D (Ω) is the set of holomorphic maps from Ω to D and F ∗ : T Ω → T D 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 ) ∈ T Ω. 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 D 2 and λ = (0 , 0) , v = (1 , 1) then z 1 and z 2 are inequivalent Carath´ eodory extremal functions. There is a large class of pairwise inequivalent Carath´ eodory D ( D 2 ). extremals for ( λ, v ), indexed by [0 , 1] × ¯

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: D ( D 2 ) is expressible by a realization for- every function ϕ ∈ ¯ mula of the form ϕ ( λ ) = A + Bλ (1 − Dλ ) − 1 C � � A B where is a contractive operator on C ⊕ M . C D

D ( D 2 ) Models for ¯ D ( D 2 ) is a pair ( M 1 ⊕ M 2 , u ) A model for a function ϕ ∈ ¯ where M 1 , M 2 are Hilbert spaces and u = ( u 1 , u 2 ) is a map from D 2 to M 1 ⊕ M 2 such that, for all λ = ( λ 1 , λ 2 ) , µ = ( µ 1 , µ 2 ) ∈ D 2 , 1 − ϕ ( µ ) ϕ ( λ ) = µ 1 λ 1 ) � u 1 ( λ ) , u 1 ( µ ) � M 1 + (1 − ¯ µ 2 λ 2 ) � u 2 ( λ ) , u 2 ( µ ) � M 2 . (1 − ¯ D ( D 2 ) has a model. Theorem (Agler, 1990) Every function in ¯

The symmetrized bidisc G This is the set G def = { ( z + w, zw ) : z, w ∈ D } . p (2,1) (−2,1) s (0,−1) G ∩ R 2 in the ( s, p ) -plane

The Carath´ eodory problem in G sp | < 1 −| p | 2 } . G = { ( z + w, zw ) : | z | < 1 , | w | < 1 } = { ( s, p ) : | s − ¯ Theorem. Let ( λ, v ) ∈ TG . There exists ω ∈ T such that the rational function Φ ω ( s, p ) = 2 ωp − s 2 − ωs is a Carath´ eodory extremal function for the tangent ( λ, v ). Thus, if λ = ( s, p ) , v = ( v 1 , v 2 ), = sup | (Φ ω ) ∗ ( λ, v ) | | λ, v | car ω ∈ T v 1 (1 − ω 2 p ) − v 2 ω (2 − ωs ) � � � � = sup � . � � sp ) ω 2 − 2(1 − | p | 2 ) ω + ¯ � � ( s − ¯ s − ¯ ps ω ∈ T �

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 , � u ( λ ) , u ( µ ) � M . � 1 − Φ T ( µ ) ∗ Φ T ( λ ) 1 − ϕ ( µ ) ϕ ( λ ) = � Here, if λ = ( s, p ) ∈ G , Φ T ( λ ) def = (2 pT − 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: For a generic direction C v ∈ CP 2 , there is an Let λ ∈ G . 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 C v / ∈ C ∪ E .

Theorem (Kosi´ nski-Zwonek 2016) If ( λ, v ) ∈ TG and there is a unique ω 0 ∈ T such that Φ ω 0 is a Carath´ eodory extremal function for ( λ, v ), then every Carath´ eodory extremal function for ( λ, v ) has the form m ◦ Φ ω 0 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 − Φ T ( µ ) ∗ Φ T ( λ )) u ( λ ) , u ( µ ) � M � � � = 1 − Φ ω ( µ )Φ ω ( λ ) � dE ( ω ) u ( λ ) , u ( µ ) � M . T

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 = 1 − Φ ω ◦ k ( w )Φ ω ◦ k ( z ) � dE ( ω ) u ◦ k ( z ) , u ◦ k ( w ) � M . T Hence 1 = I 1 + I 2 where I 1 ( z, w ) = 1 − Φ ω 0 ◦ k ( w )Φ ω 0 ◦ k ( z ) � E ( { ω 0 } ) u ◦ k ( z ) , u ◦ k ( w ) � M , 1 − ¯ wz 1 − Φ ω ◦ k ( w )Φ ω ◦ k ( z ) � I 2 ( z, w ) = � dE ( ω ) u ◦ k ( z ) , u ◦ k ( w ) � . 1 − ¯ wz T \{ ω 0 } The integrand is a positive kernel of rank 1 if ω = ω 0 , and of rank 2 if ω � = ω 0 . Since the left hand side 1 is positive of rank 1, it follows that I 2 ≡ 0 and E ( ω 0 ) u ◦ k is a constant x .

Proof – end Recall the identity � � � 1 − ϕ ( µ ) ϕ ( λ ) = 1 − Φ ω ( µ )Φ ω ( λ ) � dE ( ω ) u ( λ ) , u ( µ ) � M . T Put µ = k ( w ) again, but let λ be a general point of G . We obtain 1 − ¯ wϕ ( λ ) = (1 − ¯ w Φ ω 0 ( λ )) � u ( λ ) , x � from which it follows that � u ( λ ) , x � = 1 and ϕ = Φ ω 0 .

Essentially non-unique extremals Consider a tangent of the form λ = (2 z, z 2 ) , v = 2 c (1 , z ) for some z ∈ D and c � = 0 (a royal tangent ). Φ ω is an extremal for ( λ, v ) for all ω ∈ T . The Carath´ eodory extremal functions for ( λ, v ) are the func- tions   Ψ( s, p ) 4 ( s 2 − 4 p )  1 2 s + 1 ϕ ( s, p ) = m  1 − 1 2 s Ψ( s, p ) where m ∈ Aut D and Ψ ∈ ¯ D ( G ).

Purely balanced tangents These are the tangents in TG for which Φ ω 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 Φ ζ , Φ η are the two extremal Φ ω s for ( λ, v ). Then (Φ ζ , Φ η ) is an iso- morphism of G with an open subset of D 2 . Using this fact we can write down a large class of Carath´ eodory extremals D ( D 2 ), though we do not claim that these indexed by [0 , 1] × ¯ are all extremals.

Flat tangents These are the tangents ( λ, v ) ∈ TG where ps, 1 − | p | 2 ) λ = ( s, p ) , v = c (¯ s − ¯ for some c � = 0. Φ ω 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. 453 (2017) Anal. Applic. 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