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

Transcription

1 Realization formulae for bounded holomorphic functions on certain domains and an application to the Carathéodory extremal problem Nicholas Young Leeds and Newcastle Universities Joint work with Jim Agler, UCSD and Zina Lykova, Newcastle CIRM, Trento, October 2018

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

3 The Carathéodory 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 λ, v car def = sup F (λ, v) F D(Ω) = sup F D(Ω) D v F (λ) 1 F (λ) 2. 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 Carathéodory extremal functions.

4 Non-uniqueness of Carathéodory extremals Let (λ, v) T Ω. If F : Ω D is a Carathéodory extremal for (λ, v) and m Aut D, then m F is also an extremal (because of the invariance of the Poincaré metric on D under automorphisms). We say that an extremal function F for (λ, v) is essentially unique if the only Carathéodory 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 Carathéodory extremal functions. There is a large class of pairwise inequivalent Carathéodory extremals for (λ, v), indexed by [0, 1] D(D 2 ).

5 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(D 2 ) is expressible by a realization formula of the form where [ A B C D ] ϕ(λ) = A + Bλ(1 Dλ) 1 C is a contractive operator on C M.

6 Models for D(D 2 ) A model for a function ϕ D(D 2 ) is a pair (M 1 M 2, u) 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) = (1 µ 1 λ 1 ) u 1 (λ), u 1 (µ) M1 + (1 µ 2 λ 2 ) u 2 (λ), u 2 (µ) M2. Theorem (Agler, 1990) Every function in D(D 2 ) has a model.

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

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

9 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 Φ T (µ) Φ T (λ) ) u(λ), u(µ) M. Here, if λ = (s, p) G, Φ T (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.

10 Theorem Generically, Carathéodory extremals are unique in G. That is: Let λ G. For a generic direction Cv CP 2, there is an essentially unique Carathéodory extremal function for the tangent (λ, v) T G. 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.

11 Theorem (Kosiński-Zwonek 2016) If (λ, v) T G and there is a unique ω 0 T such that Φ ω0 is a Carathéodory extremal function for (λ, v), then every Carathéodory extremal function for (λ, v) has the form m Φ ω0 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 ω de(ω). Then, for any λ, µ G, T 1 ϕ(µ)ϕ(λ) = (1 Φ T (µ) Φ T (λ))u(λ), u(µ) M = T ( 1 Φω (µ)φ ω (λ) ) de(ω)u(λ), u(µ) M.

12 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 Φω 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 ω0 k(z) 1 wz E({ω 0 })u k(z), u k(w) M, I 2 (z, w) = T\{ω 0 } 1 Φ ω k(w)φ ω k(z) 1 wz de(ω)u k(z), u k(w). 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.

13 Proof end Recall the identity 1 ϕ(µ)ϕ(λ) = T ( 1 Φω (µ)φ ω (λ) ) de(ω)u(λ), u(µ) M. Put µ = k(w) again, but let λ be a general point of G. We obtain 1 wϕ(λ) = (1 wφ ω0 (λ)) u(λ), x from which it follows that u(λ), x = 1 and ϕ = Φ ω0.

14 Essentially non-unique extremals Consider a tangent of the form λ = (2z, z 2 ), v = 2c(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 Carathéodory extremal functions for (λ, v) are the functions ϕ(s, p) = m 1 2 s Ψ(s, p) (s2 4p) 1 1 2sΨ(s, p) where m Aut D and Ψ D(G).

15 Purely balanced tangents These are the tangents in T G for which Φ ω is a Carathéodory 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 isomorphism of G with an open subset of D 2. Using this fact we can write down a large class of Carathéodory extremals indexed by [0, 1] D(D 2 ), though we do not claim that these are all extremals.

16 Flat tangents These are the tangents (λ, v) T G where for some c 0. λ = (s, p), v = c( s ps, 1 p 2 ) Φ ω is extremal for every flat tangent and every ω T. We construct a class of inequivalent Carathéodory extremal functions for flat (λ, v), parametrized by D(D).

17 References L. Kosiński and W. Zwonek, Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock, J. Geom. Analysis 26 (2016) J. Agler and N. J. Young, Realization of functions on the symmetrized bidisc, J. Math. Anal. Applic. 453 (2017) J. Agler, Z. A. Lykova and N. J. Young, Carathéodory extremal functions on the symmetrized bidisc, arxiv: The end