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Finite Blaschke products and the construction of rational Γ-inner functions

Nicholas Young

Leeds and Newcastle Universities Joint work with Jim Agler and Zinaida Lykova Bordeaux, 1st June 2015

Summary

The symmetrised bidisc is the set Γ def = {(z + w, zw) : |z| ≤ 1, |w| ≤ 1} ⊂ C2. A Γ-inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T belong to the distinguished boundary bΓ of Γ. With the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions (s, p) of degree n with the n zeros of s2−4p, and the corresponding values of s, prescribed.

Blaschke interpolation data

Consider points σ1, . . . , σ4, η1, . . . , η4 as in the diagram

η η η η

1 2 4 3

σ σ σ σ

4 3 1 2

and positive numbers ρ1, ρ2.

An interpolation problem for finite Blaschke products

Suppose given points σ1, . . . , σ4, η1, . . . , η4 and positive num- bers ρ1, ρ2 as on the preceding slide. Find if possible a Blaschke product ϕ of degree 4 with the properties ϕ(σj) = ηj for j = 1, . . . , 4 and Aϕ(σj) ≤ ρj for j = 1, 2, where Aϕ(eiθ) denotes the rate of change of the argument

• f ϕ(eiθ) with respect to θ.

The Pick matrix

associated with the foregoing interpolation problem is the matrix M =

• mij

4

i,j=1 where

mij =

          

ρj if i = j ≤ 2 1 − ¯ ηiηj 1 − ¯ σiσj

• therwise.

M ≥ 0 is necessary for the solvability of the Blaschke inter- polation problem.

Theorem

If the Pick matrix of the foregoing interpolation problem is positive definite then the problem has infinitely many so- lutions, which can be parametrised as follows. There are essentially unique polynomials a, b, c, d, of degree at most 4, such that the general solution of the problem is ϕζ(λ) = a(λ)ζ + b(λ) c(λ)ζ + d(λ) for λ ∈ D, where ζ ∈ T. There are explicit formulae for a, b, c, d, in which M−1 is prominent. See papers of Sarason, Georgijevi´ c, Bolotnikov-Dym, Chen- Hu.

Γ-inner functions

Recall that Γ def = {(z + w, zw) : |z| ≤ 1, |w| ≤ 1} ⊂ C2. The distinguished boundary of Γ is the ‘symmetrised torus’ bΓ = {(z + w, zw) : |z| = 1, |w| = 1}. A Γ-inner function is a map h ∈ Hol(D, Γ) whose boundary values lie in bΓ a.e. For each pair (ϕ, ψ) of (classical) inner functions, the func- tion (ϕ+ψ, ϕψ) is Γ-inner, but there are other Γ-inner func- tions, such as the degree 1 function (for β ∈ D) h(λ) = (β + ¯ βλ, λ). Describe all rational Γ-inner functions of degree n.

Royal nodes

The royal variety in Γ is the set R def = {(2λ, λ2) : |λ| ≤ 1} = {(s, p) ∈ Γ : s2 = 4p}. For any rational Γ-inner function h = (s, p), we say that λ ∈ D− is a royal node of h if h(λ) ∈ R, or equivalently, if s(λ)2 = 4p(λ). If h(λ) = (−2η, η2) then we say that η is the royal value corresponding to the royal node λ of h. Theorem If h = (s, p) is a rational Γ-inner function and p has degree n then h has exactly n royal nodes in D−, counted according to multiplicity.

Blaschke interpolation data again

The points σ1, . . . , σ4, η1, . . . , η4 are as in the diagram

η η η η

1 2 4 3

σ σ σ σ

4 3 1 2

and ρ1, ρ2 are positive numbers.

Γ-inner functions with prescribed royal nodes and values

Given Blaschke interpolation data σ1, . . . , σ4, η1, . . . , η4, ρ1, ρ2 as above, find a rational Γ-inner function h = (s, p) of degree 4 with royal nodes σ1, . . . , σ4 and corresponding royal values η1, . . . , η4 such that Ap(σj) = 2ρj, for j = 1, 2. Call these conditions (C).

An algorithm

• 1. Form the Pick matrix M corresponding to the Blaschke

interpolation data. If M is not positive definite then there is no Γ-inner function satisfying (C). Otherwise, let xλ

def

=

   

1 1−¯ σ1λ

. . .

1 1−¯ σ4λ

    ,

yλ

def

=

   

¯ η1 1−¯ σ1λ

. . .

¯ η4 1−¯ σ4λ

   

for λ ∈ D−.

• 2. Choose a point τ ∈ T \ {σ1, σ2}.
• 3. Find (s0, p0) ∈ bΓ such that

s0

• xλ, M−1xτ
• +
• yλ, M−1yτ
• +2
• xλ, M−1yτ
• +2p0
• yλ, M−1xτ
• is zero for all λ ∈ D.

Algorithm continued

• 4. If there is no (s0, p0) with these properties then there is

no Γ-inner function satisfying (C). Otherwise define polyno- mials a, b, c, d by explicit formulae in terms of σj, M, τ, xλ, yλ. (For example a(λ) =

4

• j=1

1 − ¯ σjλ 1 − ¯ σjτ

• 1 − (1 − ¯

τλ)

• xλ, M−1xτ
• .)
• 5. The function h = (s, p) is a Γ-inner function of degree 4

satisfying (C), where s = 22p0c − s0d s0c − 2d , p = −2p0a + s0b s0c − 2d .

Reference

Jim Agler, Zinaida Lykova and Nicholas Young, Finite Blaschke products and the construction of rational Γ-inner functions, preprint.