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 } ⊂ C 2 . 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 b Γ 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 s 2 − 4 p , and the corresponding values of s , prescribed.
Blaschke interpolation data Consider points σ 1 , . . . , σ 4 , η 1 , . . . , η 4 as in the diagram σ η 2 2 σ η 4 4 σ η 3 3 η σ 1 1 and positive numbers ρ 1 , ρ 2 .
An interpolation problem for finite Blaschke products Suppose given points σ 1 , . . . , σ 4 , η 1 , . . . , η 4 and positive num- bers ρ 1 , ρ 2 as on the preceding slide. Find if possible a Blaschke product ϕ of degree 4 with the properties ϕ ( σ j ) = η j for j = 1 , . . . , 4 and Aϕ ( σ j ) ≤ ρ j for j = 1 , 2 , where Aϕ (e iθ ) denotes the rate of change of the argument of ϕ (e iθ ) with respect to θ .
The Pick matrix associated with the foregoing interpolation problem is the � 4 � matrix M = i,j =1 where m ij if i = j ≤ 2 ρ j m ij = 1 − ¯ η i η j otherwise. 1 − ¯ σ i σ j 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 ϕ ζ ( λ ) = a ( λ ) ζ + b ( λ ) for λ ∈ D , c ( λ ) ζ + d ( λ ) 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 } ⊂ C 2 . The distinguished boundary of Γ is the ‘symmetrised torus’ b Γ = { ( z + w, zw ) : | z | = 1 , | w | = 1 } . A Γ -inner function is a map h ∈ Hol( D , Γ) whose boundary values lie in b Γ 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 = { (2 λ, λ 2 ) : | λ | ≤ 1 } = { ( s, p ) ∈ Γ : s 2 = 4 p } . 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 = 4 p ( λ ). If h ( λ ) = ( − 2 η, η 2 ) 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 σ 1 , . . . , σ 4 , η 1 , . . . , η 4 are as in the diagram σ η 2 2 σ η 4 4 σ η 3 3 η σ 1 1 and ρ 1 , ρ 2 are positive numbers.
Γ -inner functions with prescribed royal nodes and values Given Blaschke interpolation data σ 1 , . . . , σ 4 , η 1 , . . . , η 4 , ρ 1 , ρ 2 as above, find a rational Γ-inner function h = ( s, p ) of degree 4 with royal nodes σ 1 , . . . , σ 4 and corresponding royal values η 1 , . . . , η 4 such that Ap ( σ j ) = 2 ρ j , 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 ¯ η 1 1 1 − ¯ σ 1 λ 1 − ¯ σ 1 λ def def . . for λ ∈ D − . . . = . = . x λ , y λ 1 ¯ η 4 1 − ¯ σ 4 λ 1 − ¯ σ 4 λ 2. Choose a point τ ∈ T \ { σ 1 , σ 2 } . 3. Find ( s 0 , p 0 ) ∈ b Γ such that �� x λ , M − 1 x τ � � y λ , M − 1 y τ �� � x λ , M − 1 y τ � � y λ , M − 1 x τ � + +2 +2 p 0 s 0 is zero for all λ ∈ D .
Algorithm continued 4. If there is no ( s 0 , p 0 ) 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 σ j , M, τ, x λ , y λ . (For example 4 1 − ¯ σ j λ � � x λ , M − 1 x τ �� � a ( λ ) = 1 − (1 − ¯ τλ ) . ) 1 − ¯ σ j τ j =1 5. The function h = ( s, p ) is a Γ-inner function of degree 4 satisfying (C), where s = 22 p 0 c − s 0 d s 0 c − 2 d , p = − 2 p 0 a + s 0 b . s 0 c − 2 d
Reference Jim Agler, Zinaida Lykova and Nicholas Young, Finite Blaschke products and the construction of rational Γ-inner functions, preprint.
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