Interpolation problems, the symmetrized bidisc and the tetrablock

Abstract

The spectral Nevanlinna-Pick interpolation problem is to find, if it is possible, an analytic function f : D → Ck×k from the unit disc D = {z ∈ C : |z| < 1} to the space Ck×k of k × k complex matrices, which interpolates a finite number of distinct points in D to the target matrices in Ck×k subject to the spectral radius r(f(λ)) ≤ 1, for every λ ∈ D. For k = 2, this problem is connected to interpolation problem in Hol(D, Γ), where Hol(D, Γ) denotes the space of analytic functions from D to the closed symmetrized bidisc Γ = {(z1 + z2, z1z2) : z1, z2 ∈ D} ⊂ C 2 . In this thesis, we consider a special case of the three-point spectral Nevanlinna-Pick problem and give necessary and sufficient conditions for its solvability. We also study interpolation problems from D to the tetrablock. The closed tetrablock is defined to be E = {x ∈ C 3 : 1 − x1z − x2w + x3zw 6= 0 for all z, w ∈ D}. Given n distinct points λ1, · · · , λn in D and n points x 1 , · · · , x n in E, find, if is possible, an analytic function ϕ : D → E such that ϕ(λj) = x j for j = 1, · · · , n. This problem is closely connected to the µDiag-synthesis interpolation problem. For given data λj → Wj , 1 ≤ j ≤ n, where λj are distinct points in D and Wj are complex 2 × 2 matrices, find, if it is possible, an analytic matrix function F : D → C 2×2 such that F(λj) = Wj , 1 ≤ j ≤ n, and µDiag(F(λ)) ≤ 1 for all λ ∈ D. We give criteria for the solvability of such interpolation problems. Here Diag is the space of 2 × 2 diagonal matrices, and for A ∈ C2×2 , µDiag(A) := 1 inf{kXk : X ∈ Diag, 1 − AX is singular} . If 1 − AX is non-singular for all X ∈ Diag, then µDiag(A) = 0. In addition, we give a realization theorem for analytic functions from the disc to the tetrablock.