The Construction of Rational Tetra-Inner Functions

Abstract

The tetrablock is the set E = fx 2 C3 : 1 􀀀 x1z 􀀀 x2w + x3zw 6= 0 whenever jzj 1; jwj 1g: The closure of E is denoted by E. A tetra-inner function is an analytic map x from the unit disc D to E whose boundary values at almost all points of the unit circle T belong to the distinguished boundary bE of E. There is a natural notion of degree of a rational tetra-inner function x; it is simply the topological degree of the continuous map xjT from T to bE. In this thesis we give a prescription for the construction of a general rational tetra-inner function of degree n. The prescription makes use of a known solution of an interpolation problem for nite Blaschke products of given degree in terms of a Pick matrix formed from the interpolation data. Alsalhi and Lykova proved that if x = (x1; x2; x3) is a rational tetra-inner function of degree n, then x1x2􀀀x3 either is equal to 0 or has exactly n zeros in the closed unit disc D, counted with an appropriate notion of multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function x = (x1; x2; x3) consists of the points in D for which x1x2 􀀀 x3 = 0 and the values of x at these points. We also give a matricial formulation of a criterion for the solvability of a Diag-synthesis problem. The symbol Diag denotes an instance of the structured singular value of 2 2 matrix corresponding to the subspace of diagonal matrices in M2 2(C). Given distinct points 1; :::; n 2 D and target matrices W1; :::;Wn 2 M2 2(C) one seeks an analytic 2 2 matrix-valued function F on D such that F( j) = Wj for j = 1; :::; n; and Diag(F( )) < 1; for all 2 D: