Realization of Functions into the Symmetrised Bidisc

Abstract

We give two realization formulae for analytic functions from the open unit disc to the open symmetrised bidisc G. We prove that the ABCD matrices of the two realizations are equal, albeit with two different partitions. We illustrate the calculation of ABCD -matrices by finding explicitly the realizations of a certain class of extremal analytic functions from the disc to the domain G to wit the complex geodesics of G.

G is defined to be the domain

$$G = \left\{ {\left( {z_1 + z_2 ,z_1 z_2 } \right):\left| {z_1 } \right| < 1,\left| {z_2 } \right| < 1} \right\}.$$

The formulae permit the construction of analytic 2 ×2-matrix-valued functions F such that F (λ) has spectral radius no greater than 1 for every λ in the disc.