Realization of functions into the symmetrised bidisc

Jim Agler, Fang-Bo Yeh and Nicholas Young

Keywords

Kobayashi extremal problem, spectral Nevanlinna-Pick problem, interpolation, mu-synthesis, complex geodesics

Status

Appeared in the volume Reproducing kernel spaces and applications, Operator Theory: Advances and Applications, volume 143, pages 1-37, Birkha\"user (2003), edited by Daniel Alpay.

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 = { (z+w,zw): |z| < 1, |w| <1 }.

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

J. Agler's work is suported by an NSF grant in Modern Analysis. This work was also supported by NATO Collaborative grant CRG 971129


Nicholas Young
25th October 1999
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