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.