On the Geometry and Operator Theory of the Bidisc and the Symmetrized Bidisc

Abstract

This work is concerned with the geometric and operator theoretic aspects of the bidisc and the symmetrized bidisc. First we have focused on the geometry of these two do- mains. The symmetrized bidisc, a non-homogeneous domain, is partitioned into a col- lection of orbits under the action of its automorphism group. We investigate the prop- erties of these orbits and pick out some necessary properties so that the symmetrized bidisc can be characterized up to biholomorphic equivalence. As a consequence, among other things, we have given a new defining condition of the symmetrized bidisc and we have found that a biholomorphic copy of the symmetrized bidisc defined by E. Cartan. This work on the symmetrized bidisc also helps us to develop a characterization of the bidisc. Being a homogeneous domain, the bidisc’s automorphism group does not reveal much about its geometry. Using the ideas from the work on the symmetrized bidisc, we have identified a subgroup of the automorphism group of the bidisc and observed the corresponding orbits under the action of this subgroup. We have identified some prop- erties of these orbits which are sufficient to characterize the bidisc up to biholomorphic equivalence. Turning to operator theoretic work on the domains, we have focused mainly on the Schur Agler class on the bidisc and the symmetrized bidisc. Each element of the Schur Agler class on these domains has a nice representation in terms of a unitary operator, called the realization formula. We have generalized the ideas developed in the context of the bidisc and the symmetrized bidisc and applied it to the Nevanlinna problem and the interpolating sequences. It turns out, our generalization works for a number of domains, such as annulus and multiply connected domains, not only the bidisc and the symmetrized bidisc.