Distinguished varieties through the Berger-Coburn-Lebow theorem

Abstract

Distinguished algebraic varieties in C2 have been the focus of much research in recent years for good reasons. This note gives a different perspective. We find a new characterization of an algebraic variety W which is distinguished with respect to the bidisc. It is in terms of the joint spectrum of a pair of commuting linear matrix pencils. There is a known characterization of D2∩W due to a seminal work of Agler and McCarthy. We show that Agler–McCarthy characterization can be obtained from the new one and vice versa. En route, we develop a new realization formula for operator-valued contractive analytic functions on the unit disc. There is a one-to-one correspondence between operator-valued contractive holomorphic functions and canonical model triples. This pertains to the new realization formula mentioned above. Pal and Shalit gave a characterization of an algebraic variety, which is distinguished with respect to the symmetrized bidisc, in terms of a matrix of numerical radius no larger than 1. We refine their result by making the class of matrices strictly smaller. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc. At the root of our work is the Berger–Coburn–Lebow theorem characterizing a commuting tuple of isometries.