Distinguished varieties through the Berger-Coburn-Lebow theorem

Abstract

Distinguished algebraic varieties in ${ℂ}^2$ have been the focus of much research in recent years for good reasons. This note gives a different perspective.

• (1) We find a new characterization of an algebraic variety $\mathcal{𝒲}$ which is distinguished with respect to the bidisc. It is in terms of the joint spectrum of a pair of commuting linear matrix pencils.

• (2) There is a known characterization of ${\mathbb{𝔻}}^2\cap \mathcal{𝒲}$ 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.

• (3) En route, we develop a new realization formula for operator-valued contractive analytic functions on the unit disc.

• (4) 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.

• (5) 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.

• (6) 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.