#### Vol. 15, No. 2, 2022

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Distinguished varieties through the Berger-Coburn-Lebow theorem

### Tirthankar Bhattacharyya, Poornendu Kumar and Haripada Sau

Vol. 15 (2022), No. 2, 477–506
##### 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 $\mathsc{𝒲}$ 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 ${\mathbb{𝔻}}^{2}\cap \mathsc{𝒲}$ 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.

##### Keywords
distinguished varieties, commuting isometries, inner functions, linear pencils, algebraic varieties, joint spectrum
##### Mathematical Subject Classification
Primary: 47A13
Secondary: 32C25, 47A20, 47A48, 47A57