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 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 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 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.

  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.

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Keywords
distinguished varieties, commuting isometries, inner functions, linear pencils, algebraic varieties, joint spectrum
Mathematical Subject Classification
Primary: 47A13
Secondary: 32C25, 47A20, 47A48, 47A57
Milestones
Received: 26 April 2020
Revised: 24 July 2020
Accepted: 6 October 2020
Published: 12 April 2022
Authors
Tirthankar Bhattacharyya
Department of Mathematics
Indian Institute of Science
Bangalore
India
Poornendu Kumar
Department of Mathematics
Indian Institute of Science
Bangalore
India
Haripada Sau
Department of Mathematics
Indian Institute of Science Education and Research
Pune
India