Vol. 15, No. 2, 2022

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 3, 613–890
Issue 2, 309–612
Issue 1, 1–308

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
Distinguished varieties through the Berger-Coburn-Lebow theorem

Tirthankar Bhattacharyya, Poornendu Kumar and Haripada Sau

Vol. 15 (2022), No. 2, 477–506

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.

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