Bhattacharyya, T and Kumar, P and Sau, H (2022) DISTINGUISHED VARIETIES THROUGH THE BERGER-COBURN-LEBOW THEOREM. In: Analysis and PDE, 15 (2). pp. 477-506.
Full text not available from this repository.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. (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 © 2022. Analysis and PDE.All Rights Reserved.
| Item Type: | Journal Article |
|---|---|
| Publication: | Analysis and PDE |
| Publisher: | Mathematical Sciences Publishers |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 27 May 2022 05:36 |
| Last Modified: | 27 May 2022 05:36 |
| URI: | https://eprints.iisc.ac.in/id/eprint/72750 |
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