Preprint
Finite sum of squares, finite realization and noncommutative Carathéodory approximation
ArXiv.org
04 Jun 2026
Abstract
In the noncommutative polydisc, we first prove a positive sum of squares formula for a non-negative hereditary rational nc-function. The number of summands is finite. This result is used to derive a finite-dimensional realization formula for contractive nc-rational functions, where the colligation matrix is contractive. It is unitary if and only if the function is inner. Finally, we apply these results to generalize Carathéodory's classical theorem - approximating holomorphic self-maps of the unit disc by finite Blaschke products - to the setting of holomorphic functions on the noncommutative polydisc. This is in sharp contrast with the commutative situation where Carathéodory's approximation is known for Schur classes only in the unit disc and the unit bidisc.
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Details
- Title
- Finite sum of squares, finite realization and noncommutative Carathéodory approximation
- Creators
- Tirthankar Bhattacharyya - Indian Institute of Science BangaloreJames Eldred Pascoe - Drexel UniversityChandan Pradhan - University of New Mexico
- Publication Details
- ArXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022190260704721