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The Operator Systems of Hankel Matrices and Complex Polynomials
Abstract
We investigate the operator system structure of finite Hankel matrices and its connection with complex polynomials on the real line. Using polynomial matrix spectral factorization, we establish a unital complete order isomorphism between the operator system of (n + 1) x (n + 1) Hankel matrices and the dual of the operator system of polynomials of degree at most 2n. We also show that the operator system of complex polynomials is minimal, while the Hankel operator system is maximal. As an application, we characterize positivity of block Hankel matrices with entries in a C*-algebra in terms of completely positive maps on the polynomial operator system.
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BibTeX
@misc{sinclair2026operator,
title = {{The Operator Systems of Hankel Matrices and Complex Polynomials}},
author = {Thomas Sinclair and Shengwei Qiu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12087},
eprint = {2609.12087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12087},
}
RIS
TY - UNPB
TI - The Operator Systems of Hankel Matrices and Complex Polynomials
AU - Thomas Sinclair
AU - Shengwei Qiu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12087
UR - https://arxiv.org/abs/2609.12087
AB - We investigate the operator system structure of finite Hankel matrices and its connection with complex polynomials on the real line. Using polynomial matrix spectral factorization, we establish a unital complete order isomorphism between the operator system of (n + 1) x (n + 1) Hankel matrices and the dual of the operator system of polynomials of degree at most 2n. We also show that the operator system of complex polynomials is minimal, while the Hankel operator system is maximal. As an application, we characterize positivity of block Hankel matrices with entries in a C*-algebra in terms of completely positive maps on the polynomial operator system.
ER -