Journal of Operator Theory

Volume 93, Issue 1, Winter 2025 pp. 91-113.

Extensions of S-lemma for noncommutative polynomial

Authors: Feng Guo (1), Sizhuo Yan (2), Lihong Zhi (3)
Author institution: (1) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
(2) Key Lab of Mathematics Mechanization, AMSS, University of Chinese Academy of Sciences, Beijing 100190, China
(3) Key Lab of Mathematics Mechanization, AMSS, University of Chinese Academy of Sciences, Beijing 100190, China

Summary: We consider the problem of extending the classical S-lemma from the commutative case to noncommutative cases. Precisely, we extend the S-lemma to three kinds of noncommutative polynomials: noncommutative polynomials whose coefficients are real numbers, noncommutative matrix-valued polynomials, and hereditary noncommutative matrix-valued polynomials. Different from the commutative case, the S-lemma for noncommutative polynomials can be extended to the case involving multiple quadratic constraints.

DOI: http://dx.doi.org/10.7900/jot.2022dec21.2409
Keywords: S-lemma, noncommutative polynomial, linear matrix inequality, positive semidefinite matrix, completely positive linear map

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