Mathematics – Operator Algebras
Scientific paper
2009-07-13
Indiana Univ. Math. J. 59, No. 3, 857-874 (2010)
Mathematics
Operator Algebras
15 pages, accepted for publication in Indiana Univ. Math. J
Scientific paper
10.1512/iumj.2010.59.4107
Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all real n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and the cartesian product of S with the real projective (t-1)-space. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for non-symmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.
Klep Igor
Schweighofer Markus
No associations
LandOfFree
Pure states, positive matrix polynomials and sums of hermitian squares does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Pure states, positive matrix polynomials and sums of hermitian squares, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Pure states, positive matrix polynomials and sums of hermitian squares will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-331300