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Linear operators preserving the set of positive (nonnegative) polynomials. (English) Zbl 1182.93061

Bru, Rafael (ed.) et al., Positive systems. Proceedings of the third multidisciplinary international symposium on positive systems: theory and applications (POSTA 09), Valencia, Spain, September 2–4, 2009. Berlin: Springer (ISBN 978-3-642-02893-9/pbk; 978-3-642-02894-6/ebook). Lecture Notes in Control and Information Sciences 389, 83-90 (2009).
Summary: This note deals with linear operators preserving the set of positive (nonnegative) polynomials. Numerous works of prominent mathematicians in fact contain the exhaustive description of linear operators preserving the set of positive (nonnegative) polynomials. In spite of this, since this description was not formulated explicitly, it is almost lost for possible applications. In the paper we formulate and prove these classical results and give some applications. For example, we prove that there are no linear ordinary differential operators of order \(m \in \mathbb N\) with polynomial coefficients which map the set of nonnegative (positive) polynomials of degree \(\leq ( \lfloor\frac{m}{2}\rfloor+1 )\) into the set of nonnegative polynomials. This result is a generalization of a theorem by Guterman and Shapiro.
For the entire collection see [Zbl 1173.93001].

MSC:

93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
15B48 Positive matrices and their generalizations; cones of matrices
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