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Classifications of linear operators preserving elliptic, positive and non-negative polynomials
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Julius Borcea
Published/Copyright:
January 7, 2011
Abstract
We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer–Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [Borcea, Guterman, Shapiro, Preserving positive polynomials and beyond] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.
Received: 2008-10-17
Revised: 2009-01-12
Published Online: 2011-01-07
Published in Print: 2011-January
© Walter de Gruyter Berlin · New York 2011
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Articles in the same Issue
- Ricci soliton solvmanifolds
- Rational normal scrolls and the defining equations of Rees algebras
- Classifications of linear operators preserving elliptic, positive and non-negative polynomials
- Graded polynomial identities and exponential growth
- Un théorème de la masse positive pour le problème de Yamabe en dimension paire
- Degenerate problems with irregular obstacles
- Twisted cyclic theory, equivariant KK-theory and KMS states
- A trace formula for varieties over a discretely valued field
