Convex-Cyclic Matrices, Convex-Polynomial Interpolation & Invariant Convex Sets

DOI10.7153/OAM-11-31arXiv1507.08323MaRDI QIDQ6264046

Publication date: 29 July 2015

Abstract: We define a convex-polynomial to be one that is a convex combination of the monomials

1, z, z^{2}, l d o t s

. This paper explores the intimate connection between peaking convex-polynomials, interpolating convex-polynomials, invariant convex sets, and the dynamics of matrices. In particular, we use these intertwined relations to both prove which matrices are convex-cyclic while at the same time proving that we can prescribe the values and a finite number of the derivatives of a convex-polynomial subject to certain natural constraints. These properties are also equivalent to determining those matrices whose invariant closed convex sets are all invariant subspaces. Our characterization of the convex-cyclic matrices gives a new and correct proof of a similar result by Rezaei that was stated and proven incorrectly.

Mathematics Subject Classification ID

Norms of matrices, numerical range, applications of functional analysis to matrix theory (15A60) Cyclic vectors, hypercyclic and chaotic operators (47A16)

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