# American Institute of Mathematical Sciences

August  2018, 38(8): 4117-4132. doi: 10.3934/dcds.2018179

## Lower spectral radius and spectral mapping theorem for suprema preserving mappings

 1 Institute of Mathematics, Czech Academy of Sciences, Žitna 25, 115 67 Prague, Czech Republic 2 Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia 3 Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia

* Corresponding author: Aljoša Peperko

Received  November 2017 Published  May 2018

We study Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max-type kernel operators.

Citation: Vladimir Müller, Aljoša Peperko. Lower spectral radius and spectral mapping theorem for suprema preserving mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4117-4132. doi: 10.3934/dcds.2018179
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