On the Bonsall cone spectral radius and the approximate point spectrum

Vladimir Müller; Aljoša Peperko

doi:10.3934/dcds.2017232

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2017, Volume 37, Issue 10: 5337-5354. Doi: 10.3934/dcds.2017232

This issue Previous Article Relations between Bohl and general exponents Next Article Strichartz estimates for $N$-body Schrödinger operators with small potential interactions

On the Bonsall cone spectral radius and the approximate point spectrum

Vladimir Müller^1, and
Aljoša Peperko^2,3, ,

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
* Corresponding author: Aljoša Peperko

Received: March 2017

Revised: May 2017

Published: October 2017

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Abstract

We study the Bonsall cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone spectral radius of such maps is always included in its approximate point spectrum. Moreover, the approximate point spectrum always contains a (possibly trivial) interval. Our results apply to a large class of (nonlinear) max-type operators.

We also generalize a known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum. Under additional generalized compactness type assumptions our results imply Krein-Rutman type results.

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Mathematics Subject Classification: Primary: 47H07, 47J10, 47H10; Secondary: 47H08, 47B65, 47A10.

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