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THE ROLE OF DOMINATION AND SMOOTHING CONDITIONS IN THE THEORY OF EVENTUALLY POSITIVE SEMIGROUPS

Part of: Special classes of linear operators Groups and semigroups of linear operators, their generalizations and applications General theory of linear operators Differential equations in abstract spaces

Published online by Cambridge University Press:  29 March 2017

DANIEL DANERS Open the ORCID record for DANIEL DANERS [Opens in a new window]  and
JOCHEN GLÜCK Open the ORCID record for JOCHEN GLÜCK [Opens in a new window]
DANIEL DANERS*
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia email daniel.daners@sydney.edu.au
JOCHEN GLÜCK
Affiliation:
Institut für Angewandte Analysis, Universität Ulm, D-89069 Ulm, Germany email jochen.glueck@uni-ulm.de
*
✉daniel.daners@sydney.edu.au
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Abstract

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We carry out an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand, we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron–Frobenius type spectral theorems. We furthermore prove a Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

Keywords

one-parameter semigroups of linear operatorseventually positive semigroupdomination conditionsmoothing conditionPerron–Frobenius theory

MSC classification

Primary: 47D06: One-parameter semigroups and linear evolution equations
Secondary: 34G10: Linear equations 47A10: Spectrum, resolvent 47B65: Positive operators and order-bounded operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 286 - 298
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

J. Glück was partially supported by a scholarship within the scope of the LGFG Baden-Württemberg, Germany.

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