Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T11:11:48.215Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H. P., Moustakas, U., Nagel, R., Neubrander, F. and Schlotterbeck, U., One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (Springer, Berlin, 1986).Google Scholar
Daners, D., ‘Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator’, Positivity 18 (2014), 235–256.Google Scholar
Daners, D., Glück, J. and Kennedy, J. B., ‘Eventually and asymptotically positive semigroups on Banach lattices’, J. Differential Equations 261 (2016), 2607–2649.Google Scholar
Daners, D., Glück, J. and Kennedy, J. B., ‘Eventually positive semigroups of linear operators’, J. Math. Anal. Appl. 433 (2016), 1561–1593.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T., Linear Operators. I. General Theory, Pure and Applied Mathematics, VII (Interscience Publishers, New York, 1958).Google Scholar
Ellison, E. M., Hogben, L. and Tsatsomeros, M. J., ‘Sign patterns that require eventual positivity or require eventual nonnegativity’, Electron. J. Linear Algebra 19 (2009), 98–107.Google Scholar
Engel, K.-J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194 (Springer, New York, 2000).Google Scholar
Erickson, C., ‘Sign patterns that require eventual exponential nonnegativity’, Electron. J. Linear Algebra 30 (2015), 171–195.Google Scholar
Ferrero, A., Gazzola, F. and Grunau, H.-C., ‘Decay and eventual local positivity for biharmonic parabolic equations’, Discrete Contin. Dyn. Syst. 21 (2008), 1129–1157.Google Scholar
Gazzola, F. and Grunau, H.-C., ‘Eventual local positivity for a biharmonic heat equation in ℝ n ’, Discrete Contin. Dyn. Syst. S 1 (2008), 83–87.Google Scholar
Glück, J., ‘Growth rates and the peripheral spectrum of positive operators’, Houston J. Math. Preprint available at https://arxiv.org/abs/1512.07483.Google Scholar
Glück, J., Invariant Sets and Long Time Behaviour of Operator Semigroups, PhD Thesis, Universität Ulm, 2016.Google Scholar
Glück, J., ‘Towards a Perron–Frobenius theory for eventually positive operators’, Preprint, 2016, https://arxiv.org/abs/1609.08498.Google Scholar
Grunau, H.-C. and Sweers, G., ‘Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions’, Math. Ann. 307 (1997), 589–626.Google Scholar
Grunau, H.-C. and Sweers, G., ‘The maximum principle and positive principal eigenfunctions for polyharmonic equations’, in: Reaction Diffusion Systems (Trieste 1995), Lecture Notes in Pure and Applied Mathematics, 194 (Dekker, New York, 1998), 163–182.Google Scholar
Grunau, H.-C. and Sweers, G., ‘Sign change for the Green function and for the first eigenfunction of equations of clamped-plate type’, Arch. Ration. Mech. Anal. 150 (1999), 179–190.Google Scholar
Lotz, H. P., ‘Über das Spektrum positiver Operatoren’, Math. Z. 108 (1968), 15–32.Google Scholar
Meyer-Nieberg, P., Banach Lattices, Universitext (Springer, Berlin, 1991).Google Scholar
Noutsos, D. and Tsatsomeros, M. J., ‘Reachability and holdability of nonnegative states’, SIAM J. Matrix Anal. Appl. 30 (2008), 700–712.Google Scholar
Olesky, D. D., Tsatsomeros, M. J. and van den Driessche, P., ‘ M ∨ -matrices: a generalization of M-matrices based on eventually nonnegative matrices’, Electron. J. Linear Algebra 18 (2009), 339–351.Google Scholar
Schaefer, H. H., Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, 215 (Springer, New York, 1974).Google Scholar
Sweers, G., ‘An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign’, Arch. Math. (Basel) 107 (2016), 59–62.Google Scholar
Taylor, A. E., Introduction to Functional Analysis (John Wiley & Sons, New York; Chapman & Hall, London, 1958).Google Scholar
Yosida, K., Functional Analysis, Classics in Mathematics (Springer, Berlin, 1995).CrossRefGoogle Scholar