Stability of (eventually) positive semigroups on spaces of continuous functions

Sahiba Arora; Jochen Glück

doi:10.5802/crmath.323

Analyse fonctionnelle, Théorie des opérateurs

Stability of (eventually) positive semigroups on spaces of continuous functions

Sahiba Arora ¹ ; Jochen Glück ²

¹ Technische Universität Dresden, Institut für Analysis, Fakultät für Mathematik, 01062 Dresden, Germany
² Universität Passau, Fakultät für Informatik und Mathematik, 94032 Passau, Germany

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 771-775.

Résumé

We present a new and very short proof of the fact that, for positive $C_{0}$ -semigroups on spaces of continuous functions, the spectral and the growth bound coincide. Our argument, inspired by an idea of Vogt, makes the role of the underlying space completely transparent and also works if the space does not contain the constant functions – a situation in which all earlier proofs become technically quite involved.

We also show how the argument can be adapted to yield the same result for semigroups that are only eventually positive rather than positive.

Reçu le : 2021-12-15
Accepté le : 2022-01-04
Publié le : 2022-06-30

DOI : 10.5802/crmath.323

Classification : 47D06, 47B65, 47A10

Affiliations des auteurs :

Sahiba Arora ¹ ; Jochen Glück ²

¹ Technische Universität Dresden, Institut für Analysis, Fakultät für Mathematik, 01062 Dresden, Germany
² Universität Passau, Fakultät für Informatik und Mathematik, 94032 Passau, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2022__360_G7_771_0,
     author = {Sahiba Arora and Jochen Gl\"uck},
     title = {Stability of (eventually) positive semigroups on spaces of continuous functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {771--775},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.323},
     language = {en},
}

TY  - JOUR
AU  - Sahiba Arora
AU  - Jochen Glück
TI  - Stability of (eventually) positive semigroups on spaces of continuous functions
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 771
EP  - 775
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.323
LA  - en
ID  - CRMATH_2022__360_G7_771_0
ER  -

%0 Journal Article
%A Sahiba Arora
%A Jochen Glück
%T Stability of (eventually) positive semigroups on spaces of continuous functions
%J Comptes Rendus. Mathématique
%D 2022
%P 771-775
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.323
%G en
%F CRMATH_2022__360_G7_771_0

Sahiba Arora; Jochen Glück. Stability of (eventually) positive semigroups on spaces of continuous functions. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 771-775. doi : 10.5802/crmath.323. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.323/

Bibliographie
Cité par

[1] Davide Addona; Federica Gregorio; Abdelaziz Rhandi; Cristian Tacelli Bi-Kolmogorov type operators and weighted Rellich’s inequalities, NoDEA, Nonlinear Differ. Equ. Appl., Volume 29 (2022), 13 | DOI | MR | Zbl

[2] Wolfgang Arendt; Charles J. K. Batty; Matthias Hieber; Frank Neubrander Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Birkhäuser, 2011, xii + 539 pages | DOI | Zbl

[3] Sahiba Arora; Jochen Glück Spectrum and convergence of eventually positive operator semigroups, Semigroup Forum, Volume 103 (2021) no. 3, pp. 791-811 | DOI | MR | Zbl

[4] Charles J. K. Batty; Edward B. Davies Positive semigroups and resolvents, J. Oper. Theory, Volume 10 (1983), pp. 357-363 | MR | Zbl

[5] Simon Becker; Federica Gregorio; Delio Mugnolo Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra, J. Math. Anal. Appl., Volume 495 (2021) no. 2, p. 124748 | DOI | Zbl

[6] Daniel Daners; Jochen Glück; James B. Kennedy Eventually and asymptotically positive semigroups on Banach lattices, J. Differ. Equations, Volume 261 (2016) no. 5, pp. 2607-2649 | DOI | MR | Zbl

[7] Daniel Daners; Jochen Glück; James B. Kennedy Eventually positive semigroups of linear operators, J. Math. Anal. Appl., Volume 433 (2016) no. 2, pp. 1561-1593 | DOI | MR | Zbl

[8] Robert Denk; Markus Kunze; David Ploß The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains, Integral Equations Oper. Theory, Volume 93 (2021) no. 2, p. 13 | DOI | MR | Zbl

[9] Amru Hussein; Delio Mugnolo Laplacians with Point Interactions—Expected and Unexpected Spectral Properties, Semigroups of Operators – Theory and Applications (Springer Proceedings in Mathematics & Statistics), Volume 325 (2020), pp. 47-67 | DOI | MR | Zbl

[10] One-parameter semigroups of positive operators (Rainer Nagel, ed.), Lecture Notes in Mathematics, 1184, Springer, 1986 | DOI | Zbl

[11] Tara Prajapati; Kalyan B. Sinha; Sachi Srivastava Lyapunov property of positive $C_{0}$ -semigroups on non-commutative $L^{p}$ spaces, Oper. Matrices, Volume 13 (2019) no. 4, pp. 907-919 | DOI | MR

[12] Jan Rozendaal; Mark Veraar Stability theory for semigroups using $(L^{p}, L^{q})$ Fourier multipliers, J. Funct. Anal., Volume 275 (2018) no. 10, pp. 2845-2894 | DOI | MR | Zbl

[13] Helmut H. Schaefer Banach lattices and positive operators, Grundlehren der Mathematischen Wissenschaften, 215, Springer, 1974 | DOI | Numdam | Zbl

[14] Hendrik Vogt Stability of uniformly eventually positive $C_{0}$ -semigroups on $L_{p}$ -spaces, Proc. Am. Math. Soc., Volume 150 (2022), pp. 3513-3515 | DOI | MR

[15] Lutz Weis The Stability of Positive Semigroups on $L_{p}$ Spaces, Proc. Am. Math. Soc., Volume 123 (1995) no. 10, pp. 3089-3094 | DOI | MR | Zbl

[16] Lutz Weis A Short Proof for the Stability Theorem for Positive Semigroups on $L_{p} (μ)$ , Proc. Am. Math. Soc., Volume 126 (1998) no. 11, pp. 3253-3256 | DOI | MR | Zbl

[17] Anthony W. Wickstead Compact subsets of partially ordered Banach spaces, Math. Ann., Volume 212 (1975), pp. 271-284 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse: