Analyticity of positive semigroups is inherited under domination
HTML articles powered by AMS MathViewer
- by Jochen Glück
- Proc. Amer. Math. Soc. 151 (2023), 4793-4798
- DOI: https://doi.org/10.1090/proc/16492
- Published electronically: August 4, 2023
- HTML | PDF | Request permission
Abstract:
For positive $C_0$-semigroups $S$ and $T$ on a Banach lattice such that $S(t) \le T(t)$ for all times $t$, we prove that analyticity of $T$ implies analyticity of $S$. This answers an open problem posed by Arendt in 2004.
Our proof is based on a spectral theoretic argument: we apply spectral theory of positive operators to multiplication operators that are induced by $S$ and $T$ on a vector-valued function space.
References
- W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. MR 839450, DOI 10.1007/BFb0074922
- András Bátkai, Marjeta Kramar Fijavž, and Abdelaziz Rhandi, Positive operator semigroups, Operator Theory: Advances and Applications, vol. 257, Birkhäuser/Springer, Cham, 2017. From finite to infinite dimensions; With a foreword by Rainer Nagel and Ulf Schlotterbeck. MR 3616245, DOI 10.1007/978-3-319-42813-0
- Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
- Stephan Fackler, Regularity of semigroups via the asymptotic behaviour at zero, Semigroup Forum 87 (2013), no. 1, 1–17. MR 3079770, DOI 10.1007/s00233-013-9466-y
- Günther Greiner, Zur Perron-Frobenius-Theorie stark stetiger Halbgruppen, Math. Z. 177 (1981), no. 3, 401–423 (German). MR 618205, DOI 10.1007/BF01162072
- Retha Heymann, Eigenvalues and stability properties of multiplication operators and multiplication semigroups, Math. Nachr. 287 (2014), no. 5-6, 574–584. MR 3193937, DOI 10.1002/mana.201300046
- A. Holderrieth, Matrix multiplication operators generating one parameter semigroups, Semigroup Forum 42 (1991), no. 2, 155–166. MR 1083928, DOI 10.1007/BF02573417
- Tosio Kato, A characterization of holomorphic semigroups, Proc. Amer. Math. Soc. 25 (1970), 495–498. MR 264456, DOI 10.1090/S0002-9939-1970-0264456-0
- Heinrich P. Lotz, Über das Spektrum positiver Operatoren, Math. Z. 108 (1968), 15–32 (German). MR 240648, DOI 10.1007/BF01110453
- Rainer Nagel, Some open problems in the theory of ${C}_0$-semigroups, Interplay between $(C_0)$-semigroups and PDEs: theory and applications, Aracne, 2004, pp. 193–196.
- Frank Räbiger and Manfred P. H. Wolff, Spectral and asymptotic properties of dominated operators, J. Austral. Math. Soc. Ser. A 63 (1997), no. 1, 16–31. MR 1456587, DOI 10.1017/S144678870000029X
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 423039, DOI 10.1007/978-3-642-65970-6
Bibliographic Information
- Jochen Glück
- Affiliation: University of Wuppertal, School of Mathematics and Natural Sciences, Gaußstr. 20, 42119 Wuppertal, Germany
- ORCID: 0000-0002-0319-6913
- Email: glueck@uni-wuppertal.de
- Received by editor(s): May 2, 2022
- Received by editor(s) in revised form: March 13, 2023, and March 25, 2023
- Published electronically: August 4, 2023
- Communicated by: Adrian Ioana
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4793-4798
- MSC (2020): Primary 47D06, 47B65, 47A10
- DOI: https://doi.org/10.1090/proc/16492
- MathSciNet review: 4634882
Dedicated: Dedicated to the memory of Manfred P. H. Wolff