Abstract
Extending results of Davies and of Keicher on ℓp we show that the peripheral point spectrum of the generator of a positive bounded C0-semigroup of kernel operators on Lp is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to elliptic operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Arendt, C. Batty , M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001).
W. Arendt, A.V. Bukhvalov, Integral representation of resolvents and semigroups. Forum Math., 6 (1994), 111–135.
W. Arendt, G. Metafune, D. Pallara, Schrödinger operators with unbounded drift. J. Operator Th., 55 (2006), 185–211.
W. Arendt, Heat Kernels. ISEM course 2005/06. http://isem.science.unitn.it/wiki/Main_Page.
W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates. In: C.M. Dafermos, E. Feireisl (eds) Handbook of Differential Equations. Evolutionary Equations 1, Elsevier (2004) pp.1–85.
W. Arendt, A.F.M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Oper. Th., 38 (1997), 87–130.
D. Axmann, Struktur- und Ergodentheorie irreducibler Operatoren auf Banachverbänden, PhD thesis, Tübingen (1980).
D. Daners, Heat kernel estimates for operators with boundary conditions., Math. Nachr., 217 (2000), 13–41.
E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press (1990).
E.B. Davies: Triviality of the peripheral point spectrum, J. Evol. Equ., 5 (2005), 407 – 415.
K. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, Berlin (2000)
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differenatial Equations of Second Order, Springer, Berlin (1977).
G. Greiner, Spektrum und Asymptotik stark stetiger Halbgruppen positiver Operatoren, Sitzungsber. Heidelbg. Akad. Wiss., Math.-Naturwiss. Kl., (1982), 55–80.
V. Keicher, On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices, Arch. Math., 87 (2006), 359–367.
V. Keicher, R. Nagel, Positive semigroups behave asymptotically as rotation groups, Positivity, to appear.
U. Krengel, Ergodic Theorems, de Gruyter (1985).
P.C. Kunstmann, Lp-spectral properties of the Neumann Laplacian on horns, comets and stars. Math. Z., 242 (2002), 183–201.
P. Meyer-Nieberg, Banach Lattices, Springer, Berlin (1991)
R. Nagel et al, One-parameter Semigroups of Positive Operators, Springer LN 1184 (1986).
E. M. Ouhabaz: Analysis of Heat Equations on Domains, Princeton University Press, Oxford (2005).
H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin (1974).
M. Wolff, Triviality of the peripheral point spectrum of positive semigroups on atomic Banach lattices, Positivity, to appear.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of H.H. Schaefer
Rights and permissions
About this article
Cite this article
Arendt, W. Positive Semigroups of Kernel Operators. Positivity 12, 25–44 (2008). https://doi.org/10.1007/s11117-007-2137-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-007-2137-z