Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-06T02:02:54.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Triangularizing semigroups of positive operators on an atomic normed Riesz Space

Published online by Cambridge University Press:  20 January 2009

Roman Drnovšek
Roman Drnovšek
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia ([email protected])
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.

In the first part of the paper we prove several results on the existence of invariant closed ideals for semigroups of bounded operators on a normed Riesz space (of dimension greater than 1) possessing an atom. For instance, if S is a multiplicative semigroup of positive operators on such space that are locally quasinilpotent at the same atom, then S has a non-trivial invariant closed ideal. Furthermore, if T is a non-zero positive operator that is quasinilpotent at an atom and if S is a multiplicative semigroup of positive operators such that TS ≤ ST for all S ∈ S, then S and T have a common non-trivial invariant closed ideal. We also give a simple example of a quasinilpotent compact positive operator on the Banach lattice l∞ with no non-trivial invariant band.

The second part is devoted to the triangularizability of collections of operators on an atomic normed Riesz space L. For a semigroup S of quasinilpotent, order continuous, positive, bounded operators on L we determine a chain of invariant closed bands. If, in addition, L has order continuous norm, then this chain is maximal in the lattice of all closed subspaces of L.

Keywords

normed Riesz spacespositive operatorsinvariant idealssemigroups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 43 - 55
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Abramovich, Y. A., Aliprantis, C. D. and Burkinshaw, O., On the spectral radius of positive operators, Math. Z. 211 (1992), 593–607.CrossRefGoogle Scholar
2.Abramovich, Y. A., Aliprantis, C. D. and Burkinshaw, O., Invariant subspaces of operators on l p-spaces, J. Funct. Analysis 115 (1993), 418–424.CrossRefGoogle Scholar
3.Abramovich, Y. A., Aliprantis, C. D. and Burkinshaw, O., Invariant subspace theorems for positive operators, J. Funct. Analysis 124 (1994), 95–111.CrossRefGoogle Scholar
4.Abramovich, Y. A., Aliprantis, C. D. and Burkinshaw, O., The invariant subspace problem: some recent advances, Rend. Istit. Mat. Univ. Trieste Suppl. 29 (1998), 1–76.Google Scholar
5.Aliprantis, C. D. and Burkinshaw, O., Positive operators (Academic Press, Orlando, 1985).Google Scholar
6.Birkhoff, G., Lattice theory, AMS Colloq. Publ., vol. 25 (Providence, RI, 1967).Google Scholar
7.Choi, M. D., Nordgren, E. A., Radjavi, H., Rosenthal, P. and Zhong, Y., Triangularizing semigroups of quasinilpotent operators with non-negative entries, Indiana Univ. Math. J. 42 (1993), 15–25.CrossRefGoogle Scholar
8.Jahandideh, M. T., On the ideal-triangularizability of positive operators on Banach lattices, Proc. Am. Math. Soc. 125 (1997), 2661–1670.CrossRefGoogle Scholar
9.Jahandideh, M. T., Positive operators with p-hyperinvariant closed ideals. Preprint.Google Scholar
10.Luxemburg, W. A. J. and Zaanen, A. C., Riesz spaces I (North-Holland, Amsterdam, 1971).Google Scholar
11.Meyer-Nieberg, P., Banach lattices (Springer, 1991).CrossRefGoogle Scholar
12.Nordgren, E., Radjavi, H. and Rosenthal, P., Triangularizing semigroups of compact operators, Indiana Univ. Math. J. 33 (1984), 271–275.CrossRefGoogle Scholar
13.Radjavi, H., On reducibility of semigroups of compact operators, Indiana Univ. Math. J. 39 (1990), 499–515.CrossRefGoogle Scholar
14.Ringrose, J. R., Compact non-self-adjoint operators (Van Nostrand Reinhold Math. Studies, London, 1971).Google Scholar
15.Schaefer, H. H., Banach lattices and positive operators, Grundlehren Math. Wiss. Bd. 215 (Springer, 1974).CrossRefGoogle Scholar
16.Schaefer, H. H., Topologische Nilpotenz irreduzibler Operatoren, Math. Z. 117 (1970), 135–140.CrossRefGoogle Scholar
17.Zaanen, A. C., Riesz spaces II (North-Holland, Amsterdam, 1983).Google Scholar