ON THE SPECTRAL RADIUS OF IRREDUCIBLE AND WEAKLY IRREDUCIBLE OPERATORS IN BANACH LATPICES
- 1 January 1978
- journal article
- research article
- Published by Taylor & Francis in Quaestiones Mathematicae
- Vol. 2 (4) , 495-506
- https://doi.org/10.1080/16073606.1978.9631548
Abstract
If T is an operator on a Banach lattice E we call T weakly irreducible if E contains no non-trivial T-invariant bands. We prove that if E is order complete and if the weakly irreducible operator T > 0 is in (E′oo ⊗ E)⊥⊥ then T has positive spectral radéus. Prom this follows that Jentesch's theorem holds in arbitrary Banach function spaces. If [Ttilde] denotes the restriction of T′ to E′oo, 0 ⋚ T an order continuous operator, then T is weakly irreducible if and only if [Ttilde]: E′oo→E′oo is weakly irreducible. Finally we show that the majorizing, irreducible operator T ≥ 0, has positive spectral radius if either Tn is weakly compact or E has property (P) or T is strongly majorizing.Keywords
This publication has 1 reference indexed in Scilit:
- Banach Lattices and Positive OperatorsPublished by Springer Nature ,1974