Positivity

, Volume 17, Issue 4, pp 1085–1099 | Cite as

Multiplicative coordinate functionals and ideal-triangularizability

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Abstract

In this paper we investigate how strong is the presence of atoms in Banach lattices corresponding to ideal-triangularizability of semigroups of positive operators. In the first part of the paper we prove that a semigroup \(\fancyscript{S}\) of positive operators on an atomic Banach lattice with order continuous norm is ideal-triangularizable if and only if every coordinate functional \(\phi _{a,a}\) associated to an atom \(a\) is multiplicative on \(\fancyscript{S}\) for all atoms \(a\) in \(E\). We apply this result to the case of positive ideal-triangularizable compact operators on not necessarily atomic lattices. In the second part of the paper we prove that the spectrum of a power compact ideal-triangularizable operator \(T\) satisfies
$$\begin{aligned} \sigma (T)\backslash \{0\}=\{\varphi _a(Ta):\; a\; \text{ is} \text{ an} \text{ atom} \text{ in} \; E\}\backslash \{0\}. \end{aligned}$$
We also prove that for a positive operator from some of the trace ideals the equality above between the spectrum \(\sigma (T)\) and the set of diagonal entries implies that \(T\) is ideal-triangularizable.

Keywords

Banach lattices Positive operators Compact operators  Ideal-triangularizability Spectrum 

Mathematics Subject Classification (2000)

47A15 47B65 47A10 47B06 
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Notes

Acknowledgments

The author was supported by the Slovenian Research Agency. The author also like to thank professor Roman Drnovšek for useful comments and discussions, and to referee for helpful suggestions.

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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