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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.

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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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Correspondence to Marko Kandić.

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Kandić, M. Multiplicative coordinate functionals and ideal-triangularizability. Positivity 17, 1085–1099 (2013). https://doi.org/10.1007/s11117-013-0222-z

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