# Multiplicative coordinate functionals and ideal-triangularizability

Multiplicative coordinate functionals and ideal-triangularizability 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Multiplicative coordinate functionals and ideal-triangularizability

, Volume 17 (4) – Jan 22, 2013
15 pages

/lp/springer_journal/multiplicative-coordinate-functionals-and-ideal-triangularizability-kU0pubhDEy
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0222-z
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Jan 22, 2013

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