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Multiplicative coordinate functionals and ideal-triangularizability

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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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  1. Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. American Mathematical Society (2002)

  2. Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233. Springer, New York (2006)

  3. Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Reprint of the 1985 original. Springer, Dordrecht (2006)

  4. Drnovšek, R.: Common invariant subspaces for collections of operators. Integr. Equ. Oper. Theory 39, 253–266 (2001)

    Article  MATH  Google Scholar 

  5. Drnovšek, R.: Triangularizing semigroups of positive operators on an atomic normed Riesz spaces. Proc. Edin. Math. Soc. 43, 43–55 (2000)

    Article  MATH  Google Scholar 

  6. Drnovšek, R., Kandić, M.: Ideal-triangularizability of semigroups of positive operators. Integr. Equ. Oper. Theory 64(4), 539–552 (2009)

    Article  MATH  Google Scholar 

  7. Drnovšek, R., Kandić, M.: More on positive commutators. J. Math. Anal. Appl. 373, 580–584 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Drnovšek, R., Kokol-Bukovšek, D., Livshits, L., MacDonald, G., Omladič, M., Radjavi, H.: An irreducible semigroup of non-negative square-zero operators. Integr. Equ. Oper. Theory 42(4), 449–460 (2002)

    Article  MATH  Google Scholar 

  9. Kandić, M.: Ideal-triangularizability of upward directed sets of positive operators. Ann. Funct. Anal. 2(1), 206–219 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Keicher, V.: On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices. Arch. Math. 87, 359–367 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. König, H.: Eigenvalue distribution of compact operators. In: Operator Theory, Advances and Applications, vol. 16. Birkhäuser, Basel (1986)

  12. Konvalinka, M.: Triangularizability of polynomially compact operators. Integr. Equ. Oper. Theory 52, 271–284 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North Holland, Amsterdam (1971)

    MATH  Google Scholar 

  14. MacDonald, G., Radjavi, H.: Standard triangularization of semigroups of non-negative operators. J. Funct. Anal. 219, 161–176 (2005)

    Google Scholar 

  15. Radjavi, H., Rosenthal, P.: Simultaneous Triangularization. Universitext. Springer, New York (2000)

  16. Schep, A.R.: Positive diagonal and triangular operators. J. Oper. Theory 3, 165–178 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Wolff, M.P.H.: Triviality of the peripheral point spectrum of positive semigroups on atomic Banach lattices. Positivity 12, 185–192 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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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).

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