Abstract
In this paper we prove that a positive commutator between a positive compact operator A and a positive operator B is in the radical of the Banach algebra generated by A and B. Furthermore, on every at least three-dimensional Banach lattice we construct finite rank operators A and B satisfying \(AB\ge BA\ge 0\) such that the commutator \(AB-BA\) is not contained in the radical of the Banach algebra generated by A and B. These two results now completely answer to two open questions published in (Bračič et al., Positivity 14:431–439, 2010). We also obtain relevant results in the case of the Volterra and the Donoghue operator.
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References
Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. American Mathematical Society, Providence (2002)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators (reprint of the 1985 original). Springer, Dordrecht (2006)
Bonsall, F.F., Duncan, J.: Complete Normed Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 80. Springer, New York (1973)
Bračič, J., Drnovšek, R., Farforovskaya, Y.B., Rabkin, E.L., Zemánek, J.: On positive commutators. Positivity 14, 431–439 (2010)
Drnovšek, R.: Once more on positive commutators. Studia Math. 211, 241–245 (2012)
Drnovšek, R.: Triangularizing semigroups of positive operators on an atomic normed Riesz spaces. Proc. Edinb. Math. Soc. 43, 43–55 (2000)
Drnovšek, R., Kandić, M.: More on positive commutators. J. Math. Anal. Appl. 373, 580–584 (2011)
ter Elst, A.F.M., Sauter, M., Zemánek, J.: Generation and commutation properties of the Volterra operator. Arch. Math. 99(5), 467–479 (2012)
Gao, N.: On commuting and semi-commuting positive operators. Proc. Am. Math. Soc. 142, 2733–2745 (2014)
Halmos, P.: A Hilbert Space Problem Book, Graduate Texts in Mathematics, 2nd edn. Springer, New York (1982)
Lomonosov, V.: Invariant subspaces of the family of operators that commute with a completely continuous operator (Russian). Funkcional. Anal. i Priloen. 7(3), 55–56 (1973)
Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North Holland, Amsterdam (1971)
Radjavi, H., Rosenthal, P.: Invariant Subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 77. Springer, New York (1973)
Radjavi, H., Rosenthal, P.: Simultaneous Triangularization. Springer, New York (2000)
Schaefer, H.H.: Banach Lattices. Springer, Berlin (1974)
Sirotkin, G.: A version of the Lomonosov subspace theorem for real Banach spaces. Indiana Univ. Math. J. 54(1), 257–262 (2005)
Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1997)
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This work was supported in part by the Slovenian Research Agency.
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Kandić, M., Šivic, K. On the positive commutator in the radical. Positivity 21, 99–111 (2017). https://doi.org/10.1007/s11117-016-0409-1
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DOI: https://doi.org/10.1007/s11117-016-0409-1