On commuting and semi-commuting positive operators

Gao, Niushan

doi:10.1090/S0002-9939-2014-12002-8

On commuting and semi-commuting positive operators
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by Niushan Gao

Proc. Amer. Math. Soc. 142 (2014), 2733-2745

DOI: https://doi.org/10.1090/S0002-9939-2014-12002-8

Published electronically: May 7, 2014

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Abstract:

Let $K$ be a positive compact operator on a Banach lattice. We prove that if either $[K\rangle$ or $\langle K]$ is ideal irreducible, then $[K\rangle =\langle K]=L_+(X)\cap \{K\}’$. We also establish the Perron-Frobenius Theorem for such operators $K$. Finally, we apply our results to answer questions posed by Abramovich and Aliprantis (2002) and Bračič et al. (2010).

References

Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, On the spectral radius of positive operators, Math. Z. 211 (1992), no. 4, 593–607. MR 1191098, DOI 10.1007/BF02571448
Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, Invariant subspaces of operators on $l_p$-spaces, J. Funct. Anal. 115 (1993), no. 2, 418–424. MR 1234398, DOI 10.1006/jfan.1993.1097
Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, Invariant subspace theorems for positive operators, J. Funct. Anal. 124 (1994), no. 1, 95–111. MR 1284604, DOI 10.1006/jfan.1994.1099
Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782, DOI 10.1090/gsm/050
E. A. Alekhno, Spectral properties of band irreducible operators, Positivity IV—theory and applications, Tech. Univ. Dresden, Dresden, 2006, pp. 5–14. MR 2243476
Egor A. Alekhno, Some properties of essential spectra of a positive operator, Positivity 11 (2007), no. 3, 375–386. MR 2336203, DOI 10.1007/s11117-007-2088-4
Egor A. Alekhno, The irreducibility in ordered Banach algebras, Positivity 16 (2012), no. 1, 143–176. MR 2892579, DOI 10.1007/s11117-011-0117-9
Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Springer, Dordrecht, 2006. Reprint of the 1985 original. MR 2262133, DOI 10.1007/978-1-4020-5008-4
Razvan Anisca and Vladimir G. Troitsky, Minimal vectors of positive operators, Indiana Univ. Math. J. 54 (2005), no. 3, 861–872. MR 2151236, DOI 10.1512/iumj.2005.54.2544
Janko Bračič, Roman Drnovšek, Yuliya B. Farforovskaya, Evgueniy L. Rabkin, and Jaroslav Zemánek, On positive commutators, Positivity 14 (2010), no. 3, 431–439. MR 2680506, DOI 10.1007/s11117-009-0028-1
Ben de Pagter, Irreducible compact operators, Math. Z. 192 (1986), no. 1, 149–153. MR 835399, DOI 10.1007/BF01162028
Roman Drnovšek, Triangularizing semigroups of positive operators on an atomic normed Riesz space, Proc. Edinburgh Math. Soc. (2) 43 (2000), no. 1, 43–55. MR 1744698, DOI 10.1017/S001309150002068X
Roman Drnovšek, Common invariant subspaces for collections of operators, Integral Equations Operator Theory 39 (2001), no. 3, 253–266. MR 1818060, DOI 10.1007/BF01332655
Roman Drnovšek, Once more on positive commutators, Studia Math. 211 (2012), no. 3, 241–245. MR 3002445, DOI 10.4064/sm211-3-5
Roman Drnovšek and Marko Kandić, Ideal-triangularizability of semigroups of positive operators, Integral Equations Operator Theory 64 (2009), no. 4, 539–552. MR 2534032, DOI 10.1007/s00020-009-1705-y
Roman Drnovšek and Marko Kandić, More on positive commutators, J. Math. Anal. Appl. 373 (2011), no. 2, 580–584. MR 2720706, DOI 10.1016/j.jmaa.2010.07.056
Niushan Gao, Extensions of Perron-Frobenius theory, Positivity 17 (2013), no. 4, 965–977. MR 3129846, DOI 10.1007/s11117-012-0215-3
N. Gao, Ph.D. thesis, forthcoming.
Niushan Gao and Vladimir G. Troitsky, Irreducible semigroups of positive operators on Banach lattices, Linear Multilinear Algebra 62 (2014), no. 1, 74–95. MR 3175401, DOI 10.1080/03081087.2012.762715
Hailegebriel E. Gessesse, Invariant subspaces of super left-commutants, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1357–1361. MR 2465659, DOI 10.1090/S0002-9939-08-09673-1
Jacobus J. Grobler, Band irreducible operators, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 4, 405–409. MR 869756, DOI 10.1016/1385-7258(86)90025-9
J. J. Grobler, Spectral theory in Banach lattices, Operator theory in function spaces and Banach lattices, Oper. Theory Adv. Appl., vol. 75, Birkhäuser, Basel, 1995, pp. 133–172. MR 1322503, DOI 10.1007/978-3-0348-9076-2_{1}0
D. W. Hadwin, A. Kitover and M. Orhon, Strong monotonicity of spectral radius of positive operators, arXiv:1205.5583 [math.FA].
A. K. Kitover, A generalized Jentzsch theorem, Positivity 9 (2005), no. 3, 501–509. MR 2188534, DOI 10.1007/s11117-004-8291-7
Ivo Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math. 19 (1970), 607–628. MR 415405, DOI 10.1137/0119060
Fumio Niiro and Ikuko Sawashima, On the spectral properties of positive irreducible operators in an arbitrary Banach lattice and problems of H. H. Schaefer, Sci. Papers College Gen. Ed. Univ. Tokyo 16 (1966), 145–183. MR 205084
Heydar Radjavi, The Perron-Frobenius theorem revisited, Positivity 3 (1999), no. 4, 317–331. MR 1721557, DOI 10.1023/A:1009766219129
Heydar Radjavi and Peter Rosenthal, Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000. MR 1736065, DOI 10.1007/978-1-4612-1200-3
Gleb Sirotkin, A version of the Lomonosov invariant subspace theorem for real Banach spaces, Indiana Univ. Math. J. 54 (2005), no. 1, 257–262. MR 2126724, DOI 10.1512/iumj.2005.54.2561
Ikuko Sawashima, On spectral properties of some positive operators, Natur. Sci. Rep. Ochanomizu Univ. 15 (1964), 53–64. MR 187096
Helmut Schaefer, Some spectral properties of positive linear operators, Pacific J. Math. 10 (1960), 1009–1019. MR 115090, DOI 10.2140/pjm.1960.10.1009
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6