Ideal-triangularizability of nil-algebras generated by positive operators
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Abstract:
R. Drnovšek, D. Kokol-Bukovšek, L. Livshits, G. MacDonald, M. Omladič, and H. Radjavi constructed an irreducible set of positive nilpotent operators on $L^p[0,1)$ which is closed under multiplication, addition and multiplication by positive real scalars with the property that any finite subset is ideal-triangularizable. In this paper we prove the following:
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every algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable whenever the nilpotency index of its operators is bounded;
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every finite subset of an algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable.
References
- A. Z. Anan′in, A nil algebra with nonradical tensor square, Sibirsk. Mat. Zh. 26 (1985), no. 2, 192–194, 224 (Russian). MR 788345
- 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 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
- R. Drnovšek, D. Kokol-Bukovšek, L. Livshits, G. MacDonald, M. Omladič, and H. Radjavi, An irreducible semigroup of non-negative square-zero operators, Integral Equations Operator Theory 42 (2002), no. 4, 449–460. MR 1885443, DOI 10.1007/BF01270922
- Sandy Grabiner, The nilpotency of Banach nil algebras, Proc. Amer. Math. Soc. 21 (1969), 510. MR 236700, DOI 10.1090/S0002-9939-1969-0236700-9
- D. Hadwin, E. Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal, On simultaneous triangularization of collections of operators, Houston J. Math. 17 (1991), no. 4, 581–602. MR 1147275
- I. N. Herstein, Noncommutative rings, Carus Mathematical Monographs, vol. 15, Mathematical Association of America, Washington, DC, 1994. Reprint of the 1968 original; With an afterword by Lance W. Small. MR 1449137
- Graham Higman, On a conjecture of Nagata, Proc. Cambridge Philos. Soc. 52 (1956), 1–4. MR 73581
- Mohammed Taghi Jahandideh, On the ideal-triangularizability of positive operators on Banach lattices, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2661–2670. MR 1396983, DOI 10.1090/S0002-9939-97-03885-9
- W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1971. MR 0511676
- Masayoshi Nagata, On the nilpotency of nil-algebras, J. Math. Soc. Japan 4 (1952), 296–301. MR 53088, DOI 10.2969/jmsj/00430296
- Edmund R. Puczyłowski, Nil algebras with nonradical tensor square, Proc. Amer. Math. Soc. 103 (1988), no. 2, 401–403. MR 943055, DOI 10.1090/S0002-9939-1988-0943055-2
- Heydar Radjavi and Peter Rosenthal, Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000. MR 1736065, DOI 10.1007/978-1-4612-1200-3
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
- Yong Zhong, Irreducible semigroups of functionally positive nilpotent operators, Trans. Amer. Math. Soc. 347 (1995), no. 8, 3093–3100. MR 1264835, DOI 10.1090/S0002-9947-1995-1264835-0
Additional Information
- Marko Kandić
- Affiliation: Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
- Email: marko.kandic@fmf.uni-lj.si
- Received by editor(s): December 17, 2009
- Received by editor(s) in revised form: March 2, 2010
- Published electronically: July 12, 2010
- Additional Notes: This work was supported by the Slovenian Research Agency
- Communicated by: Nigel J. Kalton
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 485-490
- MSC (2010): Primary 47A15, 47B65; Secondary 16N40
- DOI: https://doi.org/10.1090/S0002-9939-2010-10476-8
- MathSciNet review: 2736331