Ideal-triangularizability of nil-algebras generated by positive operators

Kandić, Marko

doi:10.1090/S0002-9939-2010-10476-8

Ideal-triangularizability of nil-algebras generated by positive operators
HTML articles powered by AMS MathViewer

by Marko Kandić PDF

Proc. Amer. Math. Soc. 139 (2011), 485-490 Request permission

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:

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;
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