On reducibility of semigroups

of compact quasinilpotent operators

Author:
Roman Drnovsek

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2391-2394

MSC (1991):
Primary 47A15, 47D03

DOI:
https://doi.org/10.1090/S0002-9939-97-04108-7

MathSciNet review:
1422865

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Abstract | References | Similar Articles | Additional Information

Abstract: The following generalization of Lomonosov's invariant subspace theorem is proved. Let be a multiplicative semigroup of compact operators on a Banach space such that for every finite subset of , where denotes the Rota-Strang spectral radius. Then is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space , every semigroup of compact quasinilpotent operators on is reducible.

(B) For every complex Hilbert space , for every semigroup of compact quasinilpotent operators on , and for every finite subset of it holds that .

The question whether the assertion (A) is true was considered by Nordgren, Radjavi and Rosenthal in 1984, and it seems to be still open.

**1.**P. S. Guinand,*On quasinilpotent semigroups of operators*, Proc. Amer. Math. Soc.**86**(1982), 485-486. MR**84h:47042****2.**D. Hadwin, E. Nordgren, M. Radjabalipour, H. Radjavi and P. Rosenthal,*A nil algebra of bounded operators on Hilbert space with semisimple norm closure*, Integral Equat. Oper. Th.**9**(1986), 739-743. MR**87k:47104****3.**A. A. Jafarian, H. Radjavi, P. Rosenthal and A. R. Sourour,*Simultaneous triangularizability, near commutativity and Rota's theorem*, Trans. Amer. Math. Soc.**347**(1995), 2191-2199. MR**95i:47033****4.**A. J. Michaels,*Hilden's simple proof of Lomonosov's invariant subspace theorem*, Adv. in Math.**25**(1977), 56-58. MR**59:17893****5.**E. Nordgren, H. Radjavi and P. Rosenthal,*Triangularizing semigroups of compact operators*, Indiana Univ. Math. J.**33**(1984), 271-275. MR**85b:47047****6.**P. Rosenthal and A. So{\l}tysiak,*Formulas for the joint spectral radius of non-commuting Banach algebra elements*, Proc. Amer. Math. Soc.**123**(1995), 2705-2708. MR**95k:47008****7.**G.-C. Rota and W. G. Strang,*A note on the joint spectral radius*, Indag. Math.**22**(1960), 379-381. MR**26:5434**

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Additional Information

**Roman Drnovsek**

Affiliation:
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia

Email:
roman.drnovsek@fmf.uni-lj.si

DOI:
https://doi.org/10.1090/S0002-9939-97-04108-7

Keywords:
Invariant subspaces,
semigroups,
reducibility,
simultaneous triangularizability

Received by editor(s):
March 1, 1996

Additional Notes:
This work was supported in part by the Research Ministry of Slovenia.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society