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

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), no. 3, 485–486. MR**671220**, 10.1090/S0002-9939-1982-0671220-5**2.**Donald Hadwin, Eric Nordgren, Mehdi Radjabalipour, Heydar Radjavi, and Peter Rosenthal,*A nil algebra of bounded operators on Hilbert space with semisimple norm closure*, Integral Equations Operator Theory**9**(1986), no. 5, 739–743. MR**860869**, 10.1007/BF01195810**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), no. 6, 2191–2199. MR**1257112**, 10.1090/S0002-9947-1995-1257112-5**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.**Eric Nordgren, Heydar Radjavi, and Peter Rosenthal,*Triangularizing semigroups of compact operators*, Indiana Univ. Math. J.**33**(1984), no. 2, 271–275. MR**733900**, 10.1512/iumj.1984.33.33014**6.**P. Rosenthal and A. Sołtysiak,*Formulas for the joint spectral radius of noncommuting Banach algebra elements*, Proc. Amer. Math. Soc.**123**(1995), no. 9, 2705–2708. MR**1257123**, 10.1090/S0002-9939-1995-1257123-5**7.**Gian-Carlo Rota and Gilbert Strang,*A note on the joint spectral radius*, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math.**22**(1960), 379–381. MR**0147922**

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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:
http://dx.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