Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The following generalization of Lomonosov's invariant subspace theorem is proved. Let ${\mathcal S}$ be a multiplicative semigroup of compact operators on a Banach space such that $\hat {r} (S_1, \ldots , S_n) = 0$ for every finite subset $\{S_1, \ldots , S_n\}$ of ${\mathcal S}$, where $\hat {r}$ denotes the Rota-Strang spectral radius. Then ${\mathcal S}$ is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space ${\mathcal H}$, every semigroup of compact quasinilpotent operators on ${\mathcal H}$ is reducible.

(B) For every complex Hilbert space ${\mathcal H}$, for every semigroup of compact quasinilpotent operators on ${\mathcal H}$, and for every finite subset $\{S_1, \ldots , S_n\}$ of ${\mathcal S}$ it holds that $\hat {r}(S_1, \ldots , S_n) = 0$.

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

References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47A15, 47D03

Retrieve articles in all journals with MSC (1991): 47A15, 47D03

Additional Information

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

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

American Mathematical Society