Roman Drnovsek - On reducibility of semigroups of compact quasinilpotent operators

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Roman Drnovsek - On reducibility of semigroups of compact quasinilpotent operators

	ROMAN DRNOVSEK, Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
	On reducibility of semigroups of compact quasinilpotent operators

Using the famous Lomonosov-Hilden technique we proved the following invariant subspace theorem. Let $\cal S$ be a multiplicative semigroup of compact operators on a Banach space X of dimension greater than 1 such that $\hat r (S_1, \ldots, S_n) = 0$ for every finite subset $\{S_1, \ldots, S_n\}$ of $\cal S$ , where $\hat r$ denotes the Rota-Strang spectral radius. Then $\cal S$ is reducible, i.e., there exists a closed subspace of X, other than $\{0\}$ and X, which is invariant under every member of $\cal S$ .

This result implies that the following assertions are equivalent:

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

(B) For every complex Hilbert space ${\cal H}$ , for every semigroup of compact quasinilpotent operators on ${\cal H}$ , and for every finite subset $\{S_1, \ldots, S_n\}$ of $\cal S$ 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 (and independently by Shul'man) in 1984, and it seems to be still open.

Next: Douglas Farenick - Extremal Up: Operator Theory / Théorie Previous: Allan Donsig - Algebraic

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