Sesquitransitive and Localizing Operator Algebras |
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Authors: | Victor I. Lomonosov Heydar Radjavi Vladimir G. Troitsky |
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Affiliation: | (1) Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA;(2) Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada;(3) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada |
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Abstract: | An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x n ) in B there exists a subsequence and a bounded sequence (A k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC. |
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Keywords: | Primary 47A15 Secondary 47L10 |
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