On Semitransitive Collections of Operators |
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Authors: | J Bernik L Grunenfelder M Mastnak H Radjavi VG Troitsky |
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Institution: | (1) Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia;(2) Department of Mathematics, Dalhousie University, Halifax, NS, B3H 3J5, Canada;(3) Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada;(4) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G2G1, Canada |
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Abstract: | A collection F of operators on a vector space V is said to be
semitransitive if for every pair of nonzero vectors x and y in V
there exists a member T of F such that either Tx = y or Ty = x (or
both). We study semitransitive algebras and semigroups of operators. One
of the main results is that if the underlying field is algebraically
closed, then every semitransitive algebra of operators on a space of dimension n contains a nilpotent element of index n. Among other results on semitransitive semigroups, we show
that if the rank of nonzero members of such a semigroup acting on an
n-dimensional space is a constant k, then k divides n. |
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Keywords: | Semitransitive operator algebra A30 D03 L10 |
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