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Infinite-dimensional triangularizable algebras

  • Zachary Mesyan EMAIL logo
Veröffentlicht/Copyright: 11. August 2018

Abstract

Let Endk(V) denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define XEndk(V) to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of Endk(V) is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of Endk(V), which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields.


Communicated by Manfred Droste


Acknowledgements

I am grateful to George Bergman for a very enlightening conversation about this material, and for numerous comments on an earlier draft of this paper, which have led to significant improvements. I also would like to thank Greg Oman for a pointer to the literature, Manuel Reyes for a helpful example, and the referee for a careful reading of the manuscript.

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Received: 2018-05-07
Revised: 2018-06-27
Published Online: 2018-08-11
Published in Print: 2019-01-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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