Infinite-dimensional triangularizable algebras

Zachary Mesyan

doi:10.1515/forum-2018-0109

Article

Infinite-dimensional triangularizable algebras

Zachary Mesyan

Published/Copyright: August 11, 2018

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From the journal Forum Mathematicum Volume 31 Issue 1

Abstract

Let Endk⁢(V) denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define X⊆Endk⁢(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.

Keywords: Triangular matrix; linear transformation; simultaneous triangularization; triangularizable algebra; nilpotent semigroup; endomorphism ring; function topology

MSC 2010: 15A04; 16S50; 16W80; 47D03

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

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https://doi.org/10.1515/forum-2018-0109

Keywords for this article

Triangular matrix; linear transformation; simultaneous triangularization; triangularizable algebra; nilpotent semigroup; endomorphism ring; function topology