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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a theorem of Brickman-Fillmore
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by Antonio Hwang PDF
Proc. Amer. Math. Soc. 55 (1976), 93-94 Request permission

Abstract:

Let $V$ be a finite dimensional vector space over an arbitrary field. We show that if $\dim V \leqslant 3$ and if $A,B$ and $C$ are pairwise commuting linear transformations on $V$ such that every subspace invariant for both $A$ and $B$ is also invariant for $C$, then $C$ is a polynomial in $A$ and $B$. (Brickman and Fillmore proved that if $B = 0$ then this statement is true for any finite dimensional vector space $V$.) An example shows that this is not true for $\dim V > 3$.
References
Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 55 (1976), 93-94
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0394245-3
  • MathSciNet review: 0394245