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Proceedings of the American Mathematical Society

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



On a theorem of Brickman-Fillmore

Author: Antonio Hwang
Journal: Proc. Amer. Math. Soc. 55 (1976), 93-94
MathSciNet review: 0394245
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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$.

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Keywords: Invariant subspace
Article copyright: © Copyright 1976 American Mathematical Society