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
- L. Brickman and P. A. Fillmore, The invariant subspace lattice of a linear transformation, Canadian J. Math. 19 (1967), 810–822. MR 213378, DOI 10.4153/CJM-1967-075-4
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