Let $X$ be a Banach space and $A:X\to X$ a bounded operator. Assume that $A^2$ admits a fixed point in $X$ i.e. there exists $x_0\in X$ such that $A^2x_0=x_0$. Does this mean that $A$ has also a fixed point ?
2026-04-02 10:09:10.1775124550
$A^2$ has a fixed point implies $A$ has also a fixed point
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Other than $0$? No. Try $A = -I$.