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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 ?

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  • $\begingroup$ I assume you want $x\neq 0$ for this? $\endgroup$
    – Zach Stone
    Aug 3, 2015 at 20:23
  • $\begingroup$ $0$ is a fixed point of any linear operator. $\endgroup$ Aug 3, 2015 at 20:23

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Other than $0$? No. Try $A = -I$.

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