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Let $V$ be an inner product space finitely generated over $\mathbb{C}$ and let $\alpha$ and $\beta$ be self-adjoint endomorphisms of $V$ satisfying the condition that $\alpha\beta$ is a projection. Is $\beta\alpha$ necessarily also a projection?

I am stuck on this problem. Any hint would be appreciated.

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Hint:

If $\alpha \beta$ is a projection than we have $(\alpha \beta)^2=\alpha \beta$

so,if $\alpha$ is invertible, we have: $$ (\beta \alpha)(\beta \alpha)=\alpha^{-1}\alpha(\beta \alpha)(\beta \alpha)=\alpha^{-1}(\alpha\beta) (\alpha\beta) \alpha=\alpha^{-1}\alpha\beta \alpha=\beta \alpha $$ so: $(\beta \alpha)^2=\beta \alpha$

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  • $\begingroup$ So the answer would be no. Just in the case $\alpha$ is an automorphism, $\beta\alpha$ is a projection. Right? $\endgroup$ – Parisina Jan 3 '17 at 17:47

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