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Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that

$$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$

for $\sigma\in H_+\equiv \bigoplus\limits_{0<|\lambda|\leq c}E_\lambda$. Here $E_\lambda$ is the eigenspace of $D$ with eigenvalue $\lambda$.

I'm not sure if this question is too easy for math overflow. Could you give me some help with the details? Thanks in advance.

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    $\begingroup$ Express $\sigma$ in an orthonormal basis consisting of eigenvectors. $\endgroup$
    – user473423
    Commented Apr 10, 2022 at 7:13
  • $\begingroup$ Could you please write down your idea with some details? I can't follow you. Thanks a lot.@Echo $\endgroup$
    – Radeha Longa
    Commented Apr 10, 2022 at 7:15
  • $\begingroup$ If $\sigma=\sum_i\mu_if_i$ with $Df_i=\lambda_if_i$, then $\parallel D\sigma\parallel^2=\sum_i|\mu_i|^2|\lambda_i|^2\ge\sum_ic^2|\mu_i|^2=c^2\parallel \sigma\parallel^2$ $\endgroup$
    – user473423
    Commented Apr 10, 2022 at 11:08
  • $\begingroup$ Excuse me, I checked your calculation and I think it should be $|\lambda_i|^2\leq c^2$ for $\sigma\in H_+$. @Echo $\endgroup$
    – Radeha Longa
    Commented Apr 10, 2022 at 11:55
  • $\begingroup$ Oh sorry, I read the inequality the other way round. But see, only this way your claim is true, otherwise it is false. Or did you mean $\le$ in your claim? $\endgroup$
    – user473423
    Commented Apr 10, 2022 at 14:05

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