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On page 348 of Infinite Dimensional Dynamical Systems in Mechanics and Physics by Roger Temam, there is something I don't understand. I abstracted the question as follow: Let $T:H \rightarrow H$ be a bounded self-adjoint operator on Hilbert space $H$. If $T$ is nonnegative and have bounded inverse $T^{-1}$ then why $$ \inf_{\| \phi \|=1} \langle T\phi , \phi \rangle >0. $$ is valid?

I found a similar question here Bounded Self-adjoint Operator on Hilbert Space, however it only provide answer when $T$ is positive, is it true for nonnegative operator as well?

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  • $\begingroup$ Yes, actually every invertible nonnegative operator is positive. $\endgroup$
    – Berci
    Dec 3, 2020 at 9:22

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Let $m:=\inf_{\| \phi \|=1} \langle T\phi , \phi \rangle $ and $M:=\sup_{\| \phi \|=1} \langle T\phi , \phi \rangle $.

It is well known that for the spectrum of $T$ we have

$$ \sigma(T) \subseteq [m,M].$$

Suppose that that $m=0$, then $0 \in \sigma(T),$ a contradiction, since $T$ has a bounded inverse.

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