# A question on self-adjoint operator on Hilbert space

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?

• Yes, actually every invertible nonnegative operator is positive. Dec 3, 2020 at 9:22

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.