Let $H$ be a Hilbert space and $T$ be a bounded and positive operator on $H$. Define a real function $f$ on positive real numbers by $$f(r):=\|(r+T)^{-1}\|^{-1}-r\quad(r\in\mathbb R_+).$$ Does the image of $f$ contain a positive number?
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Since $T$ is a bounded symmetric positive operator $f( r)=\inf \sigma(T)$, a constant function. So the answer is: yes, if and only if $\inf \sigma(T)>0$, that is, $T$ is invertible.
answered Apr 30, 2018 at 7:14
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$\begingroup$ (recall that for any invertible positive operator $L$ one has $\|L^{-1}\|^{-1}=\inf\sigma(L)$) $\endgroup$ Commented Apr 30, 2018 at 7:56