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?



Does the image of $f$ contain a positive number?
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. 


