Let $A:L^{2}\left( (0,1)\right) \rightarrow L^{2}\left( (0,1)\right) $ be defined as $Af\left( x\right) =-\ln \left( x\right) f\left( \frac{x}{2}% \right) $ for every $f\in L^{2}\left( (0,1)\right) $ and $x\in \left( 0,1\right) $
The map $A$ is linear and $\ln \in L^{2}\left( (0,1)\right) $ but I have no idea if $A$ is bounded or not. That's what I'm trying to answer to.
For $n\ge 2$ let $$f_n(x)=\begin{cases}1 & 0\le x\le {1/n}\\ 0 & x>{1/n} \end{cases}$$ Then $\|f\|_2^2={1\over n}$ and $$\|Af_n\|_2^2= \int\limits_0^{2/n}\ln^2x \,dx\ge {2\over n}\ln^2(2/n)$$ Thus $${\|Af_n\|_2\over \|f_n\|_2}\ge \sqrt{2}|\ln(2/n)|\underset{n\to \infty}{\longrightarrow}\infty$$