Can anyone give me a a restricted non-void set of bounded inputs which results in bounded outputs, though the operator is not bounded on the whole space (consider L2-space).
One can consider a simple differential equation $\frac{d}{dt}y - \alpha y = \frac{d}{dt}u, \alpha >0$, where $u$ should be considered as the input and $y$ as the output.
From the control theoretic viewpoint, the above system is clearly unstable because it has a pole in RHP of s-plane. Therefore, for a bounded input the output will be unbounded as $t\to\infty$.