The operator $T$ is $\dfrac{d}{dt}$ and $$\left\{\begin{array}{lc}x'(t)-\lambda x(t)=-y(t)\\ x(0)=0\end{array}\right.$$
and the domain of $T$ is $D(T)=\{x\in L^2(0,\infty):\; x\; \text{absolutely continuous on }(0,+\infty)\}$
if $\lambda =0$ , $$x(t)=-\int_0^ty(s)ds$$
How to prouve that the $Im(T)$ is dense in $L^2(0,\infty)$.