Let $a>0$ and consider the dilation operator $\delta^a f(x)=f(ax)$ for functions defined on $(0,\infty)$. I know that if $|\lambda| \neq a^{-1/p}$, then $\lambda-\delta^a$ is injective on $L^p(0,\infty)$ (this follows by spectral arguments, note that $\|\delta^a\|=a^{-1/p}$). I'd like to now if somebody knows an example of two functions which live in different $L^p$-spaces and on which $\lambda-\delta^a$ coincides (a concrete example for some $p$ and $\lambda$ is also fine, I don't need it in general).
The motivation for this question is the compatibility of inverses. From my understanding of the matter such functions should always exist but I want to make it concrete.