Someone told me about this condition ("definition of the domain of the adjoint") for an operator $T$ acting on $C^\infty(L^2(\mathbb{R}))$ functions:
$$f\in D(T^*) \iff (\exists M_f>0 \;s.t.\; |(f,Tg)|<M_f||g||,\;\forall g\in D(T))$$
However I couldn't find this result anywhere (textbook or internet). And it also seems to me that such a result would make the problem of checking self-adjointness of operators easier that it actually is (since if you can find a $g$ for which $(\exists M_f>0 \;s.t.\; |(f,Tg)|<M_f||g||,\;\forall g\in D(T))$ doesn't hold for each $f\notin D(T)$, you can prove self-adjointness).
Has anyone encountered this? Is it a valid theorem/result in functional analysis?