mathematical physics - Implications of unbounded operators in quantum mechanics

Quantum mechanical observables of a system are represented by self - adjoint operators in a separable complex Hilbert space $\mathcal{H}$. Now I understand a lot of operators employed in quantum mechanics are unbounded operators, in nutshell these operators cannot be defined for all vectors in $\mathcal{H}$. For example according to "Stone - von Neumann", the canonical commutation relation $[P, Q] =-i\hbar I$ has no solution for $P$ and $Q$ bounded ! My basic question is :

• If the state of our system $\psi$ is for example not in the domain of $P$ (because $P$ is unbounded), i.e., if $P\cdot \psi$ does not, mathematically, make sense, what does this mean ? Does it mean we cannot extract any information about $P$ when the system is in state $\psi$ ?