Problem with the Wightman axiom about the transformation law of the fields

Question

The axiom in the title is almost always stated as $$ U(a, A)^{\dagger} \phi(x) U(a, A)=S(A) \phi\left(A^{-1}(x-a)\right) $$ or, more precisely, as $$ U(a, A) \phi_n(f) U(a, A)^{-1}=\sum_{m=1}^N S\left(A^{-1}\right)_n^m \phi_m((a, A) \cdot f) $$ where the terms and the notation can be found on Wikipedia or on Folland's book. I do not know how to interpret the summation on the right, because generally it is not possible to sum two unbounded operators unless they have the same domain. Should I interpret the equality as $$ \langle\eta|U(a, A) \phi_n(f) U(a, A)^{-1}|\theta\rangle=\sum_{m=1}^N S\left(A^{-1}\right)_n^m \langle\eta|\phi_m((a, A) \cdot f)|\theta\rangle $$ for any $\eta, \theta$ in the dense subspace $D$ of the Hilbert space?