With $$ O_{\mu}^{\lambda} = \sum_{j_pj_q} \langle p \vert \vert O^{\lambda}\vert \vert q \rangle \lambda^{-1}\left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}_{\mu}, $$ we can write $$ \langle \Phi^J_M|O^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle = \sum_{pq} \langle p \vert O_{\mu}^{\lambda} \vert q \rangle \langle \Phi_{M}^{J} \vert a^{\dagger}_pa_q \vert \Phi_{M'}^{J'}\rangle, $$ as $$ \langle \Phi^J_M|O^{\lambda}_{\mu}|\Phi^{J'}_{M'}\rangle = \sum_{j_pj_q} \langle p \vert \vert O^{\lambda} \vert\vert q \rangle \langle \Phi_{M}^{J}\vert \lambda^{-1}\left[ a_{j_p}^{\dagger}\tilde{a}_{j_q}\right]^{\lambda}_{\mu} \vert \Phi_{M'}^{J'}\rangle. $$ We have suppressed the summation over quantum numbers like \( n_p,n_q \) etc.