| \beta m_{\beta} \rangle 
  a_{\beta m_{\beta}}^{\dagger} a_{\alpha m_{\alpha}} 
. 
\label{E:spo}$$ The corresponding strength distribution is given by $$R(E) = 
\sum_{\nu} | \langle \nu | 
O_{{J}^{\pi}{M}} | 0 \rangle |^2 \delta (E - E_{\nu}) 
, 
\label{E:strd}$$ where the sum runs over states with quantum numbers ${J}^{\pi}$. The transition matrix element between the ground state and the excited state $| \nu \rangle $ \[see Eq. (\[E:nu\])\] is given by $$\begin{aligned}
\lefteqn{ \langle \nu | 
O_{{J}^{\pi}{M}} | 0 \rangle 
= } \nonumber \\ 
&&
\sum_{\alpha m_{\alpha}  \beta m_{\beta}} 
\langle \alpha m_{\alpha} | O_{{J}^{\pi}{M}} | \beta m_{\beta} \rangle 
\nonumber \\ && \mbox{} \hspace{13mm}\times  
 \langle 0 | Q_{\nu, {J}^{\pi}{M}}  a_{\beta m_{\beta}}^{\dagger} a_{\alpha m_{\alpha}} 
| 0 \rangle 
. \end{aligned}$$ After we substitute for the operator $Q$ using Eqs. (\[E:nu\]), (\[E:Aop\]), and (\[E:Nmat\]) we obtain $$\begin{aligned}
\lefteqn{ \langle \nu | 
O_{{J}^{\pi}{M}} | 0 \rangle 
= } \nonumber \\ && 
\hat{J}^{-1}  
\sum_{ph} 
[ 
                    X_{ph}^{\nu , {J}^{\pi} \ast      } 
+ (-1)^{{J}} Y_{ph}^{\nu , {J}^{\pi} \ast      } 
] 
\nonumber \\ && \mbox{} \hspace{13mm} \times  
D_{ph}^{1/2} 
\langle p || O_{{J}} || h \rangle 
. 
\label{E:nu0me}\end{aligned}$$ The s.p. transition amplitudes are renormalized by a factor $D_{ph}^{1/2}$ due to the ground-state correlations.

In the UCOM framework the operators of all observables must be 