limits_G\left(\sum_{[\xi]\in \widehat{G}}d_\xi \Vert {\sigma}_{N }(x,[\xi])\Vert_{\mathscr{S}_p}^p \right)^{\frac{1}{p}}\textnormal{d}x\\
&= \sqrt{d}_\eta \smallint\limits_G\Vert {\sigma}_{N }(x,\cdot)\Vert_{\mathscr{S}_p(\widehat{G})} \textnormal{d}x\\
&= \sqrt{d}_\eta \Vert {\sigma}_{N }(x,\cdot)\Vert_{L^1(G,\mathscr{S}_p(\widehat{G}))}.
\end{align*}
All the analysis above allows the following estimate  for the $p$-Schatten norm of $A$ as follows:
\begin{align*}
  \Vert Af \Vert_{\mathscr{S}_p} &\lesssim \sum_{[\eta]\in \widehat{G}}\sum_{r,s=1}^{d_\eta}\langle \eta\rangle^{-N}\Vert \eta_{rs}\Vert_{L^\infty(G)}\Vert\textnormal{Op}(\widehat{\sigma}_{N,\,sr }([\eta],\cdot ))\Vert_{\mathscr{S}_p}  \\
  &\lesssim \sum_{[\eta]\in \widehat{G}}\sum_{r,s=1}^{d_\eta}\langle \eta\rangle^{-N}\Vert \eta_{rs}\Vert_{L^\infty(G)}\sqrt{d}_\eta \Vert {\sigma}_{N }(x,\cdot)\Vert_{L^1(G,\mathscr{S}_p(\widehat{G}))}  \\
  &= \Vert {\sigma}_{N }(x,\cdot)\Vert_{L^1(G,\mathscr{S}_p(\widehat{G}))}  \sum_{[\eta]\in \widehat{G}}\langle \eta\rangle^{-N}\sqrt{d}_\eta \sum_{r,s=1}^{d_\eta}\Vert \eta_{rs}\Vert_{L^\infty(G)}  \\
   &\leq  \Vert {\sigma}_{N }(x,\cdot)\Vert_{L^1(G,\mathscr{S}_p(\widehat{G}))}  \sum_{[\eta]\in \widehat{G}}\langle \eta\rangle^{-N}\sqrt{d}_\eta d_\eta \sqrt{d}_\eta.
\end{align*} Thus, for $N>n$ where $n=\dim(G)$ we have that 
\begin{equation}
  \Vert Af \Vert_{\mathscr{S}_p} \lesssim    \Vert {\sigma}_{N }(x,\cdot)\Vert_{L^1(G,\mathscr{S}_p(\widehat{G}))} \sum_{[\eta]\in \widehat{G}}d_\eta^2\langle \eta\rangle^{-N}<\infty,
\end{equation} where we have use that (see Lemma 3.8 of \cite{DelRuzTrace11})
$$ \sum_{[\eta]\in \widehat{G}}d_\eta^2\langle \eta\rangle^{-s}<\infty \Longleftrightarrow s>n. $$
The proof of Theorem \ref{Michael:criterion:2} is complete.
\end{proof}
The following is a consequence of Theorem \ref{Michael:criterion:2}.
\begin{corollary}Let $G$ be a compact Lie group of dimension $n.$ Let $A:C^\infty(G)\rightarrow C^\infty(G) $ be a continuous linear operator and assume that its matrix-valued symbol satisfies the regularity condition
\begin{equation}
    \Vert (1+\mathcal{L}_G)^{\frac{N}{2}}\sigma_A(x,\cdot)\Vert_{L^p(G,\mathscr{S}_p(\widehat{G}))}=\left(\smallint\limits_G\Vert (1+\mathcal{L}_G)^{\frac{N}{2}}\sigma_A(x,\cdot)\Vert_{\mathscr{S}_p(\widehat{G})}^p\textnormal{d}x\right)^{\frac{1}{p}}<\infty
\end{equation}where $N>n.$ Then $A\in \mathscr{S}_{p}(L^2(G))$ provided that $1\leq p<\infty.$
\end{corollary}
\begin{proof}Using the H\"older inequality and that the Haar measure of $G$ is normalised we have that
\begin{align}
   \smallint\limits_G\Vert (1+\mathcal{L}_G)^{\frac{N}{2}}\sigma_A(x,\cdot)\Vert_{\mathscr{S}_p(\widehat{G})}\textnormal{d}x\leq \left( \smallint\limits_G\Vert (1+\mathcal{L}_G)^{\frac{N}{2}}\sigma_A(x,\cdot)\Vert_{\mathscr{S}_p(\widehat{G})}^p\textnormal{d}x  \right)^{\frac{1}{p}}<\infty,
\end{align}for all $N>n.$ Then, in view of Theorem \ref{Michael:criterion:2} follows the membership of $A$ in the Schatten class $\mathscr{S}_p(L^2(G)).$    
\end{proof}


\subsection{Schatten properties of elliptic operators. The $(\rho,\delta)$-case}\label{rhodeltasection}



Now we characterise the membership of elliptic pseudo-differential operators to the Schatten classes on $L^2(G).$
\begin{theorem}\label{Schatten:properties:}  Let $m\in \mathbb{R},$ $r>0, $ and let $0\leq \delta<\rho\leq 1.$ Consider an elliptic pseudo-differential operator $A\in \Psi^m_{\rho,\delta}(G\times \widehat{G})$. The following conditions are equivalent:
\begin{itemize}
    \item[(1)] $m\in (-\infty,0)$ and  $A$ belongs to the Schatten class of order $r>0:$ $A\in \mathscr{S}_r(L^2(G));$
    \item[(2)] The Bessel potential of order $m$ belongs to the Schatten class of order $r>0:$ $B_{m}:=(1+\mathcal{L}_G)^{\frac{m}{2}}\in \mathscr{S}_r(L^2(G)). $
    \item[(3)] The following summability condition holds
    \begin{equation}
        \sum_{[\xi]\in \widehat{G}}d_\xi\smallint\limits_G\|\sigma_{|A|^{\frac{r}{2}}}(x,[\xi])\|_{\textnormal{HS} }^2\textnormal{d}x<\infty;
    \end{equation}
    \item[(4)] The following summability condition holds
    \begin{equation}
        \smallint\limits_G\sum_{[\xi]\in \widehat{G}}d_\xi\|\sigma_{A}(x,[\xi])\|_{\mathscr{S}_r }^r\textnormal{d}x<\infty;
    \end{equation}
    \item[(5)] $m<-n/r.$
\end{itemize}
\end{theorem}
\begin{proof}We start with the first equivalence $(1)\Longleftrightarrow(2):$


    \begin{itemize}
    \item $(1)\Longrightarrow (2).$ Assume that $A\in \mathscr{S}_r(L^2(G)).$  Let  $B_{s}:=(1+\mathcal{L}_G)^{\frac{s}{2}}\in \mathscr{S}_r(L^2(G))$ be the Bessel potential of order $s\in \mathbb{R}.$ The global functional calculus for elliptic operators  on compact Lie groups (see \cite{RuzhanskyWirth2014}) implies that
    \begin{equation}
        (1+|A|^{\frac{1}{|m|}})^{ \pm {m}}\in \Psi^{ \pm m}_{\rho,\delta}(G\times \widehat{G}).
    \end{equation} In view of the composition properties of the pseudo-differential calculus we have that $$  B_{m}(1+|A|^{\frac{1}{|m|}})^{ -m }\in \Psi^{0}_{\rho,\delta}(G\times \widehat{G}).$$ The Calder\'on-Vaillancourt theorem (see Theorem 5.2 in \cite{RuzhanskyWirth2014}) implies that $B_{m}(1+|A|^{\frac{1}{|m|}})^{ -m }$ is a bounded operator on $L^2(G).$ Since $\mathscr{S}_{r}(L^2(G))$ is an ideal of operators on the algebra $\mathscr{B}(L^2(G)),$ we have that
    \begin{equation}
        B_{m}= B_{m}(1+|A|^{\frac{1}{|m|}})^{ -m }(1+|A|^{\frac{1}{|m|}})^{ m },
    \end{equation}belongs to the Schatten class $\mathscr{S}_r(L^2(G))$