r^{-1}\rho$ is a state such that $\,{\mathrm{Tr}}{G}\hat{\rho}\leq E/r$. So, by using concavity of the function $E\rightarrow\left[\|A\|^{G}_E\right]^2$ and Lemma \[WL\] in Section 2 we obtain $${\mathrm{Tr}}A\rho A^*=r{\mathrm{Tr}}A\hat{\rho} A^*\leq r\left[\|A\|^{G}_{E/r}\right]^2\leq \left[\|A\|^{G}_{E}\right]^2.$$

B)[^7] Show first that the inequality ${\mathrm{Tr}}{G}\rho\leq E$ can be replaced by the equality ${\mathrm{Tr}}{G}\rho=E$ in (\[ec-on\]). Assume that there exist $E>0$ and $A\in\B(\H)$ such that $$\label{assum}
\sup_{\substack{\rho\in\mathfrak{S}(\mathcal{H}):
{\mathrm{Tr}}{G}\rho=E}}\sqrt{{\mathrm{Tr}}A\rho A^*}\leq \|A\|^{G}_E-\varepsilon$$ for some $\varepsilon>0$. Let $\rho_\varepsilon$ be a state such that $\sqrt{{\mathrm{Tr}}A\rho_\varepsilon A^*}>\|A\|^{G}_E-\varepsilon/2$ and ${\mathrm{Tr}}G\rho_\varepsilon<E$. For each natural $n>E$ there exist a state $\sigma_n$ and a number $p_n\in(0,1)$ such that ${\mathrm{Tr}}G\sigma_n\in(n,+\infty)$ and ${\mathrm{Tr}}G\varrho_n=E$, where $\varrho_n=(1-p_n)\rho_\varepsilon+p_n\sigma_n$. It is clear that $p_n\rightarrow 0$ as $n\rightarrow+\infty$. Hence ${\mathrm{Tr}}A\varrho_n A^*$ tends to ${\mathrm{Tr}}A\rho_\varepsilon A^*$ contradicting (\[assum\]).

For any $\varepsilon>0$ let $\rho_\varepsilon$ be a state such that ${\mathrm{Tr}}A\rho_\varepsilon A^*>[\|A\|^{G}_E-\varepsilon]^2$ and ${\mathrm{Tr}}G\rho_\varepsilon=E$. By Corollary 1 in there is a probability measure $\mu$ on $\S(\H)$ supported by pure states such that $$\label{p-s-d}
\rho_\varepsilon=\int \sigma\mu(d\sigma)\quad \textrm{and}\quad {\mathrm{Tr}}H\sigma=E\;\;\textrm{for}\;\;\mu\textrm{-almost all}\;\;\sigma.$$ Since that function $\sigma\mapsto{\mathrm{Tr}}A\sigma A^*$ is affine and continuous, we have $$\int {\mathrm{Tr}}A\sigma A^*\mu(d\sigma)={\mathrm{Tr}}A\rho_\varepsilon A^*>[\|A\|^{G}_E-\varepsilon]^2.$$ It shows existence of a pure state $\sigma_\varepsilon$ such that ${\mathrm{Tr}}A\sigma_\varepsilon A^*>[\|A\|^{G}_E-\varepsilon]^2$ and ${\mathrm{Tr}}H\sigma_\varepsilon=E$.

C\) In this case the set of states $\rho$ satisfying the condition ${\mathrm{Tr}}{G}\rho\leq E$ is compact by Lemma \[Comp\] in Section 2. 