*For any $A\in\B(\H)$ and $E>0$ the following expressions hold $$\label{ec-on+}
 \|A\|^{G}_E=\sup_{\substack{\varphi\in\H_{\leq 1}:\langle\varphi|G|\varphi\rangle\leq E}}\|A\varphi\|=\sup_{\substack{\rho\in\T_{+,1}(\H):
{\mathrm{Tr}}{G}\rho\leq E}}\sqrt{{\mathrm{Tr}}A\rho A^*},\qquad \forall A\in \B(\H),$$ i.e. the suprema in definitions (\[ec-on-b\]) and (\[ec-on\]) can be taken, respectively, over all vectors in $\H_{\leq 1}$ satisfying the condition $\,\langle\varphi|G|\varphi\rangle\leq E$ and over all operators in $\T_{+,1}(\H)$ satisfying the condition $\,{\mathrm{Tr}}{G}\rho\leq E$.*

B\) *If the operator $G$ is unbounded then for any $A\in\B(\H)$ and $E>0$ the conditions $\,\langle\varphi|G|\varphi\rangle\leq E$ in (\[ec-on-b\]) and $\,{\mathrm{Tr}}{G}\rho\leq E$ in (\[ec-on\]) and can be replaced, respectively, by the conditions $\,\langle\varphi|G|\varphi\rangle=E$ and $\,{\mathrm{Tr}}G\rho=E$.*

C\) *If $G$ is a discrete unbounded operator (Def.\[D-H\]) then for any $A\in\B(\H)$ and $E>0$ the suprema in (\[ec-on-b\]) and (\[ec-on\]) are attainable. Moreover, if $\|A\|^{G}_{E}<\|A\|$ then the suprema in (\[ec-on-b\]) and (\[ec-on\]) are attained, respectively, at unit vector $\varphi_0$ in $\H$ such that $\,\langle\varphi_0|G|\varphi_0\rangle=E$ and at a state $\rho_0$ in $\S(\H)$ such that $\,{\mathrm{Tr}}G\rho_0=E$.*

\[en-def-r\] It is easy to see that assertion A of Proposition \[en-def\] is not valid if the operator $G$ doesn’t satisfy condition (\[H-cond\]).

*Proof of Proposition \[en-def\].* A) It suffices to show that the last expression in (\[ec-on+\]) does not exceed $\|A\|^{G}_E$. Let $\rho$ be an operator in $\T_{+,1}(\H)$ such that $\,{\mathrm{Tr}}{G}\rho\leq E$ and $r={\mathrm{Tr}}\rho$. Then $\hat{\rho}\doteq 