Hence the supremum in (\[ec-on\]) is attained at some state $\rho_0$. The condition $\|A\|^{G}_E<\|A\|$ implies, by Proposition \[ec-on-p1\], that $\|A\|^{G}_{E'}<\|A\|^{G}_E$ for any $E'<E$. This shows that ${\mathrm{Tr}}G\rho_0=E$. By using decomposition (\[p-s-d\]) of $\rho_0$ and the arguments below it is easy to show existence of a unit vector $\varphi_0$ in $\H$ such that $\,\langle\varphi_0|G|\varphi_0\rangle=E$ at which the supremum in (\[ec-on-b\]) is attained. $\square$

Consider the following norm on $\B(\H)$ depending on a positive parameter $E$: $$\label{eq-norms-2}
{\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E}=\sup\left\{\|A\varphi\|\left|\, \varphi\in \D(\sqrt{G}),\, \|\varphi\|^2+\langle\varphi|G|\varphi\rangle/E\leq 1\right.\right\}.$$ This norm naturally appears in analysis of relatively bounded operators (see Section 5). We will obtain relations between the norms $\|\cdot\|^{G}_{E}$ and ${\|\hspace{-1pt}|}\cdot{\|\hspace{-1pt}|}^{G}_{E}$ assuming that $G$ is an arbitrary positive operator satisfying condition (\[H-cond\]).

By using definitions (\[ec-on-b\]) and (\[eq-norms-2\]) it is easy to obtain $$\label{one}
\sqrt{1/2}\|A\|^{G}_{E}\leq {\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E}\leq\|A\|^{G}_{E}$$ for any $E>0$. This inequality and inequality (\[E-n-eq\]) imply that $$\label{E-n-eq+}
{\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E_1}\leq {\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E_2}\leq \sqrt{2E_2/E_1}{\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E_1}\quad\textrm{ for any } E_2>E_1>0$$ and any $A\in\B(\H)$. The above inequalities show that all the norms in the families $\{\|\cdot\|^{G}_{E}\}_{E>0}$ and $\{{\|\hspace{-1pt}|}\cdot{\|\hspace{-1pt}|}^{G}_{E}\}_{E>0}$ are equivalent to each other.

In fact, all the norms $\|\cdot\|^{G}_{E}$ and ${\|\hspace{-1pt}|}\cdot{\|\hspace{-1pt}|}^{G}_{E}$ are equivalent on the space of all linear operators on $\H$ relatively bounded w.r.t. the operator $\sqrt{G}$, and each of these norms makes this 