then the set of states $\rho$ in $\S(\H)$ satisfying the inequality ${\mathrm{Tr}}{G}\rho\leq E$ is compact for any $E\geq\inf_{\|\varphi\|=1}\langle\varphi|{G}|\varphi\rangle$.*

We will also use the following simple lemma.

\[WL\] [@W-CB] *If $f$ is a concave nonnegative function on $[0,+\infty)$ then for any positive $x< y$ and any $z\geq0$ the inequality $\,xf(z/x)\leq yf(z/y)\,$ holds.*

Operator *E*-norms on $\B(\H)$
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Let ${G}$ be a positive semidefinite operator on $\H$ with a dense domain $\mathcal{D}({G})$ such that $$\label{H-cond}
\inf\left\{{\hspace{1pt}}\|G\varphi\|\,|\,\varphi\in\mathcal{D}({G}),\|\varphi\|=1{\hspace{1pt}}\right\}=0.$$ We will assume that for any positive operator $\rho$ in $\T(\H)$ the value of ${\mathrm{Tr}}{G}\rho$ (finite or infinite) is defined as $\sup_n{\mathrm{Tr}}P_nG\rho$, where $P_n$ is the spectral projector of $G$ corresponding to the interval $[0,n]$.

For given $E>0$ consider the function on $\B(\H)$ defined as $$\label{ec-on-b}
 \|A\|^G_E\doteq \sup_{\varphi\in\H_1, \langle\varphi|G|\varphi\rangle\leq E}\|A\varphi\|,$$ where $\H_1$ is the unit sphere in $\H$ and it is assumed that $\,\langle\varphi|G|\varphi\rangle=\|\sqrt{G}\varphi\|^2\,$ if $\varphi$ lies in $\D(\sqrt{G})$ and $\,\langle\varphi|G|\varphi\rangle=+\infty\,$ otherwise. This function can be also defined as $$\label{ec-on}
 \|A\|^{G}_E\doteq \sup_{\substack{\rho\in\mathfrak{S}(\mathcal{H}):
{\mathrm{Tr}}{G}\rho\leq E}}\sqrt{{\mathrm{Tr}}A\rho A^*},$$ where the supremum is over all states $\rho$ in $\S(\H)$ satisfying the inequality ${\mathrm{Tr}}{G}\rho\leq E$.[^4]

The coincidence of the r.h.s. of (\[ec-on-b\]) and (\[ec-on\]) for any $A\in\B(\H)$ is shown in .

It is easy to see that the function $A\mapsto\|A\|^{G}_E$ is 