a norm on $\B(\H)$. Definition (\[ec-on-b\]) shows the sense of the norm $\|\cdot\|^G_E$ (as a constrained version of the operator norm $\|\cdot\|$) while definition (\[ec-on\]) is more convenient for studying its analytical properties. In particular, by using definition (\[ec-on\]) the following proposition is proved in [@CSR].[^5]

\[ec-on-p1\] *For any operator $\,A\in\B(\H)$ the following properties hold:*

a)  *$\|A\|^{G}_E$ tends to $\|A\|$ as $E\rightarrow+\infty$;*

b)  *the function $E\mapsto\left[\|A\|^{G}_E\right]^p$ is concave and nondecreasing on $\,\mathbb{R}_+$ for $p\in(0,2]$;*

c)  *$\|A\varphi\|\leq K_{\varphi}\|A\|^{G}_E$ for any unit vector $\varphi$ in $\D(\sqrt{G})$, where $K_{\varphi}=\max\{1, \|\sqrt{G}\varphi\|/\sqrt{E}\}$.*

We will call the norms $\|\!\cdot\!\|^{G}_E$ the *operator E-norms* on $\B(\H)$.[^6] Property b) in Proposition \[ec-on-p1\] shows that $$\label{E-n-eq}
\|A\|^{G}_{E_1}\leq \|A\|^{G}_{E_2}\leq \sqrt{E_2/E_1}\|A\|^{G}_{E_1}\quad\textrm{ for any } E_2>E_1>0.$$ Hence for given operator ${G}$ all the norms $\|\!\cdot\!\|^{G}_{E}$, $E>0$, are equivalent on $\B(\H)$.

\[en-gen\] The definition of the operator *E*-norm is obviously generalized to operators between different Hilbert spaces $\H$ and $\K$. It is easy to see that all the above and below results concerning properties of the operator *E*-norms remain valid (with obvious modifications) for this generalization. $\square$

Since the set $\D(\sqrt{G})$ is dense in $\H$, property c) in Proposition \[ec-on-p1\] shows that the topology generated by any of the norms $\|\!\cdot\!\|^{G}_{E}$ on bounded subsets 