space a Banach space. Moreover, the functions $E\mapsto\|A\|^G_E$ and $E\mapsto{\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^G_E$ are completely determined by each other (Remark \[comp-d-r+\] in Section 5). The main advantages of the operator *E*-norm $\|\cdot\|^{G}_{E}$ in comparison with the norm ${\|\hspace{-1pt}|}\cdot{\|\hspace{-1pt}|}^{G}_{E}$ are the following:

-   the concavity of the function $E\mapsto\left[\|A\|^{G}_{E}\right]^p$ for any $p\in(0,2]$;[^8]

-   the appearance of the norm $\|\cdot\|^{G}_{E}$ in the generalized Kretschmann-Schlingemann-Werner theorem (Section 4);

-   the simple estimation of $\|\Phi(A)\|^{G}_E$ via $\|A\|^{G}_E$, where $\Phi:\B(\H)\rightarrow\B(\H)$ is any 2-positive linear map satisfying the conditions of Proposition \[bp-en-0\]E (Section 3.2).

\[nonconc\] To show that the function $E\mapsto\left[{\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E}\right]^p$ is not concave in general for any $p\in(0,2]$ it suffices to consider two-dimensional Hilbert space $\H=\mathbb{C}^2$ and the operators $$G=\left[\begin{array}{ll}
        1 & 0\\
        0 & 0
        \end{array}\right]\quad \textrm{and}   \quad A=\left[\begin{array}{ll}
        \sqrt{2} & 0\\
        0 & 1
        \end{array}\right].$$ It is easy to see that $\,{\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E}=1\,$ if $\,E\in(0,1]\,$ and $\,{\|\hspace{-1pt}|}A{\|\hspace{-1pt}|}^{G}_{E}=\sqrt{2E/(E+1)}\,$ if $\,E>1$.

Basic properties of the operator *E*-norms
------------------------------------------

In the following proposition we collect properties of the operator *E*-norms used below.

\[bp-en-0\] *Let ${G}$ be a positive densely defined operator on a Hilbert space $\H$ satisfying condition (\[H-cond\]) and $E>0$.*

A\) *$\|A\|^{G}_E=\||A|\|^{G}_E\leq \sqrt{\|A^*A\|^{G}_E}$ for all $A\in\B(\H)$ but $\,\|A^*\|^{G}_E\neq\|A\|^{G}_E$ in general;*

B\) *For arbitrary operators $A$ and $B$ in $\B(\H)$ 