operator $\sqrt{G}$. We prove that the linear space of such operators equipped with any of these norms is a Banach space. Its subspace consisting of all operators with zero $\sqrt{G}$-bound is the completion of $\B(\H)$ w.r.t. any of the operator *E*-norms. We obtain explicit relations between the operator*E*-norms and the standard characteristics of $\sqrt{G}$-bounded operators.

The operator *E*-norms allow to obtain simple upper estimates and continuity bounds for some functions depending on $\sqrt{G}$-bounded operators used in applications.

As a basic example we consider the operators associated with the Heisenberg Commutation Relation.

Preliminaries
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Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space, $\mathfrak{B}(\mathcal{H})$ – the algebra of all bounded operators on $\mathcal{H}$ with the operator norm $\|\!\cdot\!\|$ and $\mathfrak{T}(\mathcal{H})$ – the Banach space of all trace-class operators on $\mathcal{H}$ with the trace norm $\|\!\cdot\!\|_1$ (the Schatten class of order 1) . Let $\mathfrak{T}_{+}(\mathcal{H})$ be the cone of positive operators in $\mathfrak{T}(\mathcal{H})$. Trace-class operators will be usually denoted by the Greek letters $\rho $, $\sigma $, $\omega $, ... The closed convex subsets $$\T_{+,1}(\H)=\{\rho\in \T_+(\H)\,|\, {\mathrm{Tr}}\rho\leq 1\}\quad \textrm{and} \quad \S(\H)=\{\rho\in \T_+(\H)\,|\, {\mathrm{Tr}}\rho=1\}$$ of the cone $\T_+(\H)$ are complete separable metric spaces with the metric defined by the trace norm. Operators in $\S(\H)$ 