Operator $E$-norms and their applications

In [1], a family of equivalent norms (called operator $E$-norms) on the algebra $\mathfrak{B}(\mathscr{H})$ of all bounded operators on a separable Hilbert space $\mathscr{H}$ induced by a positive densely defined operator $G$ on $\mathscr{H}$ was introduced. Depending on the operator $G$, these norms generate different topologies, in particular, the strong operator topology on bounded subsets of $\mathfrak{B}(\mathscr{H})$. It was shown that operator $E$-norms are naturally defined on the set of all linear operators relatively bounded with respect to the square root of $G$ and transform this set into a Banach space.
A generalized version of the Kretschmann-Schlingemann-Werner theorem on the continuity of the Stinespring representation of quantum channels with respect to the norm of complete boundedness with energy constraint on the set of channels and the operator $E$-norm on the set of Stinespring operators was proved [2].
In recent works [3], [4] it was shown that the operator $E$-norms are an effective tool for analyzing unbounded operators and quadratic forms on Hilbert space, and interesting applications of these norms were found in the theory of quantum dynamical semigroups.