linear operators on a separable Hilbert space $\H$, some its subalgebras and subspaces are basic objects in different fields of modern mathematics and mathematical physics . In particular, $\B(\H)$ appears as an algebra of observables in the theory of quantum systems while unital completely positive maps between such algebras called quantum channels play the role of dynamical maps in the Heisenberg picture [@H-SCI; @Watrous; @Wilde].

The variety of different topologies on $\B(\H)$, relations between them and their “physical” sense are well known for anybody who is interested in functional analysis, theory of operator algebras, mathematical and theoretical physics.

In this article we describe families of norms on $\B(\H)$ producing different topologies on $\B(\H)$, in particular, the strong operator topology on bounded subsets of $\B(\H)$. These norms depending on a positive densely defined operator ${G}$ and a positive parameter $E$ were introduced in [@CSR] for quantitative analysis of continuity of the Stinespring representation of a quantum channel with respect to the strong convergence of quantum channels and the strong operator convergence of Stinespring isometries.[^2]

Now we consider these norms (called the operator *E*-norms) in more general context (assuming that ${G}$ is an arbitrary positive operator). In Section 3 we consider equivalent 