  We consider a family of norms (called operator E-norms) on the algebra $B(H)$
of all bounded operators on a separable Hilbert space $H$ induced by a positive
densely defined operator $G$ on $H$. Each norm of this family produces the same
topology on $B(H)$ depending on $G$. By choosing different generating operator
$G$ one can obtain operator E-norms producing different topologies, in
particular, the strong operator topology on bounded subsets of $B(H)$. We
obtain a generalised version of the Kretschmann-Schlingemann-Werner theorem,
which shows continuity of the Stinespring representation of CP linear maps
w.r.t. the energy-constrained $cb$-norm (diamond norm) on the set of CP linear
maps and the operator E-norm on the set of Stinespring operators.
  The operator E-norms induced by a positive operator $G$ are well defined for
linear operators relatively bounded w.r.t. the operator $\sqrt{G}$ and the
linear space of such operators equipped with any of these norms is a Banach
space. 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 bounds and continuity bounds for some functions
depending on $\sqrt{G}$-bounded operators used in applications.
