definitions and basic properties of the operator *E*-norms. We obtain explicit relations between the operator *E*-norms and the equivalent norm on $\B(\H)$ also induced by a positive operator $G$ (which is commonly used in analysis of $\sqrt{G}$-bounded operators).

The operator *E*-norms make it possible to obtain a generalization the Kretschmann-Schlingemann-Werner theorem. The original version of this theorem presented in shows continuity of the Stinespring representation of a completely positive (CP) linear map with respect to the norm of complete boundedness ($cb$-norm in what follows)[^3] on the set of CP maps and the operator norm on the set of Stinespring operators. Our aim was to obtain a version of this theorem for other (weaker) topologies on the sets of CP maps and corresponding Stinespring operators, in particular, for the strong convergence topology on the set of CP maps and the strong operator topology on the set of Stinespring operators. By using the operator *E*-norms one can upgrade the proof of the Kretschmann-Schlingemann-Werner theorem without essential changes. The generalised version of this theorem and its corollaries are presented in Section 4.

In Section 5 the operator *E*-norms induced by a positive operator $G$ are extended to linear operators relatively bounded w.r.t. the 