---
abstract: |
    We consider a family of equivalent 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 $\H$. By choosing different generating operator $G$ one can obtain the 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 estimates and continuity bounds for some functions depending on $\sqrt{G}$-bounded operators used in applications.
author:
- 'M.E. Shirokov[^1]'
title: 'Operator *E*-norms and their use'
---

Introduction
============

The algebra $\B(\H)$ of all bounded 