Operator -norms and their use

Abstract

We consider a family of equivalent norms (called operator -norms) on the algebra of all bounded operators on a separable Hilbert space induced by a positive densely defined operator on . By choosing different generating operators we can obtain the operator -norms producing different topologies, in particular, the strong operator topology on bounded subsets of . We obtain a generalised version of the Kretschmann-Schlingemann- Werner theorem, which shows that the Stinespring representation of completely positive linear maps is continuous with respect to the energy-constrained norm of complete boundedness on the set of completely positive linear maps and the operator -norm on the set of Stinespring operators. The operator -norms induced by a positive operator are well defined for linear operators relatively bounded with respect to the operator , and the linear space of such operators equipped with any of these norms is a Banach space. We obtain explicit relations between operator -norms and the standard characteristics of -bounded operators. Operator -norms allow us to obtain simple upper bounds and continuity bounds for some functions depending on -bounded operators used in applications. Bibliography: 29 titles.