Operator $E$-norms and their use

M. E. Shirokov

Sbornik. Mathematics, Tome 211 (2020) no. 9, pp. 1323-1353

Résumé

We consider a family of equivalent norms (called operator $E$-norms) on the algebra $\mathfrak B(\mathscr H)$ of all bounded operators on a separable Hilbert space $\mathscr H$ induced by a positive densely defined operator $G$ on $\mathscr H$. By choosing different generating operators $G$ we can obtain the operator $E$-norms producing different topologies, in particular, the strong operator topology on bounded subsets of $\mathfrak B(\mathscr H)$. 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 $E$-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 with respect to 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 operator $E$-norms and the standard characteristics of $\sqrt G$-bounded operators. Operator $E$-norms allow us to obtain simple upper bounds and continuity bounds for some functions depending on $\sqrt G$-bounded operators used in applications. Bibliography: 29 titles.

Détail

Keywords: trace class operator, completely positive map, Stinespring representation, relatively bounded operator.
Mots-clés : Bures distance

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