This article is cited in 1 scientific paper (total in 1 paper)
Operator $E$-norms and their use
M. E. Shirokov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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.
trace class operator, completely positive map, Stinespring representation, Bures distance, relatively bounded operator.
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Sbornik: Mathematics, 2020, 211:9, 1323–1353
MSC: 47A30, 47B02, 46B28
Received: 10.10.2019 and 05.04.2020
M. E. Shirokov, “Operator $E$-norms and their use”, Mat. Sb., 211:9 (2020), 119–152; Sb. Math., 211:9 (2020), 1323–1353
Citation in format AMSBIB
\paper Operator $E$-norms and their use
\jour Mat. Sb.
\jour Sb. Math.
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S. W. Weis, M. E. Shirokov, “Extreme points of the set of quantum states with bounded energy”, Russian Math. Surveys, 76:1 (2021), 190–192
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