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Mat. Sb., 2020, Volume 211, Number 9, Pages 119–152 (Mi msb9336)  

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

Abstract: 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.

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

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).


DOI: https://doi.org/10.4213/sm9336

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English version:
Sbornik: Mathematics, 2020, 211:9, 1323–1353

Bibliographic databases:

UDC: 517.982.22+517.983.24+519.248.3
MSC: 47A30, 47B02, 46B28
Received: 10.10.2019 and 05.04.2020

Citation: 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
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\pages 119--152
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\pages 1323--1353
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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