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Sbornik: Mathematics, 2020, Volume 211, Issue 9, Pages 1323–1353
DOI: https://doi.org/10.1070/SM9336
(Mi sm9336)
 

This article is cited in 3 scientific papers (total in 3 papers)

Operator $E$-norms and their use

M. E. Shirokov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
References:
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).
Received: 10.10.2019 and 05.04.2020
Bibliographic databases:
Document Type: Article
UDC: 517.982.22+517.983.24+519.248.3
MSC: 47A30, 47B02, 46B28
Language: English
Original paper language: Russian
Citation: M. E. Shirokov, “Operator $E$-norms and their use”, Sb. Math., 211:9 (2020), 1323–1353
Citation in format AMSBIB
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\by M.~E.~Shirokov
\paper Operator $E$-norms and their use
\jour Sb. Math.
\yr 2020
\vol 211
\issue 9
\pages 1323--1353
\mathnet{http://mi.mathnet.ru//eng/sm9336}
\crossref{https://doi.org/10.1070/SM9336}
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\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2020SbMat.211.1323S}
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\elib{https://elibrary.ru/item.asp?id=45098657}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85097218170}
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  • https://doi.org/10.1070/SM9336
  • https://www.mathnet.ru/eng/sm/v211/i9/p119
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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