Il Nuovo Cimento B (1965-1970)

, Volume 65, Issue 2, pp 229–238 | Cite as

On the commutant of irreducible sets of semi-linear operators

  • R. Ascoli
  • G. Teppati
Article
First Online:
Received:

Summary

We formulate some propositions concerning the commutant of an irreducible set of semi-linear operators of a vector space over an arbitrary field. Afterwards we apply such propositions to the case of a finite-dimensional vector space over the real, the complex or the quaternion fields. We present an exemplification of the cases that may occur. As a further application we give a simple result about the set of the Hermitian operators that commute with irreducible sets of semi-linear operators of a finite-dimensional real, complex, or quaternion scalar-product space.

Keywords

Vector Space Division Algebra Complex Field Frobenius Theorem Quaternion Case 
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О коммутации неприводимых систем кваэилинейных операторов

Реэюме

Мы формулируем некоторые утверждения, касаюшиеся коммутации неприводимой системы кваэилинейных операторов векторного пространства в проиэвольном поле. После зтого мы применяем зти утверждения к случаю конечномерного векторного пространства в вешественном, комплексном или кватернионном полях. Мы предлагаем пояснение случаев, которые могут встретиться. Как дальнейщее применение, мы приводим простой реэультат относительно системы зрмитовских операторов, которые коммутируют с неприводимыми системами кваэилинейных операторов конечно-мерных вешественных, комплексных или кватернионных скалярных проиэведений пространств.

Riassunto

Si formulano alcune proposizioni riguardanti il commutante di un insieme irriducibile di operatori semilineari di uno spazio vettoriale sopra un corpo arbitrario. In seguito si applicano tali proposizioni al caso di uno spazio a dimensione finita sul corpo dei reali, dei complessi o dei quaternioni. Si presenta una esemplificazione dei casi possibili. Come ulteriore applicazione si fornisce un risultato semplice riguardante l’insieme di operatori hermitiani che commutano con insiemi irriducibili di operatori semilineari di uno spazio vettoriale a dimensione finita con prodotto scalare sul corpo dei reali, dei complessi o dei quaternioni.

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References

  1. (1).
    Sometimes called Burnside theorem; seeW. Burnside:Proc. London Math. Soc.,3, 430 (1905);I. Schur:Sitzung. Kön. Preuss. Akad. Wissen., 406 (1905).Google Scholar
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    We quoted here two different proofs of this fact that can be found in:F. Riész andB. Sz. Nagy:Leçons d’analyse functionelle (Budapest, 1954), p. 92;H. Halmos:Introduction to Hilbert Space (New York, 1957), p. 55. All these proofs require topological considerations: but this is by no means surprising, because such proofs do not make use of the fundamental theorem of algebra. For the quaternion case see also ref. (2,3), and in this latter especially the theorem on p. 139.Google Scholar

Copyright information

© Società Italiana di Fisica 1970

Authors and Affiliations

  • R. Ascoli
    • 1
    • 2
  • G. Teppati
    • 3
  1. 1.Istituto di Fisica dell’UniversitàPalermo
  2. 2.Istituto Nazionale di Fisica NucleareGruppo di PalermoSezione Siciliana
  3. 3.Istituto Nazionale di Fisica NucleareSezione di Torino

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