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      When are the products of normal operators normal?

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      Author
      Gheondea, A.
      Date
      2009
      Source Title
      Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie
      Print ISSN
      1220-3874
      Volume
      52
      Issue
      2
      Pages
      129 - 150
      Language
      English
      Type
      Article
      Item Usage Stats
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      Abstract
      Given two normal operators A and B on a Hilbert space it is known that, in general, AB is not normal. The question on characterizing those pairs of normal operators for which their products are normal has been solved for finite dimensional spaces by F.R. Gantmaher and M. G. Krein in 1930, and for compact normal operators by N.A. Wiegmann in 1949. Actually, in the afore mentioned cases, the normality of AB is equivalent with that of BA, and a more general result of F. Kittaneh implies that it is sufficient that AB be normal and compact to obtain that BA is the same. On the other hand, I. Kaplansky had shown that it may be possible that AB is normal while BA is not. When no compactness assumption is made, but both of AB and BA are supposed to be normal, the Gantmaher-KreinWiegmann Theorem can be extended by means of the spectral theory of normal operators in the von Neumann's direct integral representation.
      Keywords
      Compact operator
      Decomposable operator
      Direct integral hilbert space
      Normal operator
      Product
      Singular numbers
      Permalink
      http://hdl.handle.net/11693/22522
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