When are the products of normal operators normal?
Author
Gheondea, A.
Date
2009Source 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
ArticleItem 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 operatorDecomposable operator
Direct integral hilbert space
Normal operator
Product
Singular numbers