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On the Spectrum of Products of Operator Ideals

✍ Scribed by Hermann König

Publisher
John Wiley and Sons
Year
1979
Tongue
English
Weight
511 KB
Volume
93
Category
Article
ISSN
0025-584X

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✦ Synopsis

On the Spectrum of Products of Operator Ideals

By HERMANN KONIG *) of Bonn (Eingegangen am 10. 3.1978)

The eigenvalues of absolutely p-summing ( 2 ~p -=-) and type lp operators (0 <p-=-) in BANACH spaces are p-th power summable. I n this note, we deal with operators which can be factored as products of absolutely p-summing and type l4 operators and derive the summability properties of their eigenvalues. They belong to the LORENTZ sequence space lr,p with l / r = l / p + l / q . Theorems of this type generalize an earlier result of the author. Nevertheless the proof given here is easier than in [5]. As an application, we characterize product ideals on HILBERT spaces like IT20X,(H). This will yield that the above result is in general optimal. A similar method shows that type El maps do not factor as the product of two absolutely %summing operators.

Another proposition which has applications to the distribution of the eigenvalues of product ideals on L,-spaces is derived : We discuss the spectral properties of the maximal extension of the SUHATTEN classes s,(H) to an operator ideal.

*) Supported by the Sonderforschungsbereich 72 at the University of Bonn.

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