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On operator ideals determined by sequences

Published online by Cambridge University Press:  17 April 2009

Manuel González  and
Antonio Martinón
Manuel González
Affiliation:
Departamento de Matemáticas, Universidad de Cantabria, 39071 Santander, Spain
Antonio Martinón
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain
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Abstract

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We associate with an operator ideal 𝒜 (in the sense of Pietsch) a class of bounded sequences S𝒜 by using the 𝒜-variation of Astala. If 𝒜 and B are operator ideals, and we define (𝒜, B) as the class of operators which map a sequence of S𝒜 into a sequence of SB, we obtain the following:

Theorem. If Tn: X → Y is a sequence of operators and for every sequence (xn) ⊂ X in S𝒜 there exists p such that (Tpxn) belongs to SB, then Tm ∈ (𝒜, B) for some m.

The compact operators, weakly compact operators and some other operator ideals can be represented as (𝒜, B). Hence several results of Tacon and other authors are a consequence of this theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 2 , October 1991 , pp. 285 - 295
Copyright
Copyright © Australian Mathematical Society 1991

References

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