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On operator ideals determined by sequences
Published online by Cambridge University Press: 17 April 2009
Abstract
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
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