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Lifting unconditionally converging series and semigroups of operators

Published online by Cambridge University Press:  17 April 2009

Manuel González  and
Antonio Martínez-Abejón
Manuel González
Affiliation:
Departamento de MatemáticasFacultad de CienciasUniversidad de Cantabria39071 Santander, Spain
Antonio Martínez-Abejón
Affiliation:
Departamento de MatemáticasFacultead de CienciasUniversidad de Oviedo33007 Oviedo, Spain
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Abstract

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We introduce and study two semigroups of operators u+ and u_, defined in terms of unconditionally converging series. We prove a lifting result for unconditionally converging series that allows us to show examples of operators in u+. We obtain perturbative characterisations for these semigroups and, as a consequence, we derive characterisations for some classes of Banach spaces in terms of the semigroups. If u+(X, Y) is non-empty and every copy of c0 in Y is complemented, then the same is true in X. We solve the perturbation class problem for the semigroup u_, and we show that a Banach space X contains no copies of ℓ∞ if and only if for every equivalent norm |·| on X, the semiembeddings of (X, |·|) belong to u+.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 135 - 146
Copyright
Copyright © Australian Mathematical Society 1998

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