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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Daugavet's equation and operators on $ L\sp 1(\mu)$


Author: James R. Holub
Journal: Proc. Amer. Math. Soc. 100 (1987), 295-300
MSC: Primary 47B38; Secondary 47B05
MathSciNet review: 884469
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Abstract: Generalizing a result of Babenko and Pichugov, it is shown that if $ T$ is a weakly compact operator on $ {L^1}(\mu )$, where $ \mu $ is a $ \sigma $-finite nonatomic measure, then $ \left\Vert {I + T} \right\Vert = 1 + \left\Vert T \right\Vert$. A characterization of all operators $ T$ on $ {L^1}(\mu )$ having this property is also given.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0884469-8
PII: S 0002-9939(1987)0884469-8
Keywords: Weakly compact operator, Daugavet's equation, operator norm equalities
Article copyright: © Copyright 1987 American Mathematical Society