On sums of narrow operators on Köthe function spaces

We find necessary and sufficient conditions on a Köthe Banach space E on [0,1] and a Banach space X under which a sum of two narrow operators from E to X is narrow. Using this condition, we prove that, given a Köthe Banach space E on [0,1], there exist a Banach space X and narrow operators T1,T2:E→X...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2013-08, Vol.404 (2), p.554-561
Main Authors: Mykhaylyuk, V.V., Popov, M.M.
Format: Article
Language:English
Subjects:
Köthe function space
Narrow operator
The Banach space [formula omitted]
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Summary:We find necessary and sufficient conditions on a Köthe Banach space E on [0,1] and a Banach space X under which a sum of two narrow operators from E to X is narrow. Using this condition, we prove that, given a Köthe Banach space E on [0,1], there exist a Banach space X and narrow operators T1,T2:E→X with non-narrow sum T=T1+T2. In particular, this answers in the negative, a question of V.M. Kadets and the second named author, of whether for every Banach space X a sum of two narrow operators from L1 to X must be narrow. Another result asserts that for every 1
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2013.03.008