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Norm‐attaining operators between Marcinkiewicz and Lorentz spaces
Author(s) -
Acosta María D.,
Kamińska Anna
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn030
Subject(s) - mathematics , lorentz space , norm (philosophy) , operator norm , bounded operator , pure mathematics , lorentz transformation , interpolation space , bounded function , banach space , finite rank operator , compact operator , operator (biology) , approximation property , operator theory , mathematical analysis , functional analysis , extension (predicate logic) , law , classical mechanics , political science , physics , repressor , chemistry , computer science , biochemistry , transcription factor , programming language , gene
Bishop and Phelps proved that the set of norm‐attaining functionals on any Banach space is dense in the topological dual. After that, the study of the same kind of problems for operators was initiated by Lindenstrauss, and several general positive results were proved. It was then consistently continued for different classes of spaces including L 1 (μ) or C ( K ). Here a similar problem is studied in the context of classical interpolation Marcinkiewicz and Lorentz spaces,M W 0and Λ 1, v , in both the real and the complex cases. We show that if wv ∈ L 1 then the identity operator between these spaces is bounded, but it is not possible to approximate it by norm‐attaining operators. We also prove that every compact operator fromM W 0to Λ 1, v can be approximated by finite‐rank norm‐attaining operators.