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A Version of James' Theorem for Numerical Radius
Author(s) -
Acosta María D.,
Galán Manuel Ruiz
Publication year - 1999
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/s0024609398004585
Subject(s) - mathematics , converse , banach space , rank (graph theory) , bounded function , reflexivity , bounded operator , radius , operator (biology) , space (punctuation) , pure mathematics , mathematical analysis , discrete mathematics , combinatorics , geometry , computer security , computer science , social science , biochemistry , chemistry , linguistics , philosophy , repressor , sociology , transcription factor , gene
We prove that if all the rank‐one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank‐one operator not attaining the numerical radius. 1991 Mathematics Subject Classification 47A12, 46B10.