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Denseness of numerical radius attaining operators. Renorming and embedding results. (English) Zbl 0725.47004

We prove that a strong geometrical condition (the unit ball is the balanced convex hull of a uniformly strongly exposed set), introduced by J. Lindenstrauss to deal with norm-attaining operators, is also sufficient for the denseness of numerical radius attaining operators. As a consequence, we show that every WCG Banach space can be equivalently renormed in such a way that the denseness of the numerical radius attaining operators is satisfied. We also obtain that every Banach space X is linearly isometric to a 1-complemented subspace of a Banach space Y with the same density character and such that the numerical radius attaining operators on Y are dense.

MSC:

47A12 Numerical range, numerical radius
46B10 Duality and reflexivity in normed linear and Banach spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
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