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We study the Daugavet property in tensor products of Banach spaces. We show that
has the Daugavet property when
are purely non-atomic measures. Also, we show that
has the Daugavet property provided
-preduals with the Daugavet property, in particular, spaces of continuous functions with this property. With the same techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.
We study the stability of disjointness preservers on Banach lattices. In many cases, we
prove that an “almost disjointness preserving” operator is well approximable by a disjointness preserving
one. However, this approximation is not always possible, as our examples show.
We characterize, in the context of rearrangement invariant spaces, the optimal range space for a class of monotone operators related to the Hardy operator. The connection between the optimal range and the optimal domain for these operators is carefully analysed.
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