arXiv subjects
Publications and source records attributed to Samir Hamad.
We will provide an example of a Banach space with the Daugavet property which does not possess the weak operator Daugavet property. We will also show that every separable Banach space with the Daugavet property contains a non-poor subspace isomorphic to $\ell^1$ while noticing that every separable subspace of $L^\infty[0,1]$ is poor.
We discuss an example of a non-complete normed space with the Daugavet property such that the norm is G\^ateaux differentiable at every nonzero point. In contrast, we note that the dual norm of a normed space with the Daugavet property is not G\^ateaux differentiable at any point. Furthermore, we show that quasilacunary M\"untz spaces form a natural class of subspaces of $C[0,1]$, isomorphic to $c_0$, for which the corresponding quotient spaces fail to have the Daugavet property. At the same time, the slice diameter two property is preserved under this construction.
We show that $W^{1,1}(\mathbb{R}^2)$ has the Daugavet property when endowed with the norm induced by the $L^1$-norm of the gradient, but fails to have the slice diameter two property when equipped with the usual Sobolev norm.