WORTH property, García-Falset coefficient and Opial property of infinite sums (Q2791941)

There are many geometric properties of Banach spaces that appear in articles considering fixed points of nonexpansive mappings. In the article under review, the author considers the stability of some of these properties under absolute sums of Banach spaces.NEWLINENEWLINEIf \(E\) is a subspace of the space of real-valued functions on an index set \(I\) containing all functions of finite support is equipped with an absolute normalized norm and \((X_i)_{i\in I}\) is a family of Banach spaces, \(\left( \bigoplus_{i\in I} X_i\right)_E = \left\{ (x_i)_{i\in I}:( \| x_i\| )_{i\in I}\in E\right\}\) endowed with the norm \(\| (x_i) \| = \| (\| x_i\| ) \| _E\) is a Banach space. The author proves, under a mild condition on \(E\), that \(\left( \bigoplus_{i\in I} X_i\right)_E\) has WORTH (a weak orthogonality condition related to the weak fixed point property) if and only if each \(X_i\) has WORTH. The author also gives conditions which ensure that the García-Falset coefficient of \(\left( \bigoplus_{i\in I} X_i\right)_E\) is less than \(2\), a condition strong enough to imply the weak fixed point property. A variety of Opial conditions are also shown to be stable under \(\ell^p\)-sums. For example, for \(1\leq p<\infty\), the \(\ell^p\)-sum of a family of Banach spaces with the Opial condition or the nonstrict Opial condition has the corresponding property. Finally, the author proves that, even though \(L^p[0,1]\), \(p\neq 2\), fails to satisfy an Opial condition, if \(X\) is a Banach space with an Opial condition, the Lebesgue-Bochner spaces \(L^p(\mu, X)\) exhibit some Opial-like properties.