Explicit characterizations of orthogonality in \(\ell_S^p(C)\)
A family \((x_\alpha)_{\alpha\in A}\) of vectors in a Banach space \(E\) is said to be orthogonal if \[ \Big\|\sum_{\alpha\in A} c_\alpha x_\alpha\Big\|= \Big\|\sum_{\alpha\in A} |c_\alpha|x_\alpha\Big\| \] whenever \(\sum_{\alpha\in A} c_\alpha x_\alpha\) converges in \(E\). For a basic sequence, that means that it is \(1\)-unconditional. The author studies orthogonality in the complex spaces \(\ell_p(I)\). When \(p\) is not an even integer, he asks whether two elements \(a\) and \(b\) of \(\ell_p(I)\) have disjoint supports when they are orthogonal and gives some partial answers. When \(p\) is an even integer, \(a=(a_i)_{i\in I}\) and \(b=(b_i)_{i\in I}\) can be orthogonal without having disjoint supports and if \(a_i\neq 0\), \(b_i\not= 0\) for every \(i\in I\), the author shows that \(I\) must have at least \(p/2+1\) elements.
scientific article; zbMATH DE number 1836767