Uniform \(L_p(w)\) spaces (Q1347362)

The subject of paper is inspired by \textit{N. J. Kalton}'s theorem [Isr. J. Math. 26, 126-136 (1977; Zbl 0348.47016)] that a non-zero continuous linear operator from \(L_p\), \(0<p<1\), into a topological vector space cannot be compact (even strictly singular), and by the Kalton-Shapiro example [\textit{N. J. Kalton} and \textit{J. H. Shapiro}, Isr. J. Math. 20, 282-291 (1975; Zbl 0305.46008)] of a trivial-dual space which admits compact operators. In 1981, Roberts (unpublished) introduced a natural class of trivial-dual spaces having compact operators-uniform \(L_p(w)\) spaces. The author of paper under review continues the investigations of these spaces. Uniform \(L_p(w)\) spaces are indexed by a scalar sequence \(c_n\). He proves that if \(c_n\uparrow \infty\), the corresponding \(L_p(w)\) admits compact operators, and there is no non-zero operator from \(L_p\) into \(L_p(w)\). He shows (answering a question of Roberts) that for any separable trivial-dual \(p\)-Banach space \(X\) there is an \(L_p(w)\) with \(L(X,L_p(w))=\{0\}\).