Basicity of weighted shift operators on locally convex spaces (Q2493341)

\textit{N.~K.\ Nikol'skij} [Изв.\ Акад.\ Наук СССР, Сер.\ Мат. 32, 1123--1137 (1968; Zbl 0182.17302); English translation in: Math.\ USSR, Izv.\ 2, 1077--1089 (1968; Zbl 0184.35102)] introduced the notion of a basis operator and found criteria for a weighted shift operator to be a basis operator on the space \(\ell^p\), \(1\leq p<\infty\), or \(c_0\). The authors extend those results to the Köthe echelon spaces \(\lambda^p(A)\), \(1\leq p<\infty\), and prove separately necessary and sufficient conditions for a weighted shift operator to be a basis operator. For some particular cases (the space of analytic functions in the open unit disk and in the complex plane; the weighted spaces \(\ell^p\), \(1\leq p<\infty\), and the weighted space \(c_0\)), the results are stated explicitly.