Existence of unconditional bases in spaces of polynomials and holomorphic functions. (Q2785228)

The authors continue their study of whether spaces of polynomials or holomorphic functions on a locally convex space have an unconditional property [J. Funct. Anal. 181, No. 1, 119--145 (2001; Zbl 0986.46031)]. Let \(E\) be a Montel Köthe echelon or coechelon space of order \(p\), \(1 < p \leq \infty.\) The main result is that \(E\) is nuclear if and only if for some (equivalently, for every) \(m \geq 2,\) the space of \(m\)-homogeneous polynomials on \(E,~{\mathcal P}(^mE),\) endowed with the compact open topology \(\tau_0\) has an unconditional basis, and which is in turn equivalent to the requirement that the space of holomorphic functions on \(E,~{\mathcal H}(E),\) endowed with the bornological topology associated with \(\tau_0\) has an unconditional basis. The main idea, of interest in itself, is the extension of the Gordon-Lewis property [\textit{Y. Gordon} and \textit{D. R. Lewis}, Acta Math. 133, 27--48 (1974; Zbl 0291.47017)]: Every countably barrelled locally convex space with an unconditional basis has the Gordon-Lewis property.