The Fréchet space ces(p+), 1 < p < infty

Abstract

[EN] The Banach spaces ces(p), 1 < p < infinity, were intensively studied by G. Bennett and others. The largest solid Banach lattice in C-N which contains l(p) and which the Cesaro operator C : C-N -> C-N maps into l(P) is ces(p). For each 1 <= p < infinity, the (positive) operator C also maps the Frechet space l(p+) = boolean AND(q > p) l(q) into itself. It is shown that the largest solid Frechet lattice in C-N which contains l(p+) and which C maps into l(p+) is precisely ces(p+) := boolean AND(q > p) ces(q). Although the spaces l(p+) are well understood, it seems that the spaces ces(p+) have not been considered at all. A detailed study of the Frechet spaces ces(p+),1 <= p < infinity, is undertaken. They are very different to the Frechet spaces l(p+) which generate them in the above sense. We prove that each ces(p+) is a power series space of finite type and order one, and that all the spaces ces(p+), 1 <= p < infinity, are isomorphic. (C) 2017 Elsevier Inc. All rights reserved.