Locally convex spaces with Toeplitz decompositions

A Toeplitz decomposition of a locally convex space E into subspaces (E-k) w ith continuous projections (P-k) is a decomposition of every x is an elemen t of E as x = Sigma(k) P(k)x where ordinary summability has been replaced b y summability with respect to an infinite and row-finite matrix. We extend to the setting of Toeplitz decompositions a number of results about the loc ally convex structure of a space with a Schauder decomposition. Namely, we give some necessary or sufficient conditions for being reflexive, a Montel space or a Schwartz space. Roughly speaking, each of these locally convex p roperties is linked to a property of the convergence of the decomposition. We apply these results to study some structural questions in projective ten sor products and spaces with Cesaro bases.