Barrelledness of Spaces with Toeplitz Decompositions

A Toeplitz decomposition of a locally convex space E into subspaces (E_k) with projections (P_k) is a decomposition of every x  E as x = ∑_kP_kx, where ordinary summability has been replaced by summability with respect to an infinite and lower triangular regular matrix. We extend to the setting of Toeplitz decompositions a couple of results about barrelledness of Schauder decompositions. The first result, given for Schauder decompositions by Noll and Stadler, links the barrelledness of a normed space E to the barrelledness of the pieces E_k via the fact that E′ is big enough so as to coincide with its summability dual. Our second theorem, given for Schauder decompositions by Dı́az and Miñarro, links the quasibarrelledness of an ₀-quasibarrelled (in particular, (DF)) space E to the quasibarrelledness of the pieces E_k via the fact that the decomposition is simple.