Barrelledness of spaces with Toeplitz decompositions

A Toeplitz decomposition of a locally convex space E into subspaces (E-k) w ith projections (P-k) is a decomposition of every x is an element of E as x = Sigma(k)P(k)x, where ordinary summability has been replaced by summabili ty with respect to an infinite and lower triangular regular matrix. We exte nd to the setting of Toeplitz decompositions a couple of results about barr elledness of Schauder decompositions. The first result, given for Schauder decompositions by Noll and Stadler, links the barrelledness of a normed spa ce E to the barrelledness of the pieces E-k via the fact that E' is big eno ugh so as to coincide with its summability dual. Our second theorem, given for Schauder decompositions by Diaz and Minarro, links the quasibarrelledne ss of an N-0-quasibarrelled (in particular, (DF)) space E to the quasibarre lledness of the pieces E-k via the fact that the decomposition is simple. ( C) 1999 Academic Press.