Perfect matrix methods

Abstract

Let e i = ( δ i j ) j = 1 ∞ , Δ = ( e i ) i = 1 ∞ {e_i} = ({\delta _{ij}})_{j = 1}^\infty ,\Delta = ({e_i})_{i = 1}^\infty and let A be an infinite matrix which maps E into E where E is an FK-space with Schauder basis Δ \Delta . Necessary and sufficient conditions in terms of the matrix A are obtained for E to be dense in the summability space E A = { x | A x ∈ E } {E_A} = \{ x | {Ax \in E\} } and conditions are obtained to guarantee that E A {E_A} has Schauder basis Δ \Delta . Finally it is shown that if weak and strong sequential convergence coincide in E then in E A {E_A} the series ∑ k x k e k \sum {_k{x_k}{e_k}} converges to x strongly if and only if it converges to x weakly.