A note on triangular operators on Smooth Sequence Spaces

Abstract

For a scalar sequence {(\theta_n)}_{n \in \mathbb{N}}, let C be the matrix defined by c_n^k = \theta_{n-k+1} if n > k, c_n^k = 0 if n < k. The map between K\"{o}the spaces \lambda(A) and \lambda(B) is called a Cauchy Product map if it is determined by the triangular matrix C. In this note we introduced some necessary and sufficient conditions for a Cauchy Product map on a nuclear K\"{o}the space \lambda(A) to nuclear G_1-space \lambda(B) to be linear and continuous. Its transpose is also considered.