turkmath.org

The scalar field will be either the real or complex numbers. Suppose that λ is a solid sequence space over the scalar field and A is an infinite lower triangular matrix with non-negative entries and positive entries on the main diagonal such that each of its columns is in λ. For each positive integer k, the kth predecessor of λ with respect to A is the solid vector space of scalar sequences x such that Ak |x| is an element of λ. We denote this space by Λk and λ itself will be denoted by Λ0. Under reasonable assumptions, these spaces inherit some topological properties from λ. We are interested in a projective limit of the infinite product of the Λk consisting of sequences of sequences (x(k)) satisfying A x(k) =x(k-1) for each k>0. We show that for interesting classes of situations including the cases when λ =lp for some p>1 and A is the Cesaro matrix, the space of our interest can be non-trivial.