ON SPACES DERIVABLE FROM A SOLID SEQUENCE SPACE AND A NON-NEGATIVE LOWER TRIANGULAR MATRIX

The scalar field will be either the real or complex numbers. Suppose that lambda 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 lambda. For each positive integer k, the kth predecessor of lambda with respect to A is the solid vector space of scalar sequences x such that A(k)vertical bar x vertical bar is an element of lambda. We denote this space by Lambda(k) and lambda itself will be denoted by Lambda(0). Under reasonable assumptions, these spaces inherit some topological properties from lambda. We are interested in a projective limit of the infinite product of the Lambda(k) consisting of sequences of sequences (x((k))) satisfying Ax((k)) = x((k-1)) for each k > 0. We show that for interesting classes of situations including the cases when lambda = l(p) for some p > 1 and A is the Cesaro matrix, the space of our interest can be non-trivial.