Abstract
Let E be the sequence space defined and studied by Tripathy and Mahanta (Soochow J Math 29: 379-391, 2003) which is invariant under the doubling operator D : x = (x(0), x(1), x(2), ...) -> y = x(0), x(0), x(1), x(1), x(2), x(2), ...). Using the approximation numbers (alpha(n)(T))(infinity)(n=0) of operators from a Banach space X into a Banach space Y, we give the sufficient conditions on E such that the finite rank operators are dense in the complete space of operators U-E(app) (X, Y), where U-E(app)(X, Y) :={T is an element of L(X, Y) : (alpha(n)(T))(infinity)(n=0) is an element of E} When M(t) = t(p), 1 <= p < infinity with sup(s) phi(s) < infinity our results coincide with that known for the space l(p).