Abstract
Let E be a Norlund sequence space which is invariant under the doubling operator
D : x = (x(0), x(1), x(2),...) bar right arrow y = (x(0), x(0), x(1), x(1), x(2), x(2),...).
Using the approximation numbers (alpha n(T))(n=0)(infinity) of operators from a Banach space X into a Banach space Y, we give the sufficient (not necessary) conditions on E such that the components
U-E(app) (X, Y) := {T is an element of L(X, Y) : (alpha(n)(T))(n=0)(infinity) is an element of E}
form an operator ideal, the finite rank operators are dense in the complete space of operators U-E(app) (X, Y) which is a longstanding open problem. Finally we give an answer for Rhoades (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 59(3-4):238-241, 1975) about the linearity of E-type spaces (U-F(app) (X, Y)), and we conclude under a few conditions that every compact operator would be approximated by finite rank operators. Our results agree with those in (J. Inequal. Appl., 2013, doi:10.1186/1029-242x-2013-186) for the space ces((p(n))), where (p(n)) is a sequence of positive reals.