Operator ideal of Norlund-type sequence spaces

Abstract

Let E be a Norlund sequence space which is invariant under the doubling operator (Formula presented.) Using the approximation numbers (Formula presented.) 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 (Formula presented.) form an operator ideal, the finite rank operators are dense in the complete space of operators U<inf>E</inf>^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<inf>E</inf>^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((pn)), where (pn) is a sequence of positive reals.