Some properties of pre-quasi operator ideal of type generalized Cesaro sequence space defined by weighted means

Let E be a generalized Cesaro sequence space defined by weighted means and by using s-numbers 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 S-E(X, Y) := {T is an element of L(X, Y) : ((S-n(T))n=0 infinity is an element of E}, of the class S-E form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal S-E, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small and the pre-quasi Banach operator ideal constructed by s-numbers is simple Banach space. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to this sequence space.