Pre-quasi simple Banach operator ideal generated by \(s\)-numbers (Q2188100)

Summary: Let \(E\) be a weighted Nakano sequence space or generalized Cesàro sequence space defined by weighted mean 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 \(S_E( X , Y):=\{ T \in L ( X , Y) : ( s_n ( T))_{n = 0}^\infty \in E\}\) 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 finally, the pre-quasi Banach operator ideal constructed by \(s\)-numbers is simple Banach space.