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

Abstract

Let E be a generalized Cesáro 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 SE(X,Y):=TL(X,Y):((sn(T))n=0∞E, SE(X, Y):=\Big\T\in L(X, Y):((sn(T))n=0&infty;\in E\Big\, \end of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, 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.