No abstract

In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spaces cesφ, where φ is an Orlicz function such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers which belong to cesφ forms an operator ideal. The completeness and denseness of its ideal components are specified and Scesφ constructs a pre-quasi Banach operator ideal. Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are explained. Moreover, we have presented the sufficient conditions on cesφ such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for cesp  (1<p<∞).

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$$\begin{array}{} \displaystyle S_{E}(X, Y):=\Big\{T\in L(X, Y):((s_{n}(T))_{n=0}^{\infty}\in E\Big\}, \end{array}$$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.

An idealIis a hereditary and additive family of subsets of positive integersℕ. In this paper, we will introduce the concept of generalized randomn-normed space as an extension of randomn-normed space. Also, we study the concept of lacunary mean (L)-ideal convergence andL-ideal Cauchy for sequences of complex numbers in the generalized randomn-norm. We introduceIL-limit points andIL-cluster points. Furthermore, Cauchy andIL-Cauchy sequences in this construction are given. Finally, we find relations among these concepts.