Small operator ideals formed by s numbers on generalized Cesáro and Orlicz sequence spaces

Abstract

In this article, we establish sufficient conditions on the generalized Cesáro and Orlicz sequence spaces such that the class of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers belonging to generates an operator ideal. The components of as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. Some inclusion relations between the operator ideals as well as the inclusion relations for their duals are obtained. Finally, we show that the operator ideal formed by and approximation numbers is small under certain conditions.

Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces

“…Finally, we show that the prequasi Banach operator ideal formed by the approximation numbers and is small under certain conditions. These results coincide with that known for , (1 < < ∞) in [3]. Furthermore we give some examples which support our main results.…”

“…(1) for all ∈ Ω( , ), ( ) ≥ 0 and ( ) = 0 if and only if = 0, Theorem 29 (see [3]). Every quasi norm on the ideal Ω is a pre-quasi norm on the ideal Ω.…”

“…In [2], Faried and Bakery examined the operator ideals developed by generalized Cesáro, Orlicz sequence spaces ℓ , and the approximation numbers. In [3], Faried and Bakery studied the operator ideals constructed by − numbers, generalized Cesáro and Orlicz sequence spaces ℓ and show that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions. Also summation process and sequences spaces applications are closely related to Korovkin type approximation theorems and linear positive operators studied by Costarelli and Vinti [4] and Altomare [5].…”

The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces

“…Bakery and Mohammed investigated the operator ideals generated by s-numbers and Orlicz-Cesáro mean sequence spaces. In [34], Faried and Bakery introduced the concept of pre-quasi operator ideal which is more general than the quasi operator ideal, studied the operator ideals formed by s-numbers, generalized Cesáro and Orlicz sequence spaces ℓ M and showed that the operator ideal formed by approximation numbers and the previous sequence spaces is small under certain conditions. Many articles have been published on the above topics, sequence spaces defined by Orlicz function, multiplier sequences, spectra of matrix operators etc in the recent past by Tripathy et al [35][36][37][38][39][40][41].…”

Eigenvalues of s-type operators on $C(p)$ equipped with a pre-quasi norm

“…Makarov and Faried (see [20]) proved that, for any infinite dimensional Banach spaces X, Y and for any q > p > 0, S app p (X, Y ) is strictly contained in S app q (X, Y ). Faried and Bakery (see [21]) introduced the concept of pre-quasi-operator ideal which is more general than the usual classes of operator ideal, they studied the operator ideals formed by s-numbers, generalized Cesáro and Orlicz sequence spaces M , and showed that the operator ideal formed by approximation numbers and the previous sequence spaces is small under certain conditions. The aim of this article to study the concept of pre-quasi-norm on C(p) which is more general than the usual norm, and give the conditions on C(p) equipped with the prequasi-norm to be Banach space.…”