Abstract
In this article, we introduce the concept of pre-quasi norm on E (Orlicz sequence space), which is more general than the usual norm, and give the conditions on E equipped with the pre-quasi norm to be Banach space. We give the necessity and sufficient conditions on E equipped with the pre-quasi norm such that the multiplication operator defined on E is a bounded, approximable, invertible, Fredholm, and closed range operator. The components of pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained for different Orlicz functions are determined. Furthermore, we give the sufficient conditions on E equipped with a pre-modular such that the pre-quasi Banach operator ideal constructed by s-numbers and E is simple and its components are closed. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to E.
Highlights
Throughout the paper, we denote the space of all bounded linear operators from a Banach space X into a Banach space Y by L(X, Y ), and if X = Y, we write L(X), the space of all real sequences is denoted by w, the real numbers R, the complex numbers C, N = {0, 1, 2, . . .}, the space of null sequences by C0, and the space of bounded sequences by ∞
The multiplication operators on Lp-spaces are related to the composition operators; this means that the properties of composition operators on Lp-spaces can be stated by the properties of multiplication operators
Makarov and Faried [14] proved that the quasi-operator ideal formed by the sequence of approximation numbers is strictly contained for different powers, i.e., for any infinite dimensional Banach spaces X, Y and for any q > p > 0, it is true that Sappp(X, Y ) Saqpp(X, Y ) L(X, Y )
Summary
Makarov and Faried [14] proved that the quasi-operator ideal formed by the sequence of approximation numbers is strictly contained for different powers, i.e., for any infinite dimensional Banach spaces X, Y and for any q > p > 0, it is true that Sappp(X, Y ) Saqpp(X, Y ) L(X, Y ). We give the necessity and sufficient conditions on E equipped with the pre-quasi norm such that the multiplication operator defined on E is a bounded, approximable, invertible, Fredholm, and closed range operator.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
