Some properties of pre-quasi norm on Orlicz sequence space

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Abstract In this article, we introduce the concept of pre-quasi norm onE(Orlicz sequence space), which is more general than the usual norm, and give the conditions onEequipped with the pre-quasi norm to be Banach space. We give the necessity and sufficient conditions onEequipped with the pre-quasi norm such that the multiplication operator defined onEis a bounded, approximable, invertible, Fredholm, and closed range operator. The components of pre-quasi operator ideal formed by the sequence ofs-numbers andEis strictly contained for different Orlicz functions are determined. Furthermore, we give the sufficient conditions onEequipped with a pre-modular such that the pre-quasi Banach operator ideal constructed bys-numbers andEis simple and its components are closed. Finally the pre-quasi operator ideal formed by the sequence ofs-numbers andEis strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs toE.

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