Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals

In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l (p,q,M) (X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l (M) (X) for p = q and also Lorentz sequence spaces for M(x) = x (q) for q a parts per thousand yen 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l (p,q,M) and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p a parts per thousand yen q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.