Optimal upper and lower sequence spaces with applications

Abstract

We study the optimal upper $X_U$ and lower $X_L$ sequence spaces that can be assigned to each Banach lattice $X$. These spaces are symmetric, have the Fatou property and the unit vector basis has in these spaces very special properties. Determined by the order structure of $X$ the spaces $X_U$ and $X_L$ turn out to be very useful when studying Banach lattices. Among other results, in terms of these constructions, we identify Banach lattices that satisfy equal-norm upper and lower $p$-estimates, give a characterization of $L_p(\mu)$-spaces, derive some properties of the tensor product operator in Lorentz and Orlicz spaces, identify Orlicz spaces in which the unit vector basis is upper (resp. lower) semi-homogeneous.