Non-squareness properties of Orlicz-Lorentz sequence spaces

In this paper criteria for non-squareness properties (non-squareness, local uniform non-squareness and uniform non-squareness) of Orlicz-Lorentz sequence spaces lambda(phi,omega) and of their n-dimensional subspaces lambda(n)(phi,omega) (n >= 2) as well as of the subspaces (lambda(phi,omega))(a) of all order continuous elements in lambda(phi,omega) are given. Since degenerate Orlicz functions phi and degenerate weight sequences w are also admitted, these investigations concern the most possible wide class of Orlicz-Lorentz sequence spaces. Finally, as immediate consequences, criteria for all non-squareness properties of Orlicz sequence spaces, which complete the results of Sundaresan (1966) [53], Hudzik (1985) [23], Hudzik (1985) [24], are deduced. Iris worth recalling that uniform non-squareness is an important property, because it implies super-reflexivity as well as the fixed point property (see James (1964) [31], James (1972) [33] and Garcia-Falser et al. (2006) [19]). (C) 2012 Elsevier Inc. All rights reserved.