Paweł Kolwicz

Local rotundity structure of Calderón-Lozanovskii spaces★

Abstract General results saying that a point x of the unit sphere S(E) of a Kothe space E is an extreme point (a strongly extreme point) [an SU-point] of B(E) if and only if ‖x‖ is an extreme point (a strongly extreme point) [an SU-point] of B(E+) and ‖x‖ is a UM-point (a ULUM-point) [nothing more] of E are proved. These results are applied to get criteria for extreme points and SU-points of the unit ball in Caldern-Lozanovski spaces which refer to problem XII from [5]. Strongly extreme points in these spaces are also discussed.

Non-squareness properties of Orlicz-Lorentz function spaces

In this paper, criteria for non-squareness and uniform non-squareness of Orlicz-Lorentz function spaces Λφ,ω are given. Since degenerated Orlicz functions φ and degenerated weight functions ω are also admitted, this investigation concerns the most possible wide class of Orlicz-Lorentz function spaces.It is worth recalling that uniform non-squareness is an important property, because it implies super-reflexivity as well as the fixed point property (see James in Ann. Math. 80:542-550, 1964; Pacific J. Math. 41:409-419, 1972 and García-Falset et al. in J. Funct. Anal. 233:494-514, 2006).MSC:46B20, 46B42, 46A80, 46E30.

Points of nonsquareness of Lorentz spaces Γp,w

Criteria for nonsquare points of the Lorentz spaces of maximal functions Γp,w are presented under an arbitrary (also degenerated) nonnegative weight function w. The criteria for nonsquareness of Lorentz spaces Γp,w and of their subspaces (Γp,w)a of all order continuous elements, proved directly in (Kolwicz and Panfil in Indag. Math. 24:254-263, 2013), are deduced.MSC: 46E30, 46B20, 46B42.