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In this work we study the non-commutative Hardy-Littlewoodmaximal operator on Lorentz spacesofτ-measurable operators. Non-commutative maximal inequalities were studied, in particular,in [1–3]. Another version of the (non-commutative) Hardy-Littlewood maximal operator was in-troduced by T. Bekjan [4]. Later J. Shao investigated the Hardy-Littlewood maximal operator onnon-commutative Lorentz spaces associated with finite atomless von Neumann algebra (see [5]).

Properties of continuous positively homogeneous operators of degree via various functions (e.g. measures of noncompactness) on all bounded subsets of a Banach space are studied. Necessary and sufficient conditions for these functions to vanish on the image of the unit ball under positively homogeneous operators are given. In particular, we give criteria for the complete continuity of the Fréchet derivative in an arbitrary Banach space and criteria for operators, acting in regular spaces, to be improving.

Geometric characteristics of regular spaces are determined. Examples of regular spaces are the Lebesgue and Lorentz spaces, in particular. For the Lorentz spaces an inequality for arbitrary subsets, connecting the measures of noncompactness and are proved. A new class of - condensing operators in the Lorentz spaces among partially-additive operators linear operators, in particular, is obtained.