(untitled) (ҚазҰУ Хабаршысы. Математика, механика, информатика сериясы, 2020)
In this work we study the non-commutative Hardy-Littlewood maximal operator on Lorentz spaces of τ-measurable operators. Non-commutative maximal inequalities were studied, in particular, in MJ1, JQ, TM. Another version of the (non-commutative) Hardy-Littlewood maximal operator was introduced by T. Bekjan TB. Later J. Shao investigated the Hardy-Littlewood maximal operator on non-commutative Lorentz spaces associated with finite atomless von Neumann algebra (see Sh). Namely, for an operator T affiliated with a semi-finite von Neumann algebra M, the Hardy-Littlewood maximal operator of T is defined by MA(x)=_r>01τ(E_[x-r, x+r](|A|))τ(|A|E_[x-r, x+r](|A|)), x≥0. While the classical Hardy-Littlewood maximal operator of a Lebesgue measurable function f:R→R, denoted by Mf(x), is defined as Mf(x)=_r>01m([x-r, x+r])_[x-r,x+r]|f(t)|dt, where m is a Lebesgue measure on (-∞,∞) SW. In view of spectral theory, |A| is represented as |A|=_σ(|A|)tdE_t, and MA(|A|) is represented as MA(x). Thus, for the operator A, Bekjan’s consideration is that MA(|A|) is defined as the operator analogue of the Hardy-Littlewood maximal operator in the classical case. Our purpose is to investigate the non-commutative Hardy-Littlewood maximal operator M in the sense of T. Bekjan (see TB). In particular, we obtain boundedness of the non-commutative Hardy-Littlewood maximal operator in non-commutative Lorentz spaces.