In this paper, we investigate the Hardy-Littlewood maximal operator (in a sence of Bekjan ) on non-commutative symmetric spaces. We obtain an upper distributional estimate (by means of the Cesàro operator) of a generalized singular number of the non-commutative Hardy-Littlewood maximal operator. We also show boundedness of the Hardy-Littlewood maximal operator from a general non-commutative symmetric space to another. Keywords Symmetric spaces of functions and operators • Hardy-Littlewood maximal operator • von Neumann algebra • (Non-commutative) Lorentz and Marcinkiewicz spaces Mathematics Subject ClassificationCommunicated by Yong Jiao.

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