Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (2): 623-652.doi: 10.1007/s10473-022-0214-9

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Yunan CUI1, Paweƚ FORALEWSKI2, Joanna KOŃCZAK2   

  1. 1. Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China;
    2. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
  • Received:2020-09-18 Revised:2021-05-31 Online:2022-04-25 Published:2022-04-22
  • Supported by:
    The first author gratefully acknowledges the support of NSF of China (11871181).

Abstract: In this article, we consider Orlicz-Lorentz sequence spaces equipped with the Orlicz norm $\left(\lambda _{\varphi,\omega}, \Vert\cdot\Vert_{\varphi, \omega}^{O}\right)$ generated by any Orlicz function and any non-increasing weight sequence. As far as we know, research on such a general case is conducted for the first time. After showing that the Orlicz norm is equal to the Amemiya norm in general and giving some important properties of this norm, we study the problem of existence of order isomorphically isometric copies of $l^{\infty}$ in the space $\left(\lambda _{\varphi,\omega}, \Vert\cdot\Vert_{\varphi, \omega}^{O}\right)$ and we find criteria for order continuity and monotonicity properties of this space. We also find criteria for monotonicity properties of $n$-dimensional subspaces $\lambda _{\varphi,\omega}^{n}$ ($n\geq 2$) and the subspace $\left(\lambda _{\varphi,\omega}\right)_{a}$ of order continuous elements of $\lambda _{\varphi,\omega}$. Finally, as an immediate consequence of the criteria considered in this article, the properties of Orlicz sequence spaces equipped with the Orlicz norm are deduced.

Key words: Orlicz-Lorentz spaces, Banach symmetric spaces, Orlicz norm, order continuity and copies of l, monotonicity properties

CLC Number: 

  • 46B20