【作者】 程娜；

【导师】 陈滋利；

【作者基本信息】 西南交通大学 ， 交通信息工程及控制， 2010， 博士

【摘要】 在有关Banach格及其上的算子理论的研究中,一般是关于空间性质和算子性质两个方面。空间性质主要讨论空间拓扑结构,序结构,以及两者的相容性等问题,诸如序连续性,KB-性质,Schur性质等。算子的性质主要讨论的是算子的控制性,分解性,格序性,算子与算子之间的关系,以及算子空间序结,拓扑结构,以及两者的相容性等问题。目前有关算子性质研究主要集中在特殊算子类,如紧算子,弱紧算子,序弱紧算子,AM-紧算子,Dunford-Pettis算子,半紧算子与M-弱紧算子(L-弱紧算子)等。算子性质的研究是Banach格及其上的算子理论的研究中的重要问题。本文讨论了Banach格上几类特殊算子的性质。主要内容分为如下五部分：第一部分研究了Banach格上AM-紧算子和Dunford-Pettis算子的分解性质。得到了Banach格上的AM-紧算子通过具有序连续范数的Banach格进行分解,并且其中个分解因子是AM-紧算子的结论。更深入的,利用AM-紧算子的分解性质得到了Banach格上正的AM-紧算子一些重要结论。在AM-紧算子分解性质的基础上考虑了Dunford-Pettis算子的分解性质,得到了在一定条件下Banach格上的Dunford-Pettis算子通过具有序连续范数的Banach格进行分解,并且其中一个分解因子是Dunford-Pettis算子的结论。第二部分是本文的主要结果,在Banach格上引入了新的算子b-AM-紧算子。首先深入说明Banach格上紧算子,AM-紧算子,b-AM-紧算子的关系,得到了Banach格上每个连续算子是b-AM-紧算子的充要条件,以及在一定条件下Banach格上每个b-弱紧算子是b-AM-紧算子的重要结论。其次讨论了Banach格上b-AM-紧算子的控制性。给出例子说明在一般情况下Banach格上被b-AM-紧算子控制的正算子不一定是b-AM-紧算子,在此基础上给出了Banach格上被b-AM-紧算子控制的正算子是b-AM-紧算子的条件,更深入的,发现了Banach格上被b-AM-紧算子控制的正算子的平方是b-AM-紧算子的结论。接着研究了Banach格上b-AM-紧算子的共轭性。给出例子说明Banach格上b-AM-紧算子的共轭算子不一定是b-AM-紧算子,且若共轭算子是b-AM-紧算子,其算子本身不一定是b-AM-紧算子。其次考虑了Banach格上b-AM-紧算子的共轭算子是b-AM-紧算子的条件,得到了Banach格上共轭算子是b-AM-紧算子,其算子本身是b-AM-紧算子的充要条件。最后考虑了如果Banach格上b-AM-紧算子的共轭算子是b-AM-紧算子,此时的Banach格具有的特殊性质,且给出了如果Banach格上共轭算了是b-AM-紧算子,其算子本身是b-AM-紧算子,此时的Banach格具有的特殊性质。更深入的,考虑了Banach格上b-AM-紧算子的格序性质。给出例子说明Banach格上正则b-AM-紧算子的模不一定是b-AM-紧算子,即Banach格上所有正则b-AM-紧算子空间不构成向量格,此外还发现了Banach格上所有正则b-AM-紧算子空间构成向量格的充要条件。最后证明了Banach格上所有正则b-AM-紧算子空间在b-AM-范数下是完备的结论。第三部分研究了正的Dunford-Pettis算子与L-弱紧算子,M-弱紧算子的关系。得到了Banach格上正的Dunford-Pettis算子是L-弱紧算子的充要条件,且给出了Banach格上正的Dunford-Pettis算子是M-弱紧算子的必要条件。第四部分考虑了正的半紧算子与L-弱紧算子,M-弱紧算子的关系。得到了Banach格上正的半紧算子是L-弱紧算子充要条件,且获得了Banach格上正的半紧算子是M-弱紧算子充要条件。第五部分研究了Banach格上正的b-弱紧算子是Dunford-Pettis算子的充分条件以及正的b-弱紧算子是Dunford-Pettis算子的必要条件,即用Banach格上b-弱紧算子的Dunford-Pettis性质来刻画Banach格上的序连续范数。更深入的,考虑了Banach格上正的半紧算子是b-弱紧算子的充要条件。更多还原

【Abstract】 The operator property is a famous problem in the history of operator theory. This dissertation is devoted to the study of properties of several special operators on Banach lattice and divided into the following five parts:The first part is devoted to the AM-compact operators factor through Banach lattice with order continuous norm and one AM-compact factor. As consequence, we obtain some interesting results on positive AM-compact operators. And the similar question for Dunford-Pettis operator is also considered.Main results are included in the second part, firstly, we give several characterizations of b-AM-compact operators. Secondly, we study several properties about the problems of domination in the class of positive b-AM-compact operators. Moreover, we present the condition that if the dual operator T’:F’→E’is b-AM-compact, then the positive operator T:E→F is b-AM-compact. We also give the condition for the adjoint operators of each b-AM-compact operators being also b-AM-compact. Finally, several equivalent conditions characterizing when the space of regular b-AM-compact operator is Dedekindσ-complete are given.In the third part, we present some necessary and sufficient conditions for positive Dunford-Pettis operators being L-weakly compact. We also give a necessary condition under which each positive Dunford-Pettis operator is M-weakly compact.In the fourth part, we present some necessary and sufficient conditions for positive semi-compact operators being L-weakly compact and M-weakly compact respectively.Finally, we investigate the sufficient condition under which each positive b-weakly compact operator is Dunford-Pettis. We also investigate the necessary condition on which each positive b-weakly compact operator is Dunford-Pettis. Necessary condition on which each positive b-weakly compact operator is weakly compact is also considered. We present a necessary and sufficient condition under which each positive semi-compact operator is b-weakly compact.■更多还原