论文标题
度量固定点定理及其某些应用
A metric fixed point theorem and some of its applications
论文作者
论文摘要
在存在锥体双托架的假设下证明了同量函数的一般固定点定理。即使对于Banach空间以及非局部紧凑的CAT（0）空间和注射式空间的异构体，也是新的。对非培训CAT（0）空间的作用的例子来自对差异群体，异常转化和紧凑的Kähler歧管的研究。固定点定理的一个特殊情况提供了一种新颖的平均千古定理，在希尔伯特空间中，这意味着冯·诺伊曼的定理。该定理可容纳经典的无定点等距图，例如Kakutani，Edelstein，Alspach和Prus的等距图。此外，从主要定理以及一些独立关注的几何论点，可以推断出，希尔伯特空间的每个有界的可逆操作员都可以在正运算符的空间上承认一个非平凡的不变的度量功能。这是朝着不变子空间问题的方向发展的结果，尽管其全部含义取决于此类度量功能的未来确定。
A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann's theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.
