公设定点定理及其一些应用

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-02-07 DOI:10.1007/s00039-024-00658-x
Anders Karlsson
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引用次数: 0

摘要

在存在圆锥二梳齿的假设下,证明了以度量函数为单位的等距线的一般定点定理。这对于巴拿赫空间凸集的等距以及非局部紧凑 CAT(0)-spaces 和注入空间都是新的。在非完全 CAT(0)-spaces 上的作用的例子来自于对衍射群、双向变换和紧凑凯勒流形的研究。定点定理的一个特例提供了一个新颖的均值遍历定理,在希尔伯特空间情况下隐含着冯-诺依曼定理。该定理适用于经典的无定点等距映射,如角谷、埃德尔斯坦、阿尔斯帕赫和普鲁斯的映射。此外,根据主定理和一些独立的几何论证,我们可以推导出希尔伯特空间的每个有界可逆算子在正算子空间上都有一个非难不变度量函数。这是不变子空间问题方向上的一个结果,尽管其全部意义取决于将来对这类度量函数的确定。
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A Metric Fixed Point Theorem and Some of Its Applications

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 for isometries of convex sets 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.

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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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