[Paper Review] Weak$^*$ Fixed Point Property and Polyhedrality in Lindenstrauss Spaces

Emanuele Casini, Enrico Miglierina+1 more|arXiv (Cornell University)|Apr 26, 2016

Optimization and Variational Analysis1 citations

TL;DR

This paper establishes geometric characterizations of the weak* fixed point property and its stable variant in duals of separable Lindenstrauss spaces by analyzing the structure of the dual unit ball. It proves that a strengthened polyhedral condition—introduced by Fonf and Veselá—equivalents the stable w*-fixed point property, and links the weaker condition to polyhedrality in the predual, refining the hierarchy of polyhedral properties in this class of spaces.

ABSTRACT

The aim of this paper is to study the $w^*$-fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of $w^*$-closed subsets of the dual sphere is equivalent to the $w^*$-fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable $w^*$-fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.

Motivation & Objective

• To characterize the w*-fixed point property in duals of separable Lindenstrauss spaces using geometric properties of the dual unit ball.
• To establish an equivalence between a stronger geometric condition on the dual ball and the stable w*-fixed point property.
• To relate the introduced geometric assumptions to polyhedral structures in the predual space.
• To refine the hierarchy of polyhedrality notions within the framework of Lindenstrauss spaces.
• To improve an existing result on norm-preserving compact extensions of compact operators as a by-product.

Proposed method

• Introduce a geometric condition on w*-closed subsets of the dual unit sphere to characterize the w*-fixed point property.
• Define and analyze a stronger geometric property of the dual unit ball, inspired by Fonf and Veseló’s notion of polyhedrality.
• Establish equivalence between this stronger property and the stable w*-fixed point property.
• Relate the weaker geometric assumption to polyhedral properties in the predual space via duality.
• Use duality and geometric functional analysis techniques to derive implications between polyhedrality concepts.
• Apply the results to extend compact operators while preserving the norm, improving a classical result.

Experimental results

Research questions

• RQ1 What geometric conditions on the dual unit ball characterize the w*-fixed point property for nonexpansive mappings?
• RQ2 How does the stable w*-fixed point property relate to a strengthened polyhedral condition in the dual space?
• RQ3 Can the weaker geometric assumption linked to the w*-fixed point property be connected to polyhedral structure in the predual?
• RQ4 What is the hierarchical relationship between various notions of polyhedrality in Lindenstrauss spaces?
• RQ5 Can the geometric framework lead to improvements in the extension theory of compact operators?

Key findings

• A geometric condition on w*-closed subsets of the dual sphere is equivalent to the w*-fixed point property in duals of separable Lindenstrauss spaces.
• A stronger geometric property of the dual unit ball is equivalent to the stable w*-fixed point property.
• The stronger geometric condition corresponds to a polyhedral-type property introduced by Fonf and Veseló, thus linking stability to polyhedral structure.
• The weaker geometric assumption is shown to be related to a polyhedral concept in the predual space, establishing a duality link.
• A hierarchy of polyhedrality notions is established within the class of Lindenstrauss spaces.
• An improvement is obtained for the norm-preserving compact extension of compact operators as a by-product of the main results.