On Coarse Isometries between Banach Spaces

Lixin Cheng ³ÌÁ¢ÐÂ  (Xiamen University)

14:45-15:45, Oct 30, 2017   Science Building A1510

Abstract:

A mapping $f$ from a Banach space $X$ into a Banach space $Y$ is said to be a Lindenstrauss-Szankowski provided $\lim_{t\rightarrow+\infty}\frac{\varepsilon(t)}t=0$, where $$\varepsilon(t)=\sup\{\big|\|f(x)-f(y)\|-\|x-y\|: \|x-y\|\vee\|f(x)-f(y)\|\leq t\},\;\;t\geq0.$$ Lindenstrauss and Szankowski first studied such mappings and showed the following remarkable representation theorem for a surjective coarse isometry $f: X\rightarrow Y$ with the additional assumption. $$\int_1^\infty\frac{\varepsilon(t)}{t^2}dt\le \infty$$ There is a linear surjective isometry $U: X\rightarrow Y$ so that $$\|f(x)-Ux\|=o(\|x\|),\;\;{\rm as\;}\;\|x\|\rightarrow\infty.$$ Nevertheless, the research of this topic has stopped for over 30 years, because they have given examples showing that this theorem fails without the integral convergence assumption above, even if both $X$ and $Y$ are the real line $\mathbb R$. In this talk, we shall show that, in certain Banach spaces, the results analogous to the Lindenstrauss-Szankowski theorem may hold without the integral convergence assumption£¬ even without the surjectivity on the mappings, but we substitute $w^*$-topology and the free ultrafilter limits on $\mathbb N$ for the norm-topology and the usual limits, respectively.