论文标题
在保留半米产品的非线性映射上
On non-linear mappings preserving the semi-inner product
论文作者
论文摘要
我们说，如果每个映射$ f：x \ to x $在$ x $上保存半米产品的每个映射$ f：x \ to x $是线性的，则具有平稳的规范空间$ x $具有属性（SL）。众所周知，每个希尔伯特空间都有该特性（SL），并且每个有限维平滑的范数空间都是如此。在本文中，我们建立了有关属性（SL）的几个新结果。我们给出了一个简单的示例，说明了平滑且严格的凸出banach空间，该空间与空间$ \ ell_p $同构，但没有属性（SL）。此外，我们根据商的子空间提供了反射光滑的Banach空间中属性（SL）的表征。结果，我们证明了$ \ ell_p $每$ 1 <p <\ infty $具有属性（SL）。最后，使用Gowers-Maurey空间的变体，我们构建了一个无限尺寸均匀平滑的Banach Space $ x $，以使每个光滑的Banach空间同构至$ x $都有该属性（SL）。
We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ preserving the semi-inner product on $X$ is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space $\ell_p$, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space $\ell_p$ have the property (SL) for every $1 < p < \infty$. Finally, using a variant of the Gowers-Maurey space, we construct an infinite-dimensional uniformly smooth Banach space $X$ such that every smooth Banach space isomorphic to $X$ has the property (SL).
