论文标题
基于规范导数的复杂规范空间中的正交关系
An orthogonality relation in complex normed spaces based on norm derivatives
论文作者
论文摘要
令$ x $为一个复杂的规范空间。基于右规范导数$ρ_ {_ {+}} $，我们定义了mapping $ρ_{_ {_ {\ infty}} $ by \ begin \ begin {equation*}ρ_{_ {_ {_ {\ infty}}}}} \frac1π\ int_0^{2π} e^{iθ}ρ_{_ {+}}}（x，x，e^{i^{iθ} y）dθ\ quad（x，y \ y \ in x）。 \ end {equation*}映射$ρ_ {_ {\ infty}} $对$ x $的某些几何属性具有很好的响应。例如，我们证明$ρ_ {_ {\ infty}}（x，x，y）=ρ_{_ {_ {\ infty}}}（y，x）$ for ALL $ x，y in x $ in x $，并且仅当$ x $是内在产品空间。此外，我们定义了$ x $中的$ρ_ {_ {\ infty}} $ - 正交性，并显示一个线性映射保留$ρ_{_ {\ infty}} $ - 正交性必须是iSOMETRY的scalar多个。还讨论了复杂规范空间的几何形状中的许多具有挑战性的问题。
Let $X$ be a complex normed space. Based on the right norm derivative $ρ_{_{+}}$, we define a mapping $ρ_{_{\infty}}$ by \begin{equation*} ρ_{_{\infty}}(x,y) = \frac1π\int_0^{2π}e^{iθ}ρ_{_{+}}(x,e^{iθ}y)dθ\quad(x,y\in X). \end{equation*} The mapping $ρ_{_{\infty}}$ has a good response to some geometrical properties of $X$. For instance, we prove that $ρ_{_{\infty}}(x,y)=ρ_{_{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. In addition, we define a $ρ_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $ρ_{_{\infty}}$-orthogonality has to be a scalar multiple of an isometry. A number of challenging problems in the geometry of complex normed spaces are also discussed.
