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An orthogonality relation in complex normed spaces based on norm derivatives
Functional Analysis · Submitted 24 Apr 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In addition, we define a $\rho_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $\rho_{_{\infty}}$-orthogonality has to be a scalar multiple of an isometry.
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The abstract, sentence by sentence
Let $X$ be a complex normed space. Based on the right norm derivative $\rho_{_{+}}$, we define a mapping $\rho_{_{\infty}}$ by \begin{equation*} \rho_{_{\infty}}(x,y) = \frac1\pi\int_0^{2\pi}e^{i\theta}\rho_{_{+}}(x,e^{i\theta}y)d\theta \quad(x,y\in X). \end{equation*} The mapping $\rho_{_{\infty}}$ has a good response to some geometrical properties of $X$. For instance, we prove that $\rho_{_{\infty}}(x,y)=\rho_{_{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. In addition, we define a $\rho_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $\rho_{_{\infty}}$-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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $X$ be a complex normed space. | background 1.00 | 0.07 | 0.01 | asserted 0.64 |
| s1 | Based on the right norm derivative $\rho_{_{+}}$, we define a mapping $\rho_{_{\infty}}$ by \begin{equation*} \rho_{_{\infty}}(x,y) = \frac1\pi\int_0^{2\pi}e^{i\theta}\rho_{_{+}}(x,e^{i\theta}y)d\theta \quad(x,y\in X). | contribution 0.65 | 0.90 | 0.06 | asserted 0.98 |
| s2 | \end{equation*} The mapping $\rho_{_{\infty}}$ has a good response to some geometrical properties of $X$. | interpretation 0.63 | 0.61 | 0.06 | asserted 0.73 |
| s3 | For instance, we prove that $\rho_{_{\infty}}(x,y)=\rho_{_{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. | result 1.00 | 0.93 | 0.29 | asserted 1.00 |
| s4 | In addition, we define a $\rho_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $\rho_{_{\infty}}$-orthogonality has to be a scalar multiple of an isometry. | result 0.81 | 0.95 | 0.44 | asserted 1.00 |
| s5 | A number of challenging problems in the geometry of complex normed spaces are also discussed. | contribution 0.68 | 0.53 | 0.04 | asserted 0.74 |
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Original abstract
Let $X$ be a complex normed space. Based on the right norm derivative $\rho_{_{+}}$, we define a mapping $\rho_{_{\infty}}$ by \begin{equation*} \rho_{_{\infty}}(x,y) = \frac1\pi\int_0^{2\pi}e^{i\theta}\rho_{_{+}}(x,e^{i\theta}y)d\theta \quad(x,y\in X). \end{equation*} The mapping $\rho_{_{\infty}}$ has a good response to some geometrical properties of $X$. For instance, we prove that $\rho_{_{\infty}}(x,y)=\rho_{_{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. In addition, we define a $\rho_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $\rho_{_{\infty}}$-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.
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