Geodesic
On the Symmetrizations of $\varepsilon$-Isometries on Positive Cones of Continuous Function Spaces
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 1, pp. 93-97

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $K$ be a compact Hausdorff space, $C(K)$ be the real Banach space of all continuous functions on $K$ endowed with the supremum norm, and $C(K)^+$ be the positive cone of $C(K)$. A weak stability result for the symmetrization $\Theta=(f(\,\boldsymbol\cdot\,)-f(-\;\boldsymbol\cdot\,)/2$ of a general $\varepsilon$-isometry $f$ from $C(K)^+\cup-C(K)^+$ to a Banach space $Y$ is obtained: For any element $k\in K$, there exists a $\phi\in S_{Y^\ast}$ such that \begin{equation*} |\langle\delta_k,x\rangle-\langle\phi,\Theta(x)\rangle|\le3\varepsilon/2\quad\text{for all }\,x\in C(K)^+\cup-C(K)^+. \end{equation*} This result is used to prove new stability theorems for the symmetrization $\Theta$ of $f$.
Keywords: symmetrization of $\varepsilon$-isometry, stability, function space. 
@article{FAA_2021_55_1_a7,
     author = {Longfa Sun},
     title = {On the {Symmetrizations} of $\varepsilon${-Isometries} on {Positive} {Cones} of {Continuous} {Function} {Spaces}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {93--97},
     publisher = {mathdoc},
     volume = {55},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a7/}
}
TY  - JOUR
AU  - Longfa Sun
TI  - On the Symmetrizations of $\varepsilon$-Isometries on Positive Cones of Continuous Function Spaces
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2021
SP  - 93
EP  - 97
VL  - 55
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a7/
LA  - ru
ID  - FAA_2021_55_1_a7
ER  - 
%0 Journal Article
%A Longfa Sun
%T On the Symmetrizations of $\varepsilon$-Isometries on Positive Cones of Continuous Function Spaces
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2021
%P 93-97
%V 55
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a7/
%G ru
%F FAA_2021_55_1_a7
Longfa Sun. On the Symmetrizations of $\varepsilon$-Isometries on Positive Cones of Continuous Function Spaces. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 1, pp. 93-97. http://geodesic.mathdoc.fr/item/FAA_2021_55_1_a7/