Functional Analysis and Its Applications ( IF 0.4 ) Pub Date : 2021-09-08 , DOI: 10.1134/s0016266321010081 Longfa Sun 1
Abstract
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
$$|\langle\delta_k,x\rangle-\langle\phi,\Theta(x)\rangle|\le3\varepsilon/2\quad\text{for all }\,x\in C(K)^+\cup-C(K)^+.$$This result is used to prove new stability theorems for the symmetrization \(\Theta\) of \(f\).
中文翻译:
连续函数空间正锥体上$$\varepsilon$$-等距的对称化
摘要
设\(K\)是一个紧致的 Hausdorff 空间,\(C(K)\)是\(K\)上的所有连续函数的实巴拿赫空间,并赋予了最高范数,并且\(C(K)^+ \)是\(C(K)\)的正锥体。一般\(\varepsilon\)的对称化\(\Theta=(f(\,\boldsymbol\cdot\,)-f(-\;\boldsymbol\cdot\,)/2\)的弱稳定性结果-isometry \(f\)从\(C(K)^+\cup-C(K)^+\)到 Banach 空间\(Y\)得到: 对于任何元素\(k\in K\),存在一个\(\phi\in S_{Y^\ast}\)使得
$$|\langle\delta_k,x\rangle-\langle\phi,\Theta(x)\rangle|\le3\varepsilon/2\quad\text{for all }\,x\in C(K)^+ \cup-C(K)^+.$$该结果用于证明\(f\)的对称化\(\Theta\)的新稳定性定理。