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On the Symmetrizations of \(\varepsilon\)-Isometries on Positive Cones of Continuous Function Spaces

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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 \(f\).

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Acknowledgments

The author is grateful to the referee for constructive comments and helpful suggestions.

Funding

The author is supported by the Fundamental Research Funds for the Central Universities 2019MS121.

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Correspondence to Longfa Sun.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 93–97 https://doi.org/10.4213/faa3751.

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Sun, L. On the Symmetrizations of \(\varepsilon\)-Isometries on Positive Cones of Continuous Function Spaces. Funct Anal Its Appl 55, 75–79 (2021). https://doi.org/10.1134/S0016266321010081

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  • DOI: https://doi.org/10.1134/S0016266321010081

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