On Perturbed Isometries Between the Positive Cones of Certain Continuous Function Spaces
Tóm tắt
Let X, Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C(X) be the real Banach space of all continuous functions on X, and
$$C_+(X)$$
be the positive cone of C(X). In this paper, we show that if there exists a
$$\delta $$
-surjective
$$\varepsilon $$
-isometry
$$F: C_+(X)\rightarrow C_+(Y)$$
, then X and Y are homeomorphic. Moreover, we show that there exists a unique additive surjective isometry
$$V:C_+(X)\rightarrow C_+(Y)$$
(the restriction of a linear surjective isometry
$$U:C(X)\rightarrow C(Y)$$
induced by the homeomorphism) such that
$$\begin{aligned} \Vert F(f)-V(f)\Vert \le 2\varepsilon ,\quad \textrm{for}\; \textrm{all}\;\;f\in C_+(X). \end{aligned}$$
This can be regarded as a localized generalization of the Banach–Stone theorem for compact Hausdorff perfectly normal spaces.
Tài liệu tham khảo
Alestalo, P., Trotsenko, D.A., Väisälä, J.: Isometric approximation. Isr. J. Math. 125, 61–82 (2001)
Banach, S.: Théorie des Opérations Lineaires, Reprinted. Clelsea Publishing Company, New York (1963)
Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis I. Amer. Math. Soc. Colloquium Publications, Vol. 48. Amer. Math. Soc., Providence (2000)
Cheng, L., Cheng, Q., Tu, K., Zhang, J.: A universal theorem for stability of \(\varepsilon \)-isometries on Banach spaces. J. Funct. Anal. 269(1), 199–214 (2015)
Cheng, L., Dong, Y., Zhang, W.: On stability of nonsurjective \(\varepsilon \)-isometries of Banach spaces. J. Funct. Anal. 264(3), 713–734 (2013)
Cheng, L., Shen, Q., Zhang, W., Zhou, Y.: More on stability of almost surjective \(\varepsilon \)-isometries of Banach spaces. Sci. China Math. 60(2), 277–284 (2017)
Dilworth, S.J.: Approximate isometries on finite-dimensional normed spaces. Bull. Lond. Math. Soc. 31, 471–476 (1999)
Dugundji, J.: Topology. Allyn and Bacon, Boston (1966)
Galego, E.M., da Silva, A.L.P.: An optimal nonlinear extension of Banach–Stone theorem. J. Funct. Anal. 271, 2166–2176 (2016)
Garrido, M.I., Jaramillo, J.A.: Variations on the Banach–Stone theorem. Extracta Math. 17, 351–383 (2002)
Hyers, D.H., Ulam, S.M.: On approximate isometries. Bull. Am. Math. Soc. 51, 288–292 (1945)
Mazur, S., Ulam, S.: Sur les transformations isométriques d’espaces vectoriels normés. C. R. Acad. Sci. Paris 194, 946–948 (1932)
Omladič, M., Šemrl, P.: On non linear perturbations of isometries. Math. Ann. 303, 617–628 (1995)
Phelps, R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)
Šemrl, P., Väisälä, J.: Nonsurjective nearisometries of Banach spaces. J. Funct. Anal. 198, 268–278 (2003)
Stone, M.H.: Applications of the theory of boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)
Sun, L.: Hyers–Ulam stability of \(\varepsilon \)-isometries between the positive cones of \(L^p\)-spaces. J. Math. Anal. Appl. (2020). https://doi.org/10.1016/j.jmaa.2020.124014
Sun, L.: A note on stability of non-surjective \(\varepsilon \)-isometries between the positive cones of \(L^p\)-spaces. Indian J. Pure Appl. Math. (2021). https://doi.org/10.1007/s13226-021-00047-2
Väisälä, J.: Isometric approximation property in Euclidean spaces. Isr. J. Math. 128, 1–27 (2002)
Väisälä, J.: Isometric approximation property of unbounded sets. Result Math. 43, 359–372 (2003)
Vestfrid, I.A.: \(\varepsilon \)-isometries in Euclidean spaces. Nonlinear Anal. 63, 1191–1198 (2005)
Vestfrid, I.A.: \(\varepsilon \)-isometries in \(\ell _\infty ^n\). Nonlinear Funct. Anal. Appl. 12, 433–438 (2007)
Vestfrid, I.A.: Near-isometries on unit sphere. Ukr. Math. J. 72, 663–670 (2020)
Zhou, Y., Zhang, Z., Liu, C.: On linear isometries and \(\varepsilon \)-isometries between Banach spaces. J. Math. Anal. App. 435, 754–764 (2016)
Zhou, Y., Zhang, Z., Liu, C.: On representation of isometric embeddings between Hausdorff metric spaces of compact convex subsets. Houston J. Math. 44, 917–925 (2018)