Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T14:02:19.743Z Has data issue: false hasContentIssue false
  • English
  • Français

AN EXTENSION OF THE BANACH–STONE THEOREM

Part of: Calculus on manifolds; nonlinear operators Linear function spaces and their duals Special classes of linear operators

Published online by Cambridge University Press:  08 November 2017

MOHAMMED BACHIR
MOHAMMED BACHIR*
Affiliation:
Laboratoire SAMM 4543, Université Paris 1, Panthéon-Sorbonne, Centre P.M.F., 90 rue Tolbiac, 75634 Paris cedex 13, France email Mohammed.Bachir@univ-paris1.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish an extension of the Banach–Stone theorem to a class of isomorphisms more general than isometries in a noncompact framework. Some applications are given. In particular, we give a canonical representation of some (not necessarily linear) operators between products of function spaces. Our results are established for an abstract class of function spaces included in the space of all continuous and bounded functions defined on a complete metric space.

Keywords

Banach–Stone theoremfunction spacescomposition operatorisometries and isomorphismsFréchet and Gâteaux differentiability on Banach spacesduality

MSC classification

Primary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Secondary: 47B33: Composition operators 58C20: Differentiation theory (Gateaux, Fréchet, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 105 , Issue 1 , August 2018 , pp. 1 - 23
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Amir, D., ‘On isomorphisms of continuous function spaces’, Israel J. Math. 3 (1966), 205–210.CrossRefGoogle Scholar
Araujo, J., ‘Realcompactness and spaces of vector-valued functions’, Fund. Math. 172 (2002), 27–40.Google Scholar
Araujo, J., ‘The noncompact Banach–Stone theorem’, J. Operator Theory 55(2) (2006), 285–294.Google Scholar
Araujo, J. and Font, J. J., ‘Linear isometries between subspaces of continuous functions’, Trans. Amer. Math. Soc. 349 (1997), 413–428.Google Scholar
Bachir, M., ‘Sur la differentiabilité générique et le théoréme de Banach–Stone’, C. R. Math. Acad. Sci. Paris 330 (2000), 687–690.Google Scholar
Bachir, M., ‘A non-convex analogue to Fenchel duality’, J. Funct. Anal. 181 (2001), 300–312.Google Scholar
Bachir, M., ‘The inf-convolution as a law of monoid. An analogue to the Banach–Stone theorem’, J. Math. Anal. Appl. 420(1) (2014), 145–166.Google Scholar
Bachir, M. and Lancien, G., ‘On the composition of differentiable functions’, Canad. Math. Bull. 46(4) (2003), 481–494.Google Scholar
Banach, S., Théorie des Opérations Lineaires, Monografie Matematyczne, 1 (Instytut Matematyczny Polskiej Akademi Nauk, Warszawa, 1932).Google Scholar
Behrends, E., M-Structure and the Banach–Stone Theorem (Springer, Berlin, Heidelberg, 1978).Google Scholar
Behrends, E. and Cambern, M., ‘An isomorphic Banach–Stone theorem’, Studia Math. 90 (1988), 15–26.Google Scholar
Cambern, M., ‘Isomorphisms of C 0(Y) onto C 0(X)’, Pacific J. Math. 35 (1970), 307–312.CrossRefGoogle Scholar
Cengiz, B., ‘A generalization of the Banach–Stone theorem’, Proc. Amer. Math. Soc. 50 (1973), 426–430.Google Scholar
Deville, R., Godefroy, G. and Zizler, V., ‘A smooth variational principle with applications to Hamilton–Jacobi equations in infinite dimensions’, J. Funct. Anal. 111 (1993), 197–212.Google Scholar
Deville, R., Godefroy, G. and Zizler, V., Smoothness and Renormings in Banach Spaces, Pitman Monographs, 64 (Longman, London, 1993).Google Scholar
Deville, R. and Revalski, J. P., ‘Porosity of ill-posed problems’, Proc. Amer. Math. Soc. 128 (2000), 1117–1124.Google Scholar
Garrido, M. I., Gómez, J. and Jaramillo, J. A., ‘Homomorphisms on function algebras’, Canad. J. Math. 46 (1994), 734–745.Google Scholar
Garrido, M. I. and Jaramillo, J. A., ‘A Banach–Stone theorem for uniformly continuous functions’, Monatsh. Math. 131 (2000), 189–192.Google Scholar
Garrido, M. I. and Jaramillo, J. A., ‘Variations on the Banach–Stone theorem’, Extracta Math. 17(3) (2002), 351–383.Google Scholar
Godefroy, G. and Kalton, N., ‘Lipschitz-free Banach spaces’, Studia Math. 159(1) (2003), 121–141.CrossRefGoogle Scholar
Gutiérrez, J. M. and Llavona, J. G., ‘Composition operators between algebras of differentiable functions’, Trans. Amer. Math. Soc. 338(2) (1993), 769–782.Google Scholar
Jaramillo, J. A., Jimenez-Sevilla, M. and Sanchez-Gonzalez, L., ‘Characterization of a Banach–Finsler manifold in terms of the algebras of smooth functions’, Proc. Amer. Math. Soc. 142 (2014), 1075–1087.Google Scholar
Jarosz, K., ‘Small isomorphisms of C (X, E) spaces’, Pacific J. Math. 138 (1989), 295–315.Google Scholar
Jarosz, K. and Pathak, V. D., Isometries and Small Bound Isomorphisms of Function Spaces, Lecture Notes in Pure and Applied Mathematics, 136 (Marcel Dekker Inc., New York, 1992).Google Scholar
Milutin, A. A., ‘Isomorphisms of spaces of continuous functions on compacta of the power of the continuum’, Teor. Funktsi. Funktsional. Anal. Prilozhen. 2 (1966), 150–156; (in Russian).Google Scholar
Stone, M. H., ‘Applications of the theory of Boolean rings to general topology’, Trans. Amer. Math. Soc. 41 (1937), 375–481.Google Scholar
Vieira, D. M., ‘Theorems of Banach–Stone type for algebras of holomorphic functions on infinite dimensional spaces’, Math. Proc. R. Ir. Acad. 106 (2006), 97–113.CrossRefGoogle Scholar
Weaver, N., ‘Isometries of noncompact Lipschitz spaces’, Canad. Math. Bull. 38 (1995), 242–249.Google Scholar