We characterize finite codimensional linear isometries on two spaces, C (n)[0; 1] and Lip [0; 1], where C (n)[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C (n)[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.
@article{bwmeta1.element.doi-10_2478_s11533-010-0082-8,
author = {Hironao Koshimizu},
title = {Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {139-146},
zbl = {1243.46006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0082-8}
}
Hironao Koshimizu. Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions. Open Mathematics, Tome 9 (2011) pp. 139-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0082-8/
[1] Araujo J., On the separability problem for isometric shifts on C(X), J. Funct. Anal., 2009, 256(4), 1106–1117 http://dx.doi.org/10.1016/j.jfa.2008.11.013 | Zbl 1165.46009
[2] Araujo J., Font J.J., Codimension 1 linear isometries on function algebras, Proc. Amer. Math. Soc., 1999, 127(8), 2273–2281 http://dx.doi.org/10.1090/S0002-9939-99-04718-8 | Zbl 0918.47031
[3] Cambern M., Isometries of certain Banach algebras, Studia Math., 1965, 25, 217–225 | Zbl 0196.42702
[4] Cambern M., Pathak V.D., Isometries of spaces of differentiable functions, Math. Japon., 1981, 26(3), 253–260 | Zbl 0464.46028
[5] Crownover R.M., Commutants of shifts on Banach spaces, Michigan Math. J., 1972, 19(3), 233–247 http://dx.doi.org/10.1307/mmj/1029000896 | Zbl 0228.47016
[6] Font J.J., Isometries between function algebras with finite codimensional range, Manuscripta Math., 1999, 100(1), 13–21 http://dx.doi.org/10.1007/s002290050192 | Zbl 0939.47023
[7] Font J.J., On weighted composition operators between spaces of measurable functions, Quaest. Math., 1999, 22(2), 143–148 http://dx.doi.org/10.1080/16073606.1999.9632068 | Zbl 0945.47017
[8] Gutek A., Hart D., Jamison J., Rajagopalan M., Shift operators on Banach spaces, J. Funct. Anal., 1991, 101(1), 97–119 http://dx.doi.org/10.1016/0022-1236(91)90150-4 | Zbl 0818.47028
[9] Holub J.R., On shift operators, Canad. Math. Bull., 1988, 31(1), 85–94 http://dx.doi.org/10.4153/CMB-1988-013-8 | Zbl 0658.47032
[10] Izuchi K., Douglas algebras which admit codimension 1 linear isometries, Proc. Amer. Math. Soc., 2001, 129(7), 2069–2074 http://dx.doi.org/10.1090/S0002-9939-00-05842-1 | Zbl 0977.46029
[11] Jarosz K., Isometries in semisimple, commutative Banach algebras, Proc. Amer. Math. Soc., 1985, 94(1), 65–71 | Zbl 0577.46052
[12] Jarosz K., Pathak V.D., Isometries between function spaces, Trans. Amer. Math. Soc., 1988, 305(1), 193–206 http://dx.doi.org/10.1090/S0002-9947-1988-0920154-7 | Zbl 0649.46024
[13] Jiménez-Vargas A., Villegas-Vallecillos M., Into linear isometries between spaces of Lipschitz functions, Houston J. Math., 2008, 34(4), 1165–1184 | Zbl 1169.46004
[14] Kasuga K., There are no codimension 1 linear isometries on the ball and polydisk algebras, Sci. Math. Jpn., 2001, 54(2), 387–390 and 2002, 56(1), 69–70 | Zbl 0994.32004
[15] Matsumoto T., Watanabe S., Surjective linear isometries of the domain of a *-derivation equipped with the Cambern norm, Math. Z., 1999, 230(1), 185–200 http://dx.doi.org/10.1007/PL00004686 | Zbl 0927.47021
[16] Pathak V.D., Isometries of C (n)[0; 1], Pacific J. Math., 1981, 94(1), 211–222 | Zbl 0393.46042
[17] Rajagopalan M., Sundaresan K., Backward shifts on Banach spaces C(X), J. Math. Anal. Appl., 1996, 202(2), 485–491 http://dx.doi.org/10.1006/jmaa.1996.0329 | Zbl 0878.47014
[18] Rajagopalan M., Sundaresan K., Backward shifts on Banach spaces C(X), II, In: Proceedings of the Tennessee Topology Conference, Nashville, 1996, World Scientific, River Edge, 1997, 199–205 | Zbl 0913.46017
[19] Rao N.V., Roy A.K., Linear isometries of some function spaces, Pacific J. Math., 1971, 38(1), 177–192 | Zbl 0218.46026
[20] Rudin W., Functional Analysis, 2nd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1991
[21] Takagi H., Koshimizu H., Ariizumi H., Backward shifts on function algebras, J. Math. Anal. Appl. (in press), DOI: 10.1016/j.jmaa.2010.10.012 | Zbl 1217.47066
[22] Takahasi S.-E., Okayasu T., A note on finite codimensional linear isometries of C(X) into C(Y), Internat. J. Math. Math. Sci., 1995, 18(4), 677–680 http://dx.doi.org/10.1155/S016117129500086X | Zbl 0844.46008
[23] Takayama T., Wada J., Isometric shift operators on the disc algebra, Tokyo J. Math., 1998, 21(1), 115–120 http://dx.doi.org/10.3836/tjm/1270041989 | Zbl 0929.47018