### Linear Isometries on Spaces of Continulously Differentiable and Lipschitz Continuous Functions

Hironao Koshimizue
Source: Nihonkai Math. J. Volume 22, Number 2 (2011), 39-47.

#### Abstract

We characterize the surjective linear isometries on $C^{(n)} [0, 1]$ and Lip$[0, 1]$. Here $C^{(n)} [0, 1]$ denotes the Banach space of $n$-times continuously differentiable functions on $[0, 1]$ equipped with the norm \begin{equation*} \| f \| = \sum_{k=0}^{n-1}|f^{(k)} (0)| + \sup _{x \in [0, 1]} | f^{(n)} (x) | \quad (f \in C^{(n)} [0, 1]) , \end{equation*} and Lip$[0, 1]$ denotes the Banach space of Lipschitz continuous functions on $[0, 1]$ equipped with the norm \begin{equation*} \| f \| = |f(0)| + \mathop{\operatorname{ess \, sup}} _{x \in [0, 1]} | f' (x) | \quad (f \in {\rm Lip}[0, 1]). \end{equation*}

First Page:
Primary Subjects: 46B04
Secondary Subjects: 46E15
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Permanent link to this document: http://projecteuclid.org/euclid.nihmj/1339696713
Zentralblatt MATH identifier: 06050968
Mathematical Reviews number (MathSciNet): MR2952819

### References

A. Browder, Introduction to Function Algebras, Benjamin, 1969.
Mathematical Reviews (MathSciNet): MR246125
Zentralblatt MATH: 0199.46103
M. Cambern, Isometries of certain Banach algebras, Studia Math., 25 (1965), 217–225.
Mathematical Reviews (MathSciNet): MR172129
M. Cambern and V. D. Pathak, Isometries of spaces of differentiable functions, Math. Japon., 26 (1981), 253–260.
Mathematical Reviews (MathSciNet): MR624212
Zentralblatt MATH: 0464.46028
J. B. Conway. A Course in Functional Analysis, 2nd ed., Springer-Varlag, 1990.
Mathematical Reviews (MathSciNet): MR1070713
R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC, 2003.
Mathematical Reviews (MathSciNet): MR1957004
J. J. Font, On weighted composition operators between spaces of measurable functions, Quaest. Math., 22 (1999), 143–148.
Mathematical Reviews (MathSciNet): MR1728511
Digital Object Identifier: doi:10.1080/16073606.1999.9632068
K. Jarosz and V. D. Pathak, Isometries between function spaces, Trans. Amer. Math. Soc., 305 (1988), 193–206.
Mathematical Reviews (MathSciNet): MR920154
Zentralblatt MATH: 0649.46024
Digital Object Identifier: doi:10.1090/S0002-9947-1988-0920154-7
A. Jiménez-Vargas and M. Villegas-Vallecillos, Into linear isometries between spaces of Lipschitz functions, Houston J. Math., 34 (2008), 1165–1184.
Mathematical Reviews (MathSciNet): MR2465373
Zentralblatt MATH: 1169.46004
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. I, Academic Press, New York, 1983.
Mathematical Reviews (MathSciNet): MR719020
Zentralblatt MATH: 0888.46039
H. Koshimizu, Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions, Cent. Eur. J. Math., 9 (2011), 139–146.
Mathematical Reviews (MathSciNet): MR2753887
Zentralblatt MATH: 05866010
Digital Object Identifier: doi:10.2478/s11533-010-0082-8
T. Matsumoto and S. Watanabe, Surjective linear isometries of the domain of a *-derivation equipped with the Cambern norm, Math. Z., 230 (1999), 185–200.
Mathematical Reviews (MathSciNet): MR1671878
Zentralblatt MATH: 0927.47021
Digital Object Identifier: doi:10.1007/PL00004686
V. D. Pathak, Isometries of $C^{(n)} [0, 1]$, Pacific J. Math., 94 (1981), 211–222.
Mathematical Reviews (MathSciNet): MR625820
Project Euclid: euclid.pjm/1102735919
N. V. Rao and A. K. Roy, Linear isometries of some function spaces, Pacific J. Math., 38 (1971), 177–192.
Mathematical Reviews (MathSciNet): MR308763
Project Euclid: euclid.pjm/1102970270
W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991.
Mathematical Reviews (MathSciNet): MR1157815