Positivity

, Volume 22, Issue 1, pp 1–16 | Cite as

A characterization of nonlinear Hölder seminorm preserving bijections

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Abstract

For a compact metric space (Xd) and \(\alpha \in (0,1)\), let \(\mathrm{Lip}^\alpha (X)\) be the linear space of all complex-valued functions f on X satisfying and \(\mathrm{lip}^\alpha (X)\) be the subspace of \(\mathrm{Lip}^\alpha (X)\) consisting of functions f with \(\lim \frac{f(x)-f(y)}{d^\alpha (x,y)} =0\) as \(d(x,y) \rightarrow 0\). In this paper, we give a characterization of a bijective map \(T:\mathrm{lip}^\alpha (X)\longrightarrow \mathrm{lip}^\alpha (Y)\), not necessarily linear, which is an isometry with respect to the Hölder seminorm \(L(\cdot )\). It is shown that there exist \(K_0>0\), a surjective map \(\Psi : Y \longrightarrow X\) with \(d^\alpha (y,z)= K_0 \, d^\alpha (\Psi (y),\Psi (z))\) for all \(y,z\in Y\), and a function \(\Lambda : \mathrm{lip}^\alpha (X) \longrightarrow {\mathbb {C}}\) (which is linear or real-linear if T is so) such that either
$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0\, f(\Psi (y))+\Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y) \end{aligned}$$
or
$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0 \,\overline{f(\Psi (y))}+ \Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y), \end{aligned}$$
where \(\tau =e^{i\theta }\) for some \(\theta \in [0,\pi )\).

Keywords

Hölder seminorm preserving maps Lipschitz functions Isometry Extreme point Weighted composition operator 

Mathematics Subject Classification

Primary 47B38 Secondary 47B33 

Notes

Acknowledgements

The authors would like to thank the referee for his/her helpful comments. The first and the second authors were partially supported by Iran National Science Foundation: INSF (Grant No. 95002593).

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Arya Jamshidi
    • 1
  • Fereshteh Sady
    • 1
  • Azin Golbaharan
    • 2
  1. 1.Department of Pure Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran
  2. 2.Department of Mathematics, Faculty of Mathematical Sciences and ComputerKharazmi UniversityTehranIran

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