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

A characterization of nonlinear Hölder seminorm preserving bijections

  • Arya Jamshidi
  • Fereshteh Sady
  • Azin Golbaharan


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
$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0\, f(\Psi (y))+\Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y) \end{aligned}$$
$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0 \,\overline{f(\Psi (y))}+ \Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y), \end{aligned}$$
where \(\tau =e^{i\theta }\) for some \(\theta \in [0,\pi )\).


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

Mathematics Subject Classification

Primary 47B38 Secondary 47B33 



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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