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Into linear isometries between spaces of Lipschitz functions. (English) Zbl 1169.46004

Let \(X\) and \(Y\) be compact metric spaces and let \({\mathbb K}\) be either \({\mathbb R}\) or \({\mathbb C}\). W. Holsztynski [Stud. Math. 26, 133–136 (1966; Zbl 0156.36903)] proved an important generalization of the classical Banach-Stone theorem: if \(T\) is a linear isometry from \(C(X)\) into \(C(Y)\), then there exists a closed subset \(Y_0\) of \(Y\), a continuous map \(\varphi\) from \(Y_0\) onto \(X\) and a function \({\tau\in C(Y)}\), \({|\tau(y)|=1}\) for all \({y\in Y_0}\), \({\|\tau\|=1}\), such that
\[ Tf(y)=\tau(y) f(\varphi(y)) \quad (f\in C(X),\;y\in Y_0). \]
The object of the paper under review is to show that Holsztynski’s theorem has a natural formulation in the context of the spaces \(\text{Lip}(X)\). The space \(\text{Lip}(X)\) is the Banach space of all Lipschitz functions \(f\) from \(X\) into \({\mathbb K}\), with the norm \({\|f\| = \max \{\|f\|_{\infty}, L(f)\} }\), where \({\|f\|_{\infty}}\) is the supremum norm of \(f\) and \(L(f)\) is the Lipschitz constant of \(f\). The authors consider linear isometries \(T\) from \(\text{Lip}(X)\) into \(\text{Lip}(Y)\) for which \(T1_X\) is a contraction (i.e., \(L(T1_X)<1\)), where \(1_X\) denotes the function constantly equal to \(1\) on \(X\).
The main theorem (Theorem 2.4) states that any linear isometry \(T\) from \(\text{Lip}(X)\) into \(\text{Lip}(Y)\) for which \(T1_X\) is a contraction, is essentially a weighted composition operator
\[ Tf(y)=\tau(y)f(\varphi(y)) \quad (f\in \text{Lip}(X),\;y\in Y_0), \]
where \(Y_0\) is a closed subset of \(Y\), \(\varphi\) is a Lipschitz map from \(Y_0\) onto \(X\) with \({L(\varphi)\leq \max\{1, \text{diam}(X)\}}\) and \(\tau\) is a function of \(\text{Lip}(Y)\) with \(\|\tau\|=1\) and \({|\tau(y)|=1}\) for all \({y\in Y_0}\). Namely, the weight function \(\tau\) is \(T1_X\). This theorem is not true when \({L(T1_X)=1}\). The authors also improve the representation in the case of onto linear isometries and classify codimension 1 linear isometries in two types.

MSC:

46B04 Isometric theory of Banach spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
54E35 Metric spaces, metrizability
26A16 Lipschitz (Hölder) classes
47B38 Linear operators on function spaces (general)

Citations:

Zbl 0156.36903
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