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Biseparating maps on generalized Lipschitz spaces. (English) Zbl 1194.46040

Let $X,Y$ be complete metric spaces and $E,F$ be Banach spaces. In the paper under review, the author considers spaces of functions from $X$ into $E$ (and from $Y$ into $F$) determined by a generalized modulus of continuity. These spaces are called generalized Lipschitz spaces. This notion serves to unify the study of spaces of Lipschitz, little Lipschitz and uniformly continuous functions. The main purpose of this paper is to characterize all biseparating operators between generalized Lipschitz spaces. Two functions $f$ and $g$ defined on $X$ with values in $E$ are said to be disjoint if, for all $x\in X$, either $f\left(x\right)=0$ or $g\left(x\right)=0$. A map $T$ between vector-valued function spaces is called separating if $T$ maps disjoint functions to disjoint functions. It is biseparating if $T$ is invertible and both $T$ and ${T}^{-1}$ are separating.

A generalized Lipschitz space of real-valued functions ${\text{Lip}}_{{\Sigma }}\left(X\right)$ is said to be Lipschitz normal if, for every pair of subsets $U,V$ of $X$ with $d\left(U,V\right)>0$, there exists $f\in {\text{Lip}}_{{\Sigma }}\left(X\right)$, $0\le f\le 1$, such that $f=0$ on $U$ and $f=1$ on $V$. A generalized Lipschitz space ${\text{Lip}}_{{\Sigma }}\left(X,E\right)$ of $E$-valued functions is called Lipschitz normal if ${\text{Lip}}_{{\Sigma }}\left(X\right)$ is.

The main result of the paper is Theorem 16, which goes as follows. Suppose that ${\text{Lip}}_{{\Sigma }}\left(X,E\right)$ and ${\text{Lip}}_{{\Sigma }}\left(Y,F\right)$ are generalized Lipschitz spaces which are Lipschitz normal. If $T:{\text{Lip}}_{{\Sigma }}\left(X,E\right)\to {\text{Lip}}_{{\Sigma }}\left(Y,F\right)$ is a linear biseparating map, then there exists a homeomorphism $h:X\to Y$ and, for each $y\in Y$, a vector space isomorphism ${S}_{y}:E\to F$ such that $Tf\left(y\right)={S}_{y}\left(f\left({h}^{-1}\left(y\right)\right)\right)$ for all $y\in Y$.

Also, the continuity properties of the family ${S}_{y}$ and of the operator $T$ with respect to suitable topologies are studied. Note that the functions in generalized Lipschitz spaces (as well as the metric spaces $X$ and $Y$) may be unbounded. The usual compactification procedures for the spaces $X$ and $Y$ are not required.

##### MSC:
 46E15 Banach spaces of continuous, differentiable or analytic functions 47B38 Operators on function spaces (general) 47B33 Composition operators