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Jesus Araujo

Publications and source records attributed to Jesus Araujo.

11 recordsLinked to original sources
arXiv2010

Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions $\mathrm{Lip}(X,E)$ and $\mathrm{Lip}(Y,F)$, for strictly convex normed spaces $E$ and $F$ and metric spaces $X$ and $Y$: \begin{enumerate} \item Characterize those base spaces $X$ and $Y$ for which all isometries are weighted composition maps. \item Give a condition independent of base spaces under which all isometries are weighted composition maps. \item Provide the general form of an isometry, both when it is a weighted composition map and when it is not. \end{enumerate} In particular, we prove that requirements of completeness on $X$ and $Y$ are not necessary when $E$ and $F$ are not complete, which is in sharp contrast with results known in the scalar context.

math.FA↗
arXiv2008

Biseparating maps between Lipschitz function spaces

For complete metric spaces $X$ and $Y$, a description of linear biseparating maps between spaces of vector-valued Lipschitz functions defined on $X$ and $Y$ is provided. In particular it is proved that $X$ and $Y$ are bi-Lipschitz homeomorphic, and the automatic continuity of such maps is derived in some cases. Besides, these results are used to characterize the separating bijections between scalar-valued Lipschitz function spaces when $Y$ is compact.

math.FA↗
arXiv2008

Stability and instability of weighted composition operators

Let $ε>0$. A continuous linear operator $T:C(X) \ra C(Y)$ is said to be {\em $ε$-disjointness preserving} if $\vc (Tf)(Tg)\vd_{\infty} \le ε$, whenever $f,g\in C(X)$ satisfy $\vc f\vd_{\infty} =\vc g\vd_{\infty} =1$ and $fg\equiv 0$. In this paper we address basically two main questions: 1.- How close there must be a weighted composition operator to a given $ε$-disjointness preserving operator? 2.- How far can the set of weighted composition operators be from a given $ε$-disjointness preserving operator? We address these two questions distinguishing among three cases: $X$ infinite, $X$ finite, and $Y$ a singleton ($ε$-disjointness preserving functionals). We provide sharp stability and instability bounds for the three cases.

math.FA↗
arXiv2007

Multiplicative bijections of semigroups of interval-valued continuous functions

We characterize all compact and Hausdorff spaces $X$ which satisfy that for every multiplicative bijection $ϕ$ on $C(X, I)$, there exist a homeomorphism $μ: X \to X$ and a continuous map $p: X \to (0, +\infty)$ such that $$ϕ(f) (x) = f(μ(x))^{p(x)}$$ for every $f \in C(X,I)$ and $x \in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. {\bf 134} (2006), 1065-1075). Some related results on other semigroups of functions are also given.

math.FA↗
arXiv2007

Examples and counterexamples of type I isometric shifts

We provide examples of nonseparable spaces $X$ for which C(X) admits an isometric shift of type I, which solves in the negative a problem proposed by Gutek {\em et al.} (J. Funct. Anal. {\bf 101} (1991), 97-119). We also give two independent methods for obtaining separable examples. The first one allows us in particular to construct examples with infinitely many nonhomeomorphic components in a subset of the Hilbert space $\ell^2$. The second one applies for instance to sequences adjoined to any n-dimensional compact manifold (for $n \ge 2$) or to the Sierpiński curve. The combination of both techniques lead to different examples involving a convergent sequence adjoined to the Cantor set: one method for the case when the sequence converges to a point in the Cantor set, and the other one for the case when it converges outside.

math.FA↗
arXiv2001

Automatic continuity of biseparating maps

We prove that a biseparating map between spaces of vector-valued continuous functions is usually automatically continuous. However, we also discuss special cases when it is not true.

math.FA↗
arXiv2001

Realcompactness and spaces of vector-valued functions

It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.

math.GN↗
arXiv2001

Realcompactness and Banach-Stone theorems

For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.

math.FA↗
arXiv2001

Isometric shifts and metric spaces

Let M be a complete metric space. It is proved that if the space or scalar-valued bounded continuous functions on M admits an isometric shift, then M is separable.

math.FA↗