1. CMB Online first
|How Lipschitz Functions Characterize the Underlying Metric Spaces|
Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that both $X,Y$ are realcompact, or both $E,F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z(f)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if \[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\] or \[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\] respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim E =\dim F\lt +\infty$, is a weighted composition operator $(Tf)(y)=J_y(f(\tau(y)))$. We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.
Keywords:(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps
Categories:46E40, 54D60, 46E15
2. CMB 2010 (vol 53 pp. 466)
|Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions|
In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.
Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuity
Categories:47B38, 46E15, 46E40, 46H40, 47B33
3. CMB 2004 (vol 47 pp. 49)
|The Essential Norm of a Bloch-to-$Q_p$ Composition Operator |
The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are subspaces of $\BMOA$ for $0
4. CMB 1999 (vol 42 pp. 118)
|Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$ |
For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.
Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-ideals