zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Compact composition operators on noncompact Lipschitz spaces. (English) Zbl 1279.47042

Let (X,d) be a metric space and Lip (X,d) denote the space of all bounded Lipschitz functions f on the metric space (X,d). A mapping ϕ:XX is called supercontractive if, for ϵ>0, there exists δ>0 such that d(ϕ(x),ϕ(y))<ϵd(x,y) whenever 0<d(x,y)<δ. The authors prove that C ϕ , the composition operator defined by C ϕ f=fϕ, is compact on Lip (X,d) if and only if ϕ is supercontractive and ϕ(X) is totally bounded in X.

This result improves upon previous work of H. Kamowitz and S. Scheinberg [Stud. Math. 96, No. 3, 255–261 (1990; Zbl 0713.47030)] that studied the case of compact metric spaces.

47B33Composition operators
46E15Banach spaces of continuous, differentiable or analytic functions