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)
Composition operators on analytic Lipschitz spaces. (English) Zbl 0793.47037
Let $D$ denote the unit disk $\{z: \vert z\vert\le 1\}$ of the complex plane. If $f$ is an analytic function on $D$ and $0<q<1$, then $\Vert f\Vert\sb q$ is defined by $$\Vert f\Vert\sb q= \sup\bigl\{\vert f(z)- f(w)\vert/\vert z-w\vert\sp q;\ z\in D,\ w\in D\bigr\},$$ and the space $A\sb q$ is defined by $A\sb q=\{f: \Vert f\Vert\sb q< \infty\}$. If $\phi: D\to D$ is analytic, then the composition operator $C\sb \phi$ is defined by $C\sb \phi(f)(x)= f\circ\phi(x)= f(\phi(x))$. The main statements of this paper indicate that (i) $C\sb \phi: A\sb q\to A\sb q$ is bounded if and only if $$\sup\left\{\left({1-\vert z\vert\sp 2\over 1-\vert\phi(z)\vert\sp 2}\right)\sp{1-q}\vert \phi'(z)\vert: \vert z\vert<1\right\} <\infty;$$ and (ii) if $\lim\sb{\vert z\vert\to 1} \left({1-\vert z\vert\sp 2\over 1- \vert\phi(z)\vert\sp 2}\right)\sp{1-q}\vert \phi'(z)\vert= 0$, then $C\sb \phi: A\sb q\to A\sb q$ is $w$-compact. Notes indicated as added in the proof of the paper state that similar conclusions have been derived by different procedures in papers of {\it R. C. Roan} [Rocky Mt. J. Math., 10, 371-379 (1980; Zbl 0433.46023)] and {\it J. H. Shapiro} [Proc. Am. Math. Soc. 100, 49-57 (1987; Zbl 0622.47028)].

47B38Operators on function spaces (general)
46E15Banach spaces of continuous, differentiable or analytic functions
47B07Operators defined by compactness properties
30C20Conformal mappings of special domains
Full Text: DOI