zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Weighted composition operators on the Bloch space. (English) Zbl 0985.47022

Let D be the open disk of the complex plane. Then the Bloch space on D and the little Bloch space 0 on D are defined by

={f:fisanalyticonDandsup{(1-|z| 2 )|f ' (z)|,zD{<},

where f ' is the derivative of f;

0 ={f:fisanalyticonDand(1-|z| 2 )|f ' (z)|0as|z|1}·

With norm |·| defined by |f| =|f(0)|+sup{(1-|z| 2 )|f ' (z)|,zD}, is a Banach space and 0 is a closed subspace of . If u is an analytic function on D and ϕ:DD, then the linear weighted composition operator uCϕ is defined by

uCϕ(f)(z)=(ufϕ)(z)=u(z)f(ϕ(z))·

In the results of this paper, the authors derive characterizations for bounded and compact weighted composition operators. In particular, it is shown that

(1) uCϕ is bounded on the Bloch space if and only if

(i) sup{(1-|z| 2 )|u ' (z)|log(2/(1-|ϕ(z)| 2 )),zD{<;

(ii) sup{(1-|z| 2 )/(1-|ϕ(z)| 2 )}|u ' (z)ϕ ' (z)|,zD}<;

and

(2) uCϕ is compact on the Bloch space if and only if expressions of (1) converge to 0 as |ϕ(z)|1; and uCϕ is compact on 0 if and only if the expressions of (1) converge to 0 as |z|1.

(3) uCϕ is bounded on 0 if the conditions of (1) are satisfied and |u(z)ϕ ' (z)|(1-|z| 2 )0 as |z|1.


MSC:
47B33Composition operators
47B38Operators on function spaces (general)
47B07Operators defined by compactness properties