scholarly journals The domain space of an analytic composition operator

1999 ◽  
Vol 66 (1) ◽  
pp. 56-65 ◽  
Author(s):  
Thomas Domenig ◽  
Hans Jarchow ◽  
Reinhard Riedl
Keyword(s):  
Composition Operator ◽  
Hilbert Spaces ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Space Of Analytic Functions ◽  
Domain Space ◽  
Systematic Fashion ◽  
Order Boundedness

AbstractIn this paper we show that, for analytic composition operators between weighted Bergman spaces (including Hardy spaces) and as far as boundedness, compactness, order boundedness and certain summing properties of the adjoint are concerned, it is possible to modify domain spaces in a systematic fashion: there is a space of analytic functions which embeds continuously into each of the spaces under consideration and on which the above properties of the operator are decided.A remarkable consequence is that, in the setting of composition operators between weighted Bergman spaces, the properties in question can be identified as properties of the operator as a map between appropriately chosen Hilbert spaces.

scholarly journals COMPACT DIFFERENCES OF COMPOSITION OPERATORS

2008 ◽  
Vol 77 (1) ◽  
pp. 161-165 ◽  
Author(s):  
ELKE WOLF
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Infinite Order ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Difference

AbstractLet ϕ and ψ be analytic self-maps of the open unit disk. Each of them induces a composition operator, Cϕ and Cψ respectively, acting between weighted Bergman spaces of infinite order. We show that the difference Cϕ−Cψ is compact if and only if both operators are compact or both operators are not compact and the pseudohyperbolic distance of the functions ϕ and ψ tends to zero if ∣ϕ(z)∣→1 or ∣ψ(z)∣→1.

scholarly journals COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON BERGMAN SPACES IN THE BALL

2010 ◽  
Vol 89 (3) ◽  
pp. 407-418 ◽  
Author(s):  
XIANG DONG YANG ◽  
LE HAI KHOI
Keyword(s):  
Unit Ball ◽  
Composition Operator ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
Necessary And Sufficient Conditions ◽  
Weighted Bergman Spaces ◽  
Finite Sum ◽  
Necessary And Sufficient

AbstractWe obtain necessary and sufficient conditions for the compactness of differences of composition operators acting on the weighted Bergman spaces in the unit ball. A representation of a composition operator as a finite sum of composition operators modulo compact operators is also studied.

scholarly journals Composition operators on weighted Bergman spaces of a half-plane

2011 ◽  
Vol 54 (2) ◽  
pp. 373-379 ◽  
Author(s):  
Sam J. Elliott ◽  
Andrew Wynn
Keyword(s):  
Spectral Radius ◽  
Half Plane ◽  
Composition Operator ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Angular Derivative ◽  
The Right

AbstractWe use induction and interpolation techniques to prove that a composition operator induced by a map ϕ is bounded on the weighted Bergman space $\mathcal{A}^2_\alpha(\mathbb{H})$ of the right half-plane if and only if ϕ fixes the point at ∞ non-tangentially and if it has a finite angular derivative λ there. We further prove that in this case the norm, the essential norm and the spectral radius of the operator are all equal and are given by λ(2+α)/2.

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