Generalized Composition Operators on Weighted Hilbert Spaces of Analytic Functions

2017 ◽  
Vol 10 ◽  
pp. 1-13 ◽  
Author(s):  
Waleed Al-Rawashdeh
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Carleson Measures ◽  
Open Unit ◽  
Open Unit Disk ◽  
Admissible Weight ◽  
Spaces Of Analytic Functions

Letφbe an analytic self-map of the open unit disk D andgbe an analytic function on D. The generalized composition operator induced by the mapsgandφis defined by the integral operatorI(g,φ)f(z) =∫0zf′(φ(ς))g(ς)dς. Given an admissible weightω, the weighted Hilbert spaceHωconsists of all analytic functionsfsuch that ∥f∥2Hω= |f(0)|2+∫D|f′(z)|2ω(z)dA(z) is finite. In this paper, we characterize the boundedness and compactness of the generalized composition operators on the spaceHωusing theω-Carleson measures. Moreover, we give a lower bound for the essential norm of these operators.

scholarly journals Composition Operators on Some Banach Spaces of Harmonic Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Zygmund Space ◽  
Spaces Of Analytic Functions

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.

Boundary vs. interior conditions associated with weighted composition operators

2014 ◽  
Vol 12 (5) ◽  
Author(s):  
Kei Izuchi ◽  
Yuko Izuchi ◽  
Shûichi Ohno
Keyword(s):  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weight Functions ◽  
Open Unit ◽  
Weighted Composition Operators ◽  
Open Unit Disk ◽  
Weighted Composition ◽  
The Difference ◽  
Harmonic And Analytic Functions

AbstractAssociated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk $$\mathbb{D}$$, we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior $$\mathbb{D}$$ and on the boundary $$\partial \mathbb{D}$$ respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

scholarly journals Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions

2021 ◽  
Vol 280 (3) ◽  
pp. 108834
Author(s):  
Pascal Lefèvre ◽  
Daniel Li ◽  
Hervé Queffélec ◽  
Luis Rodríguez-Piazza
Keyword(s):  
Hilbert Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Singular Numbers ◽  
Spaces Of Analytic Functions

scholarly journals BELLWETHERS FOR BOUNDEDNESS OF COMPOSITION OPERATORS ON WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS

2009 ◽  
Vol 86 (3) ◽  
pp. 305-314 ◽  
Author(s):  
PAUL S. BOURDON
Keyword(s):  
Analytic Functions ◽  
Composition Operators ◽  
Open Unit ◽  
Sufficient Condition ◽  
Angular Derivative ◽  
Open Unit Disc ◽  
Weighted Banach Space ◽  
Growth Space ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractLet 𝔻 be the open unit disc, let v:𝔻→(0,∞) be a typical weight, and let Hv∞ be the corresponding weighted Banach space consisting of analytic functions f on 𝔻 such that $\|f\|_v:=\sup _{z\in \mathbb {D}}v(z)\lvert f(z)\rvert \less \infty $. We call Hv∞ a typical-growth space. For ϕ a holomorphic self-map of 𝔻, let Cφ denote the composition operator induced by ϕ. We say that Cφ is a bellwether for boundedness of composition operators on typical-growth spaces if for each typical weight v, Cφ acts boundedly on Hv∞ only if all composition operators act boundedly on Hv∞. We show that a sufficient condition for Cφ to be a bellwether for boundedness is that ϕ have an angular derivative of modulus less than 1 at a point on ∂𝔻. We raise the question of whether this angular-derivative condition is also necessary for Cφ to be a bellwether for boundedness.

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