New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Flavia Colonna

Open Mathematics (2013)

  • Volume: 11, Issue: 1, page 55-73
  • ISSN: 2391-5455

Abstract

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Let ψ and φ be analytic functions on the open unit disk with φ() ⊆ . We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p, 1 ≤ p < ∞, and purely in terms of the symbols ψ and φ.

How to cite

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Flavia Colonna. "New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space." Open Mathematics 11.1 (2013): 55-73. <http://eudml.org/doc/269568>.

@article{FlaviaColonna2013,
abstract = {Let ψ and φ be analytic functions on the open unit disk $\mathbb \{D\}$ with φ($\mathbb \{D\}$) ⊆ $\mathbb \{D\}$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal \{D\}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p, 1 ≤ p < ∞, and $\mathcal \{D\}$ purely in terms of the symbols ψ and φ.},
author = {Flavia Colonna},
journal = {Open Mathematics},
keywords = {Weighted composition operators; Bloch space; Hardy space; Weighted Bergman space; Dirichlet space; weighted composition operators; weighted Bergman space},
language = {eng},
number = {1},
pages = {55-73},
title = {New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space},
url = {http://eudml.org/doc/269568},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Flavia Colonna
TI - New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space
JO - Open Mathematics
PY - 2013
VL - 11
IS - 1
SP - 55
EP - 73
AB - Let ψ and φ be analytic functions on the open unit disk $\mathbb {D}$ with φ($\mathbb {D}$) ⊆ $\mathbb {D}$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal {D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p, 1 ≤ p < ∞, and $\mathcal {D}$ purely in terms of the symbols ψ and φ.
LA - eng
KW - Weighted composition operators; Bloch space; Hardy space; Weighted Bergman space; Dirichlet space; weighted composition operators; weighted Bergman space
UR - http://eudml.org/doc/269568
ER -

References

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