Norm Estimates for Weighted Composition Operators on Spaces of Holomorphic Functions

Abstract. Boundedness of weighted composition operators Wu,ϕ acting on the classical Dirichlet space D as W h,ϕ f = h (f •ϕ) is studied in terms of the multiplier space associated to the symbol ϕ, i.e., M(ϕ) = {u ∈ D : Wu,ϕ is bounded on D}. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of Wu,ϕ in D whenever ϕ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko [13] to the context of the Dirichlet space.

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“…There is a unitary mapping The following is a straightforward Corollary of [4,Thm. 3.2], given that the operator…”

Section: The Next Results Identifies Invertible Weighted Composition Omentioning

confidence: 99%

Weighted composition operators on the Dirichlet space: boundedness and spectral properties

2015

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“…We begin with the following result, which is essentially due to Kacnelson [13], see also [11,Theorem 2.1]. For completeness, we provide a proof.…”

Section: Multiplier Inclusionsmentioning

confidence: 94%

Radially Weighted Besov Spaces and the Pick Property

Aleman

Hartz

McCarthy

et al. 2019

Trends in Mathematics

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For s ∈ R the weighted Besov space on the unit ballHere R s is a power of the radial derivative operator R = d i=1 z i ∂ ∂zi , V denotes Lebesgue measure, and ω is a radial weight function not supported on any ball of radius < 1.Our results imply that for all such weights ω and ν, every bounded column multiplication operator B s ω → B t ν ⊗ ℓ 2 induces a bounded row multiplier B s ω ⊗ℓ 2 → B t ν . Furthermore we show that if a weight ω satisfies that for some α > −1 the ratio ω(z)/(1 − |z| 2 ) α is nondecreasing for t 0 < |z| < 1, then B s ω is a complete Pick space, whenever s ≥ (α + d)/2.

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“…In particular, they have found the expression for the norm of composition operators for these spaces and shown that the composition operators must never be compact in this context. Some further results concerning so-called Zen spaces (a generalisation of weighted Bergman spaces) and weighted composition operators have been obtained in [3] and we aim to extend it in this article.…”

Section: Introductionmentioning

confidence: 93%

Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane

Kucik

2017

Complex Anal. Oper. Theory

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In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of causal weighted composition operators on these spaces are related to each other and use it to show that an (unweighted) composition operator C ϕ is bounded on a Zen space if and only if ϕ has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces.

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Norm Estimates for Weighted Composition Operators on Spaces of Holomorphic Functions

Cited by 26 publications

References 28 publications

Weighted composition operators on the Dirichlet space: boundedness and spectral properties

Weighted composition operators on the Dirichlet space: boundedness and spectral properties

Radially Weighted Besov Spaces and the Pick Property

Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane

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