Difference of weighted composition operators

Choe, Boo Rim; Choi, Koeun; Koo, Hyungwoon; Yang, Jongho

doi:10.1016/j.jfa.2019.108401

Cited by 18 publications

(9 citation statements)

References 20 publications

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“…Shapiro and Sundberg [24] studied this problem in 1990. Very recently, Choe, Choi, Koo and Yang [3] have solved this problem. For more about difference operators, see [2,8,10,11,20].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Moreover, the Proof of Theorem 1 shows that the only requirement for = ( , p) > appearing in the statement is that for some constant C = C( , p, ) > 0 . If (z) = (1 − |z|) , any > +2 p is acceptable, and the choice = 2 +2 p converts (6) to (3), as a simple computation shows. Therefore Theorem 1 indeed generalizes [1, Theorem 1] for weights in D.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Compact differences of weighted composition operators

Liu

Rättyä

2020

Collect. Math.

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Compact differences of two weighted composition operators acting from the weighted Bergman space $$A^p_{\omega }$$ A ω p to another weighted Bergman space $$A^q_{\nu }$$ A ν q , where $$0<p\le q<\infty $$ 0 < p ≤ q < ∞ and $$\omega ,\nu $$ ω , ν belong to the class $${\mathcal {D}}$$ D of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proof a new description of q-Carleson measures for $$A^p_{\omega }$$ A ω p , with $$\omega \in {\mathcal {D}}$$ ω ∈ D , in terms of pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_{\alpha }$$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights.

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“…Shapiro and Sundberg [24] studied this problem in 1990. Very recently, Choe, Choi, Koo and Yang [3] have solved this problem. For more about difference operators, see [2,8,10,11,20].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Compact differences of weighted composition operators

Liu

Rättyä

2020

Collect. Math.

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“…Differences of composition operators have been studied ever since by many authors on several function spaces, see, for example, [1,2,6,9]. In 2004, Nieminen and Saksman [13] showed that the compactness of C φ − C ψ on the Hardy space H p , with 1 ≤ p < ∞, is independent of p. By using this, Choe et al [3] then characterized compact operators C φ − C ψ on Hardy spaces by using Carleson measures in 2020. Further, Moorhouse [11,12] characterized the compactness of C φ − C ψ on the standard weighted Bergman space A 2 α .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…[19,Theorem 3(iii)]. Therefore, it suffices to show that (i) and (iii) are equivalent and establish (3).…”

Section: Carleson Measuresmentioning

confidence: 95%

Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

Liu

Rättyä

2021

J Geom Anal

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Bounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

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“…The problem of compact difference was originally raised from the question about the component in the space of composition operators under the operator norm topology [24]. In the case of classical holomorphic function spaces such as the Hardy and the standard weighted Bergman space over the disk or the ball, many characterizations for the compactness of C ϕ − C ψ has been developed over the past decades involving the pseudo-hyperbolic distance, but no result has been given in the large Bergman space; see for example [4,13,15,17]. One of the difficulties compared to the standard weighted Bergman spaces is non-existence of the explicit form of the reproducing kernel in A 2 (ω).…”

Section: Introductionmentioning

confidence: 99%

Difference of Composition operators over Bergman spaces with exponential weights

Park¹

2021

Preprint

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In this paper, we obtain a complete characterization for the compact difference of two composition operators acting on Bergman spaces with weight ω = e −η , ∆η > 0 using the η-derived bounded distance of two analytic self maps inducing composition operators. In addition, we provide an example that supports our main result. We also study the topological structure of the space of all bounded composition operators on A 2 (ω) endowed with topologies which are induced by the operator norm and the Hilbert-Schmidt norm.

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Difference of weighted composition operators

Cited by 18 publications

References 20 publications

Compact differences of weighted composition operators

Compact differences of weighted composition operators

Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

Difference of Composition operators over Bergman spaces with exponential weights

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