Compact differences of weighted composition operators
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Compact differences of weighted composition operators Bin Liu1 · Jouni Rättyä1 Received: 6 March 2020 / Accepted: 28 November 2020 © The Author(s) 2020
Abstract Compact differences of two weighted composition operators acting from the weighted q p Bergman space A𝜔 to another weighted Bergman space A𝜈 , where 0 < p ≤ q < ∞ and 𝜔, 𝜈 belong to the class D of radial weights satisfying two-sided doubling conditions, are charp acterized. On the way to the proof a new description of q-Carleson measures for A𝜔 , with 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 p Bergman space A𝛼 with −1 < 𝛼 < ∞ to the setting of doubling weights. Keywords Bergman space · Doubling weight · Weighted composition operator · Compactness Mathematics Subject Classification 47B33 · 30H20
1 Introduction and main results Let H(𝔻) denote the space of analytic functions in the unit disc 𝔻 = {z ∈ ℂ ∶ |z| < 1} . For a nonnegative function 𝜔 ∈ L1 ([0, 1)) , the extension to 𝔻 , defined by 𝜔(z) = 𝜔(|z|) for all z ∈ 𝔻 , is called a radial weight. For 0 < p < ∞ and a radial weight 𝜔 , the weighted Bergp man space A𝜔 consists of f ∈ H(𝔻) such that p
‖f ‖Ap = 𝜔
∫𝔻
�f (z)�p 𝜔(z) dA(z) < ∞,
where dA(z) = dx𝜋dy is the normalized Lebesgue area measure on 𝔻 . As usual, A𝛼 stands for the classical weighted Bergman space induced by the standard radial weight 𝜔(z) = (1 − |z|2 )𝛼 , where −1 < 𝛼 < ∞. p
The first author is supported by China Scholarship Council (No.201706330108). * Bin Liu [email protected] Jouni Rättyä [email protected] 1
Department of Physics and Mathematics, University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland
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B. Liu, J. Rättyä 1
̂ = ∫|z| 𝜔(s) ds for all z ∈ 𝔻 . In this paper we always For a radial weight 𝜔 , write 𝜔(z) p � > 0 , for otherwise A𝜔 = H(𝔻) for each 0 < p < ∞ . A weight 𝜔 belongs to assume 𝜔(z) ̂ if there exists a constant C = C(𝜔) ≥ 1 such that 𝜔(r) ) for all the class D ̂ ≤ C𝜔( ̂ 1+r 2 0 ≤ r < 1 K = K(𝜔) > 1 C = C(𝜔) > 1 . (Moreover, if there exist and such that ) ̌ . In other words, 𝜔 ∈ D ̌ if 0 ≤ r < 1 for all , then we write 𝜔 ∈ D 𝜔(r) ̂ ≥ C𝜔 ̂ 1 − 1−r K there exists K = K(𝜔) > 1 and C� = C� (𝜔) > 0 such that
𝜔(r) � ≤ C�
1− 1−r K
�r
𝜔(t) dt,
0 ≤ r < 1.
� ∩D ̌ is denoted by D , and this is the class of weights that we mainly The intersection D work with. Each analytic self-map 𝜑 of 𝔻 induces the composition operator C𝜑 on H(𝔻) defined by C𝜑 f = f ◦𝜑 . The weighted composition operator induced by u ∈ H(𝔻) and 𝜑 is uC𝜑 and sends f ∈ H(𝔻) to u ⋅ f ◦𝜑 ∈ H(𝔻) . These operators have been extensively studied in a variety of function spaces (see for example [4–8, 22, 23, 25, 26]). If now 𝜓 is another analytic self-map of 𝔻 , the pair (𝜑, 𝜓) induces the operator C𝜑 − C𝜓 . One of the most important problem considering these operators is to characterize compact differences in Hardy spaces. Shapiro and Sundberg [24] studied this problem in 1990. V
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