Difference of composition operators over the half-plane
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https://doi.org/10.1007/s11425-018-9439-2
Difference of composition operators over the half-plane Changbao Pang & Maofa Wang∗ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Email: [email protected], [email protected] Received January 24, 2018; accepted September 10, 2018
Abstract
To overcome the unboundedness of the half-plane, we use Khinchine’s inequality and atom de-
composition techniques to provide joint Carleson measure characterizations when the difference of composition operators is bounded or compact from standard weighted Bergman spaces to Lebesgue spaces over the halfplane for all index choices. For applications, we obtain direct analytic characterizations of the bounded and compact differences of composition operators on such spaces. This paper concludes with a joint Carleson measure characterization when the difference of composition operators is Hilbert-Schmidt. Keywords
weighted Bergman space, joint Carleson measure, composition operator, Khinchine’s inequality,
atom decomposition, Hilbert-Schmidt MSC(2010)
47B33, 32A36
Citation: Pang C B, Wang M F. Difference of composition operators over the half-plane. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-9439-2
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Introduction
Let Π+ be the upper half of the complex plane, i.e., Π+ := {z ∈ C : ℑz > 0}. Let µ be a positive Borel measure on Π+ . For 0 < p < ∞, let Lp (µ) denote the Lebesgue space on Π+ with the measure µ, i.e., Lp (µ) comprises all measurable complex functions f defined on Π+ for which the “norm” {∫ ∥f ∥Lp :=
|f |p dµ
} p1
Π+
is finite. When 0 < p < 1, the space Lp (µ) is a complete metric space under the translation-invariant metric (f, g) 7→ ∥f − g∥pLp . When 1 6 p < ∞, the space Lp (µ) is a Banach space. In particular, L2 (µ) is a Hilbert space. For α > −1, let dAα (z) := cα (ℑz)α dA(z), α
is a constant and dA is the Lebesgue area measure on Π+ . For 0 < p < ∞, we denote where cα = 2 (α+1) π the standard weighted Bergman space by Apα (Π+ ), which comprises holomorphic functions of Lp (dAα ). It is known that each space Apα (Π+ ) is a closed subspace of Lp (dAα ). For convenience, we use ∥f ∥Apα to represent the norm of f ∈ Apα (Π+ ). * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Pang C B et al.
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Sci China Math
Let H(Π+ ) and S(Π+ ) be the sets of all holomorphic functions and holomorphic self-maps on Π+ , respectively. The composition operator Cφ is defined by Cφ f = f ◦ φ,
f ∈ H(Π+ ).
Extensive study of the theory of composition operators has been conducted during the past four decades in various settings. For various aspects of the theory of composition operators acting on holomorphic function spaces, see [11, 21]. One of the most important problems in the study of composition operators is to characterize compact differences of such operators (see [4, 5, 7, 14, 15, 18, 19, 23]). In particular, in a study [18] on weighted Bergman spaces over the unit disk,
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