A New Characterization of Differences of Weighted Composition Operators on Weighted-Type Spaces
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A New Characterization of Differences of Weighted Composition Operators on Weighted-Type Spaces Qinghua Hu1 · Songxiao Li2 · Yecheng Shi3
Received: 23 May 2016 / Accepted: 9 September 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, we give a new characterization for the boundedness, compactness and essential norm of differences of weighted composition operators between weighted-type spaces. Keywords Weighted composition operators · Difference · Weighted-type spaces · Essential norm Mathematics Subject Classification 30H99 · 47B38
Communicated by Pekka Koskela. The authors’ work was partially supported by the National Natural Science Foundation of China Grant No. 11471143 and by Macao Science and Technology Development Fund Grant No. 083/2014/A2.
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Songxiao Li [email protected] Qinghua Hu [email protected] Yecheng Shi [email protected]
1
Department of Mathematics, Shantou University, Shantou 515063, Guangdong, People’s Republic of China
2
Institute of Systems Engineering, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau
3
Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau
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Q. Hu et al.
1 Introduction Let D be the open unit disk in the complex plane C and H (D) be the class of functions analytic in D. Let N denote the set of all non-negative integers. Let σa be the Möbius a−z transformation on D defined by σa (z) = 1− az ¯ . For z, w ∈ D, the pseudo-hyperbolic distance between z and w is given by z−w . ρ(z, w) = |σw (z)| = 1 − wz ¯ It is well known that ρ(z, w) ≤ 1. Let ϕ be an analytic self-map of D. The self-map ϕ induces a linear operator Cϕ which is defined on H (D) by Cϕ ( f )(z) = f (ϕ(z)), z ∈ D. Cϕ is called the composition operator. The compactness and essential norm of composition operator on the Bloch space were studied by many authors (see, e.g., [3,8,13,14,17]). Here, the Bloch space, denoted by B = B(D), is defined as follows. B = { f ∈ H (D) : f B = | f (0)| + sup(1 − |z|2 )| f (z)| < ∞}. z∈D
In particular, Wulan et al. [14] proved that Cϕ : B → B is compact if and only if lim ϕ j B = 0.
j→∞
Let ϕ be an analytic self-map of D and u ∈ H (D). The weighted composition operator, denoted by uCϕ , is defined as follows. (uCϕ f )(z) = u(z) f (ϕ(z)), z ∈ D. Let 0 < α < ∞. An f ∈ H (D) is said to belong to the weighted-type space, denoted by Hα∞ , if f Hα∞ = sup(1 − |z|2 )α | f (z)| < ∞. z∈D
It is well known that Hα∞ is a Banach space under the norm · Hα∞ . For all z, w ∈ D, we define α (z, w) =
sup
f Hα∞ ≤1
|(1 − |z|2 )α f (z) − (1 − |w|2 )α f (w)|.
Let ϕ and ψ be analytic self-maps of D, u, v ∈ H (D). For simplicity, we denote Du,ϕ (z) =
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(1 − |z|2 )β u(z) (1 − |z|2 )β v(z) , D (z) = . v,ψ (1 − |ϕ(z)|2 )α (1 − |ψ(z)|2 )α
A New Characterization of Differences
Recently, many researchers have studied the differences of composition operators, as well as the differences of weighted composition operators on some analytic function spaces. The purpos
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