A note on weighted composition operators on the weighted Bergman space

Abstract

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.

Weighted composition operators on weighted Bergman spaces induced by double weights

“…The following lemma can be found in [18] without a proof. For the benefits of the readers, we will prove it.…”

“…Three years later, they investigated weighted composition operators between different Bergman spaces and Hardy spaces in [5]. In [13], Peláez and Rättyä characterized the Schatten class of Toeplitz operators induced by a positive Borel measure on D and the reproducing kernel of the Bergman space A 2 ω when ω ∈ D. In [18], Zhao and Hou proved that, for A p α , the finite Blaschke product is the only composition symbol that the induced weighted composition operator is bounded if and only if the weighted symbol defines a bounded multiplication operator. The similar result for Hardy space H p can be seen in [2].…”

“…Motivated by [4,5,13], under the assumption that ω ∈ D and µ is a positive Borel measure, we investigate the boundedness, compactness and essential norm of uC ϕ : A p ω → L q µ and the Schatten class of uC ϕ : A 2 ω → A 2 ω . Motivated by [18], we get that, for some ω ∈ R, X = H ∞ if and only if ϕ is a finite Blaschke product. Here…”