Compact differences of composition operators on large weighted Bergman spaces

Abstract

While there have been extensive studies regarding the theory of composition operators in standard Bergman spaces, there have not been many results pertaining to large Bergman spaces due to a lack of useful tools. In this paper, we give the characterizations of the compact differences of composition operators in Bergman spaces with the exponential type weight using a newly defined Riemannian distance. Furthermore, we give a sufficient condition for the question when two composition operators lie in the same component.

Difference of Composition operators over Bergman spaces with exponential weights

“…The following lemma can be found in [20,Lemma 3.1]. Now, combining Proposition 3.5 and 3.6, we complete the proof of Theorem 1.1. easypart Lemma 3.9.…”

“…and the following lemma which was given in [20]. Throughout this paper, we denote the operator norm T A p (ω) by T p for simplicity.…”

“…introduced in Section 2.2 and the following lemma which is given in [20]. mainresult4 Proposition 3.6.…”

“…More information on this distance will be presented in Section 2.2 and other essential materials to be used for this paper will be stated in Section 2. In fact, in [20], the current author obtained an characterization for the compact difference C ϕ −C ψ acting on Bergman spaces with the exponential weights using peak functions provided in [3], but it is not complete and needed more conditions than results given in this paper.…”

“…where the infimum is taken over all piecewise smooth curves γ connecting z and w. One may check that ρ τ,ϕ,ψ is a distance in [20,Lemma 3.3].…”