In this paper, we characterize bounded, compact or Schatten class weighted composition operators acting on Bergman spaces with the exponential type weights. Moreover, we give the proof of the necessary part for the boundedness of C φ on large weighted Bergman spaces given by [8].

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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.

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