Compact linear combination of composition operators on Bergman spaces

It is well-known that the research of linear combination of composition operators has become a topic of increasing interest. Recently, Choe, Koo and Wang proved that the compactness of combinations composition operators induced by the symbols satisfying the condition (CNC) implies that each operator is compact on the weighted Bergman space when the sum of the coefficients is not equal to zero. Motivated by that work, in this paper, we discuss which operator is compact on the weighted Bergman space when the coefficients do not satisfy the condition (CNC).