Compact difference of composition operators with smooth symbols on the polydisks

Recently, when the symbol maps satisfy certain regularity conditions, Gu and Koo proved a simple function theoretic characterization of the compact difference of two composition operators on the Hardy or weighted Bergman spaces on the unit ball. They proved that the difference of two composition operators is compact if and only if the pseudo-hyperbolic distance between the symbols maps is vanishing as the symbol maps approach the boundary. Surprisingly, we prove that this characterization fails to hold on the polydisk setting. More precisely, we show that the distance vanishing condition on the topological boundary does imply that the difference of the composition operators is compact, but the converse fails to hold. Moreover, for the sufficiency we show that the topological boundary can not be replaced by the distinguished boundary for the distance vanishing condition. For the converse, we prove that only a certain weaker distance vanishing condition holds when the difference is compact. We provide an explicit example with polynomial symbols which shows that the pseudo-hyperbolic distance vanishing condition fails to hold while the difference of the composition operators is compact. We also provide various explicit examples which show that the regularity condition or the boundedness assumption of the single composition operator is essential in our results.