Hilbert-Schmidt double differences of composition operators and non-rigid phenomenon

In the setting of the standard weighted Bergman spaces over the unit disk, compactness characterizations for linear combinations of composition operators have been known. One of those characterizations asserts that degenerate double differences, compared with each single difference, do not improve the compactness at all in the sense that a degenerate double difference is compact only when each difference is individually compact. Such a rigid phenomenon is actually known to hold for a certain broader class of linear combinations. In this paper we investigate into similar properties for Hilbert-Schmidtness with main focus on double differences. We first obtain a complete characterization for Hilbert-Schmidt double differences of composition operators. We then observe that double differences, compared with each single difference, can improve the Hilbert-Schmidtness even in the degenerate case, by constructing concrete examples of Hilbert-Schmidt double differences with each difference not being Hilbert-Schmidt. We also include some remarks concerning connection between Hilbert-Schmidtness on the standard weighted Bergman spaces and weak-to-strong boundedness on certain vector-valued weighted Bergman spaces.