Difference of Weighted Composition Operators with Series of Symbols

Following the stream of the authors in [32] we introduce complete characterizations in terms of Carleson measures for bounded/compact differences of weighted composition operators with series of symbols acting on the standard weighted Bergman spaces over the unit disk. We allow the weight functions to be non-holomorphic and unbounded. We obtain a compactness characterization for differences of unweighted composition operators acting on the Hardy spaces in terms of Carleson measures and show that compact differences of composition operators with univalent symbols on the Hardy and weighted Bergman spaces are exactly the same. We also show that an earlier characterization due to Acharyya and Wu for compact differences of weighted composition operators with bounded holomorphic weights does not extend to the case of non-holomorphic weights. Some extended explicit examples are shown.