Trace-class membership for antisymmetric sums on quotient modules of the Hardy module

We consider the quotient module Q of the Hardy module H²(S) defined by an analytic set M˜ satisfying certain conditions. Denote d=dim_CM˜. When d=1, Q was shown to be 1-essentially normal in [24]. An analogous problem for the case d≥2 was proposed in [24], which asks whether 2d-antisymmetric sums of certain module operators are in the trace class. In this paper we solve this problem in the affirmative. In the case d=1, we derive a trace formula on Q, which answers another question raised in [24].