The Berger-Shaw Theorem for multi-rationally cyclic hyponormal operators gives an estimate on the trace of the self-commutator of the operator.  If S is a rationally cyclic subnormal (or hyponormal) operator then the Berger-Shaw theorem says that the $trace[S*,S] \leq 1/pi Area[\sigma(S)]$.  That is, the trace is at most  (1/pi) times the Area of the spectrum of the operator.
In this paper we give an exact computation of the trace of the self-commutator when S is a cyclic subnormal operator.  More precisely, if S is a cyclic subnormal operator then $trace[S*,S] = (1/pi) Area[\sigma(S) - \sigma_e(S)]$.  Thus, the trace is equal to (1/pi) times the area of the spectrum minus the essential spectrum of S.  The set $G = \sigma(S) - \sigma_e(S)$ is the set of analytic bounded point evaluations for S.  If f is a bounded analytic function on G then we also compute the trace of the self-commutator of f(S) as the Dirichlet integral of f over the set G.