(page 741) The Berger-Shaw Theorem for Cyclic Subnormal Operators

Abstract. This work gives a sharp form of the Berger-Shaw Theorem for cyclic subnormal operators. That is, if S is a cyclic subnormal operator, then S has trace class self-commutator and the trace equals (1=…)Area[(S)¡e(S)]. It also characterizes those functions f such that f(S) has trace class self-commutator and computes the trace as the Dirichlet integral of f ̂ on the set of analytic bounded point evaluations for S. The technique used also gives some partial results for rationally cyclic subnormal operators. Introduction. For an operator T the self-commutator of T is [T ⁄;T] = T ⁄T ¡TT ⁄. The Berger-Shaw Theorem (see [4], p. 152) implies that every rationally cyclic hyponormal operator T has trace class self-commutator and tr[T ⁄;T] • (1=…)Area[(T)]. This work considers the case of a pure cyclic sub-normal operator S and gives another proof that S has trace class self-commutato