Extensions of the Berger-Shaw Theorem

Abstract

We show how D. Voiculescu’s proof of the Berger-Shaw trace inequality for rationally cyclic nearly hyponormal operators can be presented using only elementary operator-theoretic concepts. In addition we show that if $T$ is a hyponormal operator whose essential spectrum has zero area, then the question of whether $[{T^ * },T]$ is trace class depends only on the spectral picture of $T$. We also show how a special case of results of Helton-Howe can be derived from the BDF theory.