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 T is a hyponormal operator whose essential spectrum has zero area, then the question of whether [ T ∗ , T ] [{T^ * },T] is trace class depends only on the spectral picture of T T . We also show how a special case of results of Helton-Howe can be derived from the BDF theory.