Toeplitz Operators Associated with Measures and the Dixmier Trace on the Hardy Space

Abstract

Let $$\mu $$ be a regular Borel measure on the open unit ball B in $$\mathbf{C}^n$$. By a natural formula, it gives rise to a Toeplitz operator $$T_\mu $$ on the Hardy space $$H^2(S)$$. We characterize the membership of $$T_\mu ^s$$, $$0 < s \le 1$$, in any norm ideal $${\mathcal {C}}_\Phi $$ that satisfies condition (DQK). The same techniques allow us to compute the Dixmier trace of $$T_\mu $$ when $$T_\mu \in {\mathcal {C}}_1^+$$.