The Dixmier trace of Hankel operators on the Bergman space (Q837064)

Let \(f\) be a smooth function on the closed unit disk \(\mathbb D\) of the complex plane and let \(H_f\) be the big Hankel operator with symbol \(f\) defined on the Bergman space \(A(\mathbb D)\) by \(H_f(\varphi):=(I-P)(f\varphi)\), where \(P\) stands for the orthogonal projection from \(L^2(\mathbb D)\) onto \(A(\mathbb D)\). The main result of this paper states that \(| H_f | := (H_f^* H_f)^ {1/2}\) has finite Dixmier trace given by \((1/2\pi) \int_ {\mathbb T} | \overline{\partial} f | \), where \(\mathbb T\) is the unit circle. The paper includes a number of another interesting results, like the analog of this formula for multiply connected domains or, for harmonic \(f\), the precise regularity conditions required for the formula to hold.