The Helton-Howe measure of almost normal Toeplitz operators
Yuto Sugahara
TL;DR
The paper determines the Helton–Howe measure $P_T$ for almost normal Toeplitz operators on the Hardy space $H^2(\mathbb{T})$, connecting the measure to the signed multiplicity $m_{\Phi}$ of the harmonic extension $\Phi$ of the symbol. For smooth symbols $\phi$, the Helton–Howe density is $dP_{T_\phi} = \frac{1}{2\pi i} m_{\Phi}(x+iy)\, dx\, dy$, with the winding number linked to the index via $\mathrm{ind}(T_\phi-\lambda) = -\mathrm{wind}(\phi,\lambda)$ and $m_{\Phi}$ encoding the multiplicity of preimages. The main advance is extending to non-smooth symbols by approximating with Poisson convolutions $\phi_r$ and taking weak-* limits, yielding $dP_{T_\phi} = \mathrm{w^*-}\lim_{r\uparrow 1} \frac{1}{2\pi i} m_{\Phi_r}(x+iy)\, dx\, dy$, with a clean formula when $\int_{\mathbb{D}} |J(\Phi)| dxdy < \infty$, i.e., $dP_{T_\phi} = \frac{1}{2\pi i} m_{\Phi}(x+iy)\, dx\, dy$. Besov-space conditions on $\Re\phi$ and $\Im\phi$ provide practical sufficiency for this integrability and for almost normality, yielding a concrete trace/measure description of commutators in this Toeplitz setting. Overall, the work generalizes the winding-number-type trace formulas to a broader class of almost normal Toeplitz operators via harmonic-analytic tools.
Abstract
The Helton-Howe measure associated with an almost normal operator was constructed by Helton and Howe. It provides a trace formula that allows us to calculate the trace of commutators that would otherwise be incalculable. We will investigate almost normal Toeplitz operators and determine their Helton-Howe measures in terms of the harmonic extensions of the symbols. Our result may be thought of as a generalization of the winding number formula of the Fredholm index of a Toeplitz operator.