Spectral properties of commutators on harmonic Bergman spaces of the unit disk

Abstract

In this paper, we determine the asymptotic behavior of the singular values of the commutator ${\mathcal{C}}_u:=[M_u,{\mathbb{P}}_{\alpha }]$ acting on $L^2(\mathbb{D},d{A}_{\alpha })$, where $M_u$ is the operator of multiplication by a subharmonic function $u$ on $\mathbb{D}(R)$ and harmonic outside the origin, with $R>1$, and ${\mathbb{P}}_{\alpha }$ is the orthogonal projection onto the space ${\mathcal{H}}_{\alpha }^2(\mathbb{D})$ of harmonic functions on $\mathbb{D}$ that are square-integrable with respect to the weighted measure $d{A}_{\alpha }$. We prove that if $u(z)=U(z)+\overline{U(z)}+{\nu }_u\log |z{|}^2$, where $U$ is a holomorphic function on $\mathbb{D}(R)$ and $2{\nu }_u$ is the Lelong number of $u$ at $0$, then $$s_n({\mathcal{C}}_u)\underset{n\to +\infty }{\sim }\frac{\sqrt{\alpha +1}}{2\pi n}{\int }_{\partial \mathbb{D}}\sqrt{{\nu }_u^2+|U^{\prime }(z){|}^2}|dz|.$$ In particular, the operator ${\mathcal{C}}_u$ belongs to the Von Neumann-Schatten class ${\mathcal{C}}_p$ for any $p>1$.

14 pages, no figure