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 $\mathscr{C}_u := [M_u, \mathbb{P}_α]$ acting on $L^2(\mathbb D, dA_α)$, 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 $\mathbb{P}_α$ is the orthogonal projection onto the space $\mathscr{H}_α^2(\mathbb D)$ of harmonic functions on $\mathbb D$ that are square-integrable with respect to the weighted measure $dA_α$. We prove that if $u(z) = U(z) + \overline{U(z)} + ν_u \log|z|^2$, where $U$ is a holomorphic function on $\mathbb D(R)$ and $2ν_u$ is the Lelong number of $u$ at $0$, then $$s_n(\mathscr{C}_u) \underset{n\to+\infty}{\sim} \frac{\sqrt{α+1}}{2πn} \int_{\partial \mathbb D} \sqrt{ν_u^2 + |U'(z)|^2} \; |dz|.$$ In particular, the operator $\mathscr{C}_u$ belongs to the Von Neumann-Schatten class $\mathcal C_p$ for any $p>1$.