Positive Toeplitz operators of finite rank on the parabolic Bergman spaces (Q1951658)

Let \({\mathbb R}_+^{n+1}={\mathbb R}^n\times(0,\infty)\), \(0<\alpha\leq 1\), \(\lambda>-1\), and \(L^{(\alpha)}=\partial/\partial t+(-\partial/\partial x_1-\dots-\partial/\partial x_n)^\alpha\) for \((x_1,\dots,x_n,t)\in{\mathbb R}_+^{n+1}\). Let \(b_\alpha^2(\lambda)\) denote the parabolic Bergman space consisting of all \(L^{(\alpha)}\)-harmonic functions on \({\mathbb R}_+^{n+1}\) that are square integrable with respect to the weighted measure \(t^\lambda\,dx\,dt\) on \({\mathbb R}_+^{n+1}\). For a nonnegative Radon measure \(\mu\) on \({\mathbb R}_+^{n+1}\), let \(\widetilde{T}_\mu^\lambda\) be the corresponding Toeplitz operator. The main result of the paper says that, if there exists a dense subspace \({\mathcal D}_0\) in \(b_\alpha^2(\lambda)\) such that \({\mathcal D}_0\subset\text{Dom}(\widetilde{T}_\mu^\lambda)\) and \(\dim(\widetilde{T}_\mu^\lambda({\mathcal D}_0))<\infty\), then \(\mu\) is a finite linear combination of point masses and \(\text{rank}(\widetilde{T}_\mu^\lambda)=\#\text{supp}(\mu)=\dim(\widetilde{T}_\mu^\lambda({\mathcal D}_0))\), where \(\# A\) denotes the cardinal number of a set \(A\).