On singular values of Hankel operators on Bergman spaces

Abstract

In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces Aωφ2, where ωφ=e−φ and φ is a subharmonic function. We consider compact Hankel operators Hϕ‾, with anti-analytic symbols ϕ‾, and give estimates of the trace of h(|Hϕ‾|) for any convex function h. This allows us to give asymptotic estimates of the singular values (sn(Hϕ‾))n in terms of a nonincreasing rearrangement of |ϕ′|/Δφ with respect to an adequate measure. For the radial weights, we first prove that the critical decay of (sn(Hϕ‾))n is achieved by (sn(Hz‾))n. Namely, we establish that if sn(Hϕ‾)=o(sn(Hz‾)), then Hϕ‾=0. Then, we show that if Δφ(z)≍1(1−|z|2)2+β with β≥0, then sn(Hϕ‾)=O(sn(Hz‾)) if and only if ϕ′ belongs to the Hardy space Hp, where p=2(1+β)2+β. Finally, we compute the asymptotics of sn(Hϕ‾) whenever ϕ′∈Hp.