Invertibility of Toeplitz operators on the Bergman spaces with harmonic symbols

Abstract

We give the sufficient conditions ensuring the invertibility of Toeplitz operators on the Bergman spaces with harmonic symbols. We show that for φ=αg+βg‾, where g∈H∞, for some constants α,β∈C, if infz∈D⁡|Tφ˜(z)|=infz∈D⁡|φ(z)|>0, then Tφ is invertible on La2(D). Moreover supposing g∈C(D‾), for φ=αg+βg‾, where g∈H∞, then Tφ is invertible on La2(D) if and only ifinfz∈D⁡|Tφ˜(z)|=infz∈D⁡|φ(z)|>0. We will furthermore describe more general results with respect to the invertibility of Toeplitz operators Tφ on the Bergman spaces with bounded harmonic symbols φ=h+g‾. For bounded harmonic symbols φ=h+g‾, suppose that h,g∈C(D‾). If Tφ is invertible on La2(D) and that |h|≠|g| on the unit circle ∂D, then either Th or Tg‾ must be at least invertible on La2(D). If Tφ is invertible on La2(D) and that |h|=|g| on the unit circle ∂D, then both Th and Tg‾ must be invertible on La2(D).