The spectrum of Bergman Toeplitz operators with some harmonic symbols (Q294509)

The results presented in the paper show some peculiarities on the spectrum behaviour of Toeplitz operators \(T_h\) with bounded harmonic symbols \(h\), acting on the Bergman space on the unit disk \(\mathbb D\). The authors show that, for the affine symbol \(h(z) = \overline{z}+az+b\), with \(a\), \(b \in \mathbb C\), \(\sigma(T_h) = \mathrm{clos}[h(\mathbb D)]\). At the same time, this result does not hold for a generic bounded harmonic symbol. In particular, for \(h(z) = \overline{z}+z^2 -z\), the authors show that \([0,1) \subset \mathrm{clos}[h(\mathbb D)]\), while \([0,1) \cap \sigma(T_h) = \emptyset\).