Kernels and spectrum of Toeplitz operators on the Dirichlet space

In this paper, we study the spectrum of Toeplitz operators on the Dirichlet space D with bounded conjugate analytic symbols and give a characterization of the kernels of Toeplitz operators with harmonic symbols in P ‾ + M ( D ) . Using this characterization, we show that the spectrum of a Toeplitz operator with symbol, the sum of an analytic polynomial with degree at most two and the conjugate of linear polynomial is connected, but there are Toeplitz operators with general harmonic polynomial symbols having disconnected spectrum.