Abstract
We study some algebraic properties of Toeplitz operators on the Dirichlet spaces of planar domains. On domains with real analytic boundary, we show that Toeplitz operators with symbol vanishing near the boundary have rank at most 1. Moreover, we construct explicit examples of Toeplitz operators having exactly rank 1. This is a sharp contrast to a known result on the unit disk. Also, on simply connected domains we characterize compact Toeplitz operators in terms of the boundary vanishing property of the Berezin transform of the symbol.
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