Toeplitz operators on Bergman spaces of polyanalytic functions (Q2918788)

The authors study some algebraic properties of Toeplitz operators on the Bergman space of polyanalytic functions on the unit disk. Recall that a function \(h\) is called polyanalytic of order \(n\) if it satisfies the equation \(\partial^n h/\partial\overline{z}^n = 0\). Among the main results of the paper are a criterion for compactness of a Toeplitz operator with \(L^1\) non-negative symbol in terms of its Berezin transform, and a characterization of finite rank commutators and semi-commutators of Toeplitz operators with harmonic symbols. The paper ends with a list of open questions.