Sums of Toeplitz products with harmonic symbols. (English) Zbl 1148.47024

The authors study special combinations of Toeplitz operators with bounded harmonic symbols acting on the Bergman space over the unit disk. Let \(h^{\infty}\) be the set of all bounded harmonic functions on the unit disk. The main result of the paper gives criteria for being of finite rank or compact for an operator of the form \[ T=T_{\lambda}+\sum_{j=1}^{N}T_{u_j}T_{v_j}, \] where \(\lambda\) is a finite sum of finite products of functions from \(h^{\infty}\) and \(u_j,\, v_j \in h^{\infty}\) for each \(j\). Some interesting special forms and examples of the operator \(T\) are considered as well.


47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A36 Bergman spaces of functions in several complex variables
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