Invertibility of Toeplitz operators with polyanalytic symbols. (English) Zbl 07064106

Tikaradze, Akaki

Adv. Oper. Theory 4, No. 4, 793-801 (2019).

Summary: For a class of continuous functions including complex polynomials in \(z\) and \(\bar{z}\), we show that the corresponding Toeplitz operator on the Bergman space of the unit disc can be expressed as a quotient of certain differential operators with holomorphic coefficients. This enables us to obtain several nontrivial operator theoretic results about such Toeplitz operators, including a new criterion for invertibility of a Toeplitz operator for a class of harmonic symbols.

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A36	Bergman spaces of functions in several complex variables
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

Keywords:

Toeplitz operator; Bergman space; harmonic function