Radial operators on polyanalytic weighted Bergman spaces

Let μ_α be the Lebesgue plane measure on the unit disk with the radial weight α+1π(1-|z|2)α. Denote by An2 the space of the n-analytic functions on the unit disk D, square-integrable with respect to μ_α. Extending results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wünsche in 2005) form an orthonormal basis of An2. Using this basis, we provide the Fourier decomposition of An2 into the orthogonal sum of the subspaces associated with different frequencies. This leads to the decomposition of the von Neumann algebra of radial operators, acting in An2, into the direct sum of some matrix algebras. In other words, all radial operators are represented as matrix sequences. In particular, we represent in this form the Toeplitz operators with bounded radial symbols, acting in An2. Moreover, using ideas by Engliš (1996), we show that the set of the Toeplitz operators with bounded generating symbols is not weakly dense in B(An2).