Radial operators on polyanalytic weighted Bergman spaces

Let la be the Lebesgue plane measure on the unit disk with the radial weight athorn1 p o1 similar to jzj2THORNa. Denote by A2 n the space of the n-analytic functions on the unit disk D, square-integrable with respect to la. Extending results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wu<spacing diaeresis> nsche in 2005) form an orthonormal basis of A2 n. Using this basis, we provide the Fourier decomposition of A2 n 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 A2n, 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 A2n. Moreover, using ideas by Englis. (1996), we show that the set of the Toeplitz operators with bounded generating symbols is not weakly dense in BoA2nTHORN.