Radial operators on polyanalytic weighted Bergman spaces

TL;DR

This paper studies radial operators on polyanalytic weighted Bergman spaces, providing their spectral decomposition, representing Toeplitz operators as matrix sequences, and analyzing their density properties.

Contribution

It extends the orthogonal basis and Fourier decomposition to polyanalytic weighted Bergman spaces, and characterizes the structure of radial operators and Toeplitz operators within this framework.

Findings

Radial operators decompose into matrix sequences.

Toeplitz operators with bounded symbols are represented as matrices.

Bounded generating symbols are not weakly dense in the operator space.

Abstract

Let $\mu_\alpha$ be the Lebesgue plane measure on the unit disk with the radial weight $\frac{\alpha+1}{\pi}(1-|z|^2)^\alpha$. Denote by $\mathcal{A}^{2}_{n}$ the space of the $n$-analytic functions on the unit disk, square-integrable with respect to $\mu_\alpha$. Extending the results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and W\"{u}nsche in 2005) form an orthonormal basis of $\mathcal{A}^{2}_{n}$. Using this basis, we provide the Fourier decomposition of $\mathcal{A}^{2}_{n}$ 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 $\mathcal{A}^{2}_n$, 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 $\mathcal{A}^{2}_n$. Moreover, using ideas by Engli\v{s} (1996), we show that the set of all Toeplitz operators with bounded generating symbols is not weakly dense in $\mathcal{B}(\mathcal{A}^{2}_n)$.