Radial operators on polyanalytic Bargmann-Segal-Fock spaces

The paper considers bounded linear radial operators on the polyanalytic Fock spaces ${\mathcal{F}}_n$ and on the true-polyanalytic Fock spaces ${\mathcal{F}}_{(n)}$. The orthonormal basis of normalized complex Hermite polynomials plays a crucial role in this study; it can be obtained by the orthogonalization of monomials in $z$ and $\bar{z}$. First, using this basis, we decompose the von Neumann algebra of radial operators, acting in ${\mathcal{F}}_n$, into the direct sum of some matrix algebras, i.e. radial operators are represented as matrix sequences. Secondly, we prove that the radial operators, acting in ${\mathcal{F}}_{(n)}$, are diagonal with respect to the basis of the complex Hermite polynomials belonging to ${\mathcal{F}}_{(n)}$. We also provide direct proofs of the fundamental properties of ${\mathcal{F}}_n$ and an explicit description of the C*-algebra generated by Toeplitz operators in ${\mathcal{F}}_{(n)}$, whose generating symbols are radial, bounded, and have finite limits at infinity.