Radial operators on polyanalytic Bargmann-Segal-Fock spaces

Abstract

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 $\overline{z}$. 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.

arXiv comment26 pages; some proofs are modified in the second version

References in corpus (4)

• The polyanalytic Ginibre ensembles
• The eigenvalues of limits of radial Toeplitz operators
• Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences
• $\mathcal{D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

Cited by in corpus (5)

• Horizontal Fourier transform of the polyanalytic Fock kernel
• Translation-invariant operators in reproducing kernel Hilbert spaces
• C*-algebras generated by radial Toeplitz operators on polyanalytic weighted Bergman spaces
• Radial operators on polyanalytic weighted Bergman spaces
• Homogeneously polyanalytic kernels on the unit ball and the Siegel domain