arXiv subjects

Roberto Moisés Barrera-Castelán

Publications and source records attributed to Roberto Moisés Barrera-Castelán.

2 recordsLinked to original sources

arXiv2023

C*-algebras generated by radial Toeplitz operators on polyanalytic weighted Bergman spaces

In a previous paper (Radial operators on polyanalytic weighted Bergman spaces, Bol. Soc. Mat. Mex. 27, 43), using disk polynomials as an orthonormal basis in the $n$-analytic weighted Bergman space, we showed that for every bounded radial generating symbol $a$, the associated Toeplitz operator, acting in this space, can be identified with a matrix sequence $γ(a)$, where the entries of the matrices are certain integrals involving $a$ and Jacobi polynomials. In this paper, we suppose that the generating symbols $a$ have finite limits on the boundary and prove that the C*-algebra generated by the corresponding matrix sequences $γ(a)$ is the C*-algebra of all matrix sequences having scalar limits at infinity. We use Kaplansky's noncommutative analog of the Stone--Weierstrass theorem and some ideas from several papers by Loaiza, Lozano, Ramírez-Ortega, Ramírez-Mora, and Sánchez-Nungaray. We also prove that for $n\ge 2$, the closure of the set of matrix sequences $γ(a)$ is not equal to the generated C*-algebra.

math.OA↗

arXiv2020

Radial operators on polyanalytic weighted Bergman spaces

Let $μ_α$ be the Lebesgue plane measure on the unit disk with the radial weight $\frac{α+1}π(1-|z|^2)^α$. Denote by $\mathcal{A}^{2}_{n}$ the space of the $n$-analytic functions on the unit disk, square-integrable with respect to $μ_α$. Extending the results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wünsche 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 Engliš (1996), we show that the set of all Toeplitz operators with bounded generating symbols is not weakly dense in $\mathcal{B}(\mathcal{A}^{2}_n)$.

math.FA↗