Constructive approximation of convergent sequences by eigenvalue sequences of radial Toeplitz--Fock operators
Kevin Esmeral García, Egor A. Maximenko
TL;DR
The paper addresses the problem of approximating arbitrary convergent sequences by eigenvalue sequences $\gamma_g$ of radial Toeplitz operators on the Segal–Bargmann–Fock space. It develops a constructive Laguerre-polynomial based method to build radial symbols $a$ with $\gamma_a$ uniformly approximating a given sequence, and proves that $c_{0}(\mathbb{N}_{0})$ is the uniform closure of $\gamma(C_{0}(\mathbb{R}_{\ge0}))$. It further establishes shift-invariance properties and density results for $\gamma(L_{\infty}(\mathbb{R}_{\ge0}))$ in the slowly oscillating class $\operatorname{RO}(\mathbb{N}_{0})$, clarifying the algebraic structure of the radial Toeplitz operators. The work provides explicit Laguerre-based constructions and rigorous density results, enabling constructive approximation of eigenvalue sequences and advancing understanding of the associated C*-algebras on Fock and Bergman spaces.
Abstract
It is well known that for every measurable function $a$, essentially bounded on the positive halfline, the corresponding radial Toeplitz operator $T_a$, acting in the Segal--Bargmann--Fock space, is diagonal with respect to the canonical orthonormal basis consisting of normalized monomials. We denote by $γ_a$ the corresponding eigenvalues sequence. Given an arbitrary convergent sequence, we uniformly approximate it by sequences of the form $γ_a$ with any desired precision. We give a simple recipe for constructing $a$ in terms of Laguerre polynomials. Previously, we proved this approximation result with nonconstructive tools (Esmeral and Maximenko, ``Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences'', Complex Anal. Oper. Theory 10, 2016). In the present paper, we also include some properties of the sequences $γ_a$ and some properties of bounded sequences, uniformly continuous with respect to the sqrt-distance on natural numbers.