Eigenvalue Characterization of Radial Operators on Weighted Bergman Spaces Over the Unit Ball

Abstract

We study the so-called radial operators, and in particular radial Toeplitz operators, acting on the standard weighted Bergman space on the unit ball in \({\mathbb{C}^n}\). They turn out to be diagonal with respect to the standard monomial basis, and the elements of their eigenvalue sequences depend only on the length of multi-indexes enumerating basis elements. We explicitly characterize the eigenvalue sequences of radial Toeplitz operators by giving a solution for the weighted extension of the classical Hausdorff moment problem, and show that the norm closure of the set of all radial Toeplitz operators with bounded measurable radial symbols coincides with the C*-algebra generated by these Toeplitz operators and is isomorphic and isometric to the C*-algebra of sequences that slowly oscillate in the sense of Schmidt.