Radially-continuous operators on the Fock space
Robert Fulsche
TL;DR
This work links radially-continuous operators on the Fock space F_t^2 to band-dominated operators on l^2(N_0) by identifying A with its matrix M(A) in the standard basis. It proves A ∈ C_R(F_t^2) if and only if M(A) ∈ BDO(l^2(N_0)), and it characterizes the Toeplitz intersection T(F_t^2) ∩ C_R(F_t^2) via uniform continuity of off-diagonal entries in the metric ρ(m,n) = |sqrt(m) − sqrt(n)|. The results extend to separately-radial actions on multi-variable Fock spaces and to Bergman spaces on the disc, using analogous BD O criteria with metrics ρ_d and d. Collectively, these findings establish a concrete bridge between group-action continuity on function spaces and matrix-analytic band-dominated operator theory, enabling a band-dominated perspective on Toeplitz and limit-operator analyses in these settings.
Abstract
We study operators on the Fock space on which by adjoining the rotation operators implements a continuous action of the circle group. We prove that this class of operators can be identified with the space of band-dominated operators on $\ell^2(\mathbb N_0)$ by mapping the operators to their matrix representations with respect to the standard orthonormal basis. Further, we prove that the intersection of this class with the Toeplitz algebra of the Fock space agrees, in the same manner, with the band-dominated operators on $\ell^2(\mathbb N_0)$ such that the off-diagonals of the matrix are sequences which are uniformly continuous with respect to the square-root metric.