When do Kernels Admit Characteristic Functions?
arXiv:2607.27799
The paper proves that a reproducing kernel admits a characteristic function exactly when a Beurling-type invariant subspace condition holds, and this condition can be expressed as an Agler-type decomposition of the kernel.
Abstract
A general framework for deriving characteristic functions for reproducing kernels that do not necessarily possess the complete Pick property was recently established by Bhattacharyya and Jindal. We show that, in this setting, the existence of a characteristic function is equivalent to a Beurling-type invariant subspace condition. Combined with recent results characterizing kernels satisfying this condition, our theorem implies that the existence of a characteristic function is equivalent to a concrete Agler-type decomposition of the underlying kernels.