Toeplitz Operators on Quaternionic Fock Spaces

Abstract

We characterize boundedness and compactness of Toeplitz operators on quaternionic Fock spaces with positive measure symbols and slice-function symbols in \(\mathrm{BMO}^1\). For positive measure symbols, we derive criteria using normalized reproducing kernels and symmetric box averages, while for slice \(\mathrm{BMO}^1\) symbols, the characterizations rely on the Berezin transform. We further introduce a global quaternionic Fock space \(F_α^p\) to define Toeplitz operators with real-valued measure symbols; this space is built by integrating slice regular functions over all complex slices of \(\mathbb{H}\) and is norm-equivalent to the standard slice-based quaternionic Fock space. In the Hilbert space case \(p=2\), a slice-independent orthogonal projection exists, which allows us to define Toeplitz operators with real-valued measure symbols and slice-function symbols in a unified way.