Toeplitz Operators on Quaternionic Fock Spaces

We characterize boundedness and compactness of Toeplitz operators on quaternionic Fock spaces with positive measure symbols and slice-function symbols in ${\mathrm{B}\mathrm{M}\mathrm{O}}^1$. For positive measure symbols, we derive criteria using normalized reproducing kernels and symmetric box averages, while for slice ${\mathrm{B}\mathrm{M}\mathrm{O}}^1$ symbols, the characterizations rely on the Berezin transform. We further introduce a global quaternionic Fock space $F_{\alpha }^p$ 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.