Operator Theory on Quaternionic Fock Spaces: Carleson Measures, Berezin Transforms, and Toeplitz Operators

We develop an operator-theoretic approach to quaternionic Fock spaces, with emphasis on Carleson measures, Berezin transforms, and Toeplitz operators. We first introduce a global Gaussian $L^p$-framework on $\mathbb H$ for slice functions and show that the resulting global Fock space $F_\alpha^p$ coincides, with equivalent norms, with the quaternionic Fock space $\mathfrak F_\alpha^p$ defined by slice norms. In particular, $F_\alpha^2$ is a right quaternionic reproducing kernel Hilbert space, and this yields a slice-independent Fock projection. Within this global framework, we characterize quaternionic Fock--Carleson measures and vanishing Fock--Carleson measures by means of reproducing-kernel testing and symmetric box conditions adapted to the slice geometry of $\mathbb H$. We then study the Berezin transform of slice functions, establishing in particular its slice stability, semigroup property, and several mapping and fixed-point properties. As an application, we introduce Toeplitz operators with slice-function symbols and with measure symbols. For positive measure symbols, we obtain characterizations of boundedness and compactness in terms of Berezin-type transforms and symmetric box averages. For slice $\mathrm{BMO}^1$ symbols, we establish corresponding characterizations in terms of the Berezin transform. In contrast with the complex Fock space, the quaternionic slice geometry naturally leads to symmetric box averages, and the noncommutativity of the slice product gives rise to new algebraic phenomena for Toeplitz operators.