On Quaternionic Fock Spaces: Kernel-induced Integral Operators, Berezin Transforms and Toeplitz Operators

Abstract: In this paper, we study quaternionic Fock spaces and develop an operator-theoretic framework centered around kernel-induced integral operators, Berezin transforms and Toeplitz operators. More precisely, the following results are obtained: (i) Global quaternionic Fock structure. We introduce a global Gaussian $L^p$--norm for slice functions on $\mathbb H$ and prove that the resulting global quaternionic Fock space $F_α^p$ coincides with the slice-defined Fock space $\mathfrak F_α^p$, with equivalent norms. In particular, $F_α^2$ becomes a right quaternionic reproducing kernel Hilbert space with an explicit reproducing kernel, yielding a slice-independent Fock projection onto $F_α^2$. (ii) Kernel-induced integral operators and Fock--Carleson measures. We investigate kernel-induced integral operators and characterize quaternionic Fock--Carleson measures. These embedding theorems provide the measure-theoretic basis that underlies boundedness and compactness criteria for operators on quaternionic Fock spaces. (iii)Berezin transforms and Toeplitz operators. We define the Berezin transform for slice functions and prove its fundamental properties, including semigroup behavior and fixed-point features. Building on the slice-independent projection and the slice product, we introduce Toeplitz operators with slice-function symbols and with measure symbols, and develop their basic algebraic properties. We then obtain complete boundedness and compactness characterizations for Toeplitz operators with two natural symbol classes: positive measures and slice $\mathrm{BMO}^1$ symbols, expressed in terms of Berezin-type transforms and slice/symmetric averaging quantities.