Positive Toeplitz Operators Between Gaussian Mixed-Norm Fock Spaces

Abstract

We study positive measure-symbol Toeplitz operators between Gaussian radial–angular mixed-norm Fock spaces throughout the full range of exponents, including infinity. We obtain a complete characterization of boundedness in terms of weighted local masses of the symbol measure, or equivalently in terms of its Gaussian Berezin profile. The criterion separates the transfer of angular integrability from that of radial summability and remains valid in the quasi-Banach and endpoint regimes. We further characterize compactness and obtain the corresponding essential-norm estimates through the asymptotic behavior of the same local-mass and Berezin profiles, and determine the mappings involving the large and little outer-endpoint spaces. We also prove an independent characterization of bounded positive Toeplitz sesquilinear forms. The proofs combine atomic recovery, localized Gaussian matrix estimates, and Rademacher extraction.