Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T _μ from one Fock space ({F^{p}_{alpha}}) to another ({F^{q}_{alpha}}) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of ({T_mu: F^{p}_{alpha} to F^{q}_{alpha}}) in terms of the averaging function ({widehat{mu}_r}) and the t-Berezin transform ({widetilde{mu}_t}) respectively. Quite differently from the Bergman space case, we show that T _μ is bounded (or compact) from ({F^{p}_{alpha}}) to ({F^{q}_{alpha}}) for some p ≤ q if and only if T _μ is bounded (or compact) from ({F^{p}_{alpha}}) to ({F^{q}_{alpha}}) for all p ≤ q. In order to prove our main results on T _μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C ⁿ .