Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T (mu) from one Fock space F-alpha(p) to another F-alpha(q) for 1 < p, q < a with positive Borel measures mu as symbols. We characterize the boundedness (and compactness) of T-mu : F-alpha(p) -> F-alpha(q) in terms of the averaging function (mu) over cap (r) and the t-Berezin transform (mu) over cap (t) respectively. Quite differently from the Bergman space case, we show that T (mu) is bounded (or compact) from F-alpha(p) for some p <= q if and only if T (mu) is bounded (or compact) from F-alpha(p) to F-alpha(q) for all p <= q. In order to prove our main results on T (mu) , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C (n) .