Integral Operators Between Fock Spaces

In this paper, the authors study the integral operator S(phi)f(z) =integral(C)phi(z,w)f(w)d lambda(alpha)(w) induced by a kernel function phi(z,<middle dot>)is an element of F-alpha infinity between Fock spaces. For 1 <= p <= infinity, they prove that (phi):F-alpha(1)-> F-alpha(p) is bounded if and only if (a is an element of C)sup & Vert;S(phi)k(a)& Vert;(p,alpha)<infinity,(dagger) where k(a) is the normalized reproducing kernel of F-alpha(2); and,S-phi:F(alpha)1 -> F(alpha)(p )is compact if and only if (|a|->infinity)lim & Vert;S(phi)k(a)& Vert;(p,alpha)= 0. When 1< q <=infinity, it is also proved that the condition (dagger) is not sufficient for boundedness of S-phi:F-alpha(q)-> F-alpha(p). In the particular case phi(z,w) = e(alpha zw)phi(z-w) with phi is an element of F-alpha(2), for 1 <= q < p <infinity, they show that S-phi:F(alpha)p -> F-alpha(q) is bounded if and only i f phi= 0; for 1< p <= q <infinity, they give sufficient conditions for the boundedness or compactness of the operator S-phi:F-alpha(p)-> F-alpha(q).